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3.235 \(\int \frac {\csc ^2(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\)

Optimal. Leaf size=357 \[ \frac {3 \sqrt {b} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \sqrt {b} \left (34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cot (c+d x)}{a^3 d}-\frac {b^2 \tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 a^2 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}-\frac {b \tan (c+d x) \left (\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a^2 (9 a-17 b)}{(a-b)^3}\right )}{32 a^3 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )} \]

[Out]

-cot(d*x+c)/a^3/d+3/64*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*b^(1/2)*(20*a+15*b-34*a^(1/2)*b^(1/2
))/a^(13/4)/d/(a^(1/2)-b^(1/2))^(5/2)-3/64*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))*b^(1/2)*(20*a+15
*b+34*a^(1/2)*b^(1/2))/a^(13/4)/d/(a^(1/2)+b^(1/2))^(5/2)-1/8*b^2*tan(d*x+c)*(a*(a+3*b)+(a^2+6*a*b+b^2)*tan(d*
x+c)^2)/a^2/(a-b)^3/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)^2-1/32*b*tan(d*x+c)*(2*a^2*(9*a-17*b)/(a-b)^3+(1
8*a^2+15*a*b-13*b^2)*tan(d*x+c)^2/(a-b)^2)/a^3/d/(a+2*a*tan(d*x+c)^2+(a-b)*tan(d*x+c)^4)

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Rubi [A]  time = 1.29, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3217, 1334, 1669, 1664, 1166, 205} \[ -\frac {b^2 \tan (c+d x) \left (\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)+a (a+3 b)\right )}{8 a^2 d (a-b)^3 \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )^2}-\frac {b \tan (c+d x) \left (\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}+\frac {2 a^2 (9 a-17 b)}{(a-b)^3}\right )}{32 a^3 d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}+\frac {3 \sqrt {b} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt {a}-\sqrt {b}\right )^{5/2}}-\frac {3 \sqrt {b} \left (34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} d \left (\sqrt {a}+\sqrt {b}\right )^{5/2}}-\frac {\cot (c+d x)}{a^3 d} \]

Antiderivative was successfully verified.

[In]

Int[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

(3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(1
3/4)*(Sqrt[a] - Sqrt[b])^(5/2)*d) - (3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[
b]]*Tan[c + d*x])/a^(1/4)])/(64*a^(13/4)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - Cot[c + d*x]/(a^3*d) - (b^2*Tan[c + d*
x]*(a*(a + 3*b) + (a^2 + 6*a*b + b^2)*Tan[c + d*x]^2))/(8*a^2*(a - b)^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Ta
n[c + d*x]^4)^2) - (b*Tan[c + d*x]*((2*a^2*(9*a - 17*b))/(a - b)^3 + ((18*a^2 + 15*a*b - 13*b^2)*Tan[c + d*x]^
2)/(a - b)^2))/(32*a^3*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
 q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1334

Int[(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{f = Coe
ff[PolynomialRemainder[x^m*(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[x^m*(d +
 e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*g - f*(b^2 - 2*a*c) - c*(b*
f - 2*a*g)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*
x^4)^(p + 1)*Simp[ExpandToSum[(2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[x^m*(d + e*x^2)^q, a + b*x^2 + c*x
^4, x])/x^m + (b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g)/x^m + c*(4*p + 7)*(b*f - 2*a*g)*x^(2 - m), x], x],
 x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IGtQ[q, 1] && ILtQ[m/2, 0]

Rule 1664

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x
)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]

Rule 1669

Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainde
r[x^m*Pq, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[PolynomialRemainder[x^m*Pq, a + b*x^2 + c*x^4, x], x, 2]}, S
imp[(x*(a + b*x^2 + c*x^4)^(p + 1)*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2))/(2*a*(p + 1)*(b^2 - 4*a*c)
), x] + Dist[1/(2*a*(p + 1)*(b^2 - 4*a*c)), Int[x^m*(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[(2*a*(p + 1)*(b^2
- 4*a*c)*PolynomialQuotient[x^m*Pq, a + b*x^2 + c*x^4, x])/x^m + (b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e)
/x^m + c*(4*p + 7)*(b*d - 2*a*e)*x^(2 - m), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x^2] && GtQ[Expon[
Pq, x^2], 1] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && ILtQ[m/2, 0]

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\csc ^2(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^6}{x^2 \left (a+2 a x^2+(a-b) x^4\right )^3} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-16 a b-\frac {2 a b \left (32 a^3-96 a^2 b+97 a b^2-29 b^3\right ) x^2}{(a-b)^3}-\frac {2 b \left (48 a^4-136 a^3 b+115 a^2 b^2-30 a b^3-5 b^4\right ) x^4}{(a-b)^3}-\frac {32 a^2 (2 a-3 b) b x^6}{(a-b)^2}-\frac {16 a^2 b x^8}{a-b}}{x^2 \left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{16 a^2 b d}\\ &=-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {b \tan (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}+\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {128 a^2 b^2+\frac {8 a^2 b^2 \left (32 a^2-55 a b+26 b^2\right ) x^2}{(a-b)^2}+\frac {4 a b^2 \left (32 a^3-18 a^2 b-15 a b^2+13 b^3\right ) x^4}{(a-b)^2}}{x^2 \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d}\\ &=-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {b \tan (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}+\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {128 a b^2}{x^2}+\frac {12 a b^3 \left (2 a (3 a-2 b)+\left (26 a^2-37 a b+15 b^2\right ) x^2\right )}{(a-b)^2 \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{128 a^4 b^2 d}\\ &=-\frac {\cot (c+d x)}{a^3 d}-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {b \tan (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}+\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {(3 b) \operatorname {Subst}\left (\int \frac {2 a (3 a-2 b)+\left (26 a^2-37 a b+15 b^2\right ) x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{32 a^3 (a-b)^2 d}\\ &=-\frac {\cot (c+d x)}{a^3 d}-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {b \tan (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}+\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\left (3 \left (\sqrt {a}+\sqrt {b}\right )^3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a^3 (a-b)^2 d}-\frac {\left (3 \left (\sqrt {a}-\sqrt {b}\right )^3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{64 a^3 (a-b)^2 d}\\ &=\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}-\sqrt {b}\right )^{5/2} d}-\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{64 a^{13/4} \left (\sqrt {a}+\sqrt {b}\right )^{5/2} d}-\frac {\cot (c+d x)}{a^3 d}-\frac {b^2 \tan (c+d x) \left (a (a+3 b)+\left (a^2+6 a b+b^2\right ) \tan ^2(c+d x)\right )}{8 a^2 (a-b)^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )^2}-\frac {b \tan (c+d x) \left (\frac {2 a^2 (9 a-17 b)}{(a-b)^3}+\frac {\left (18 a^2+15 a b-13 b^2\right ) \tan ^2(c+d x)}{(a-b)^2}\right )}{32 a^3 d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 5.03, size = 357, normalized size = 1.00 \[ -\frac {\frac {4 b \sin (2 (c+d x)) \left (28 a^2+b (13 b-19 a) \cos (2 (c+d x))+3 a b-13 b^2\right )}{(a-b)^2 (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac {3 \sqrt {b} \left (34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {128 a b \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac {3 \sqrt {b} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {\sqrt {a} \sqrt {b}-a}}+64 \cot (c+d x)}{64 a^3 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[c + d*x]^2/(a - b*Sin[c + d*x]^4)^3,x]

[Out]

-1/64*((3*Sqrt[b]*(20*a + 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a
]*Sqrt[b]]])/((Sqrt[a] + Sqrt[b])^2*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)
*ArcTanh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])^2*Sqrt[-a + Sqrt
[a]*Sqrt[b]]) + 64*Cot[c + d*x] + (4*b*(28*a^2 + 3*a*b - 13*b^2 + b*(-19*a + 13*b)*Cos[2*(c + d*x)])*Sin[2*(c
+ d*x)])/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])) + (128*a*b*(2*a + b - b*Cos[2*(c
+ d*x)])*Sin[2*(c + d*x)])/((a - b)*(-8*a + 3*b - 4*b*Cos[2*(c + d*x)] + b*Cos[4*(c + d*x)])^2))/(a^3*d)

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fricas [B]  time = 3.64, size = 6323, normalized size = 17.71 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/256*(8*(32*a^2*b^2 - 83*a*b^3 + 45*b^4)*cos(d*x + c)^9 - 48*(19*a^2*b^2 - 54*a*b^3 + 30*b^4)*cos(d*x + c)^7
 - 8*(64*a^3*b - 301*a^2*b^2 + 555*a*b^3 - 270*b^4)*cos(d*x + c)^5 + 16*(55*a^3*b - 188*a^2*b^2 + 235*a*b^3 -
90*b^4)*cos(d*x + c)^3 + 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^
4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 +
3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b
- 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 -
 a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5
389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3
 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/(
(a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(1728000*a^6*b^2 - 7369920*a^5*b^3
+ 13507020*a^4*b^4 - 13573305*a^3*b^5 + 31519503/4*a^2*b^6 - 5011875/2*a*b^7 + 1366875/4*b^8 - 27/4*(256000*a^
6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(
d*x + c)^2 + 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^1
1*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*
a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 1
20*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^1
0)*d^4))*cos(d*x + c)*sin(d*x + c) + (12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6
*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b
^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*
a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^
2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*
b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b
^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^
3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b +
7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*
a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^
2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*
b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) - 3*((a^5*b^2 -
 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5
*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2
+ (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*
b^4 + 105*b^5 - (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2
355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492
300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a
^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8
*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(1728000*a^6*b^2 - 7369920*a^5*b^3 + 13507020*a^4*b^4 - 13573305*a^3*b^5
+ 31519503/4*a^2*b^6 - 5011875/2*a*b^7 + 1366875/4*b^8 - 27/4*(256000*a^6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*
b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(d*x + c)^2 - 27/2*((26*a^17 - 167*a^16
*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^11*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*
a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^
2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*
b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) + (
12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d
*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 - (a^11 - 5*a
^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^
6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((
a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45
*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)
*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 168
1*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*
a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^
6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((
a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45
*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) + 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8
 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*
x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b
^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*
a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 907
3120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a
^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 -
 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(
-1728000*a^6*b^2 + 7369920*a^5*b^3 - 13507020*a^4*b^4 + 13573305*a^3*b^5 - 31519503/4*a^2*b^6 + 5011875/2*a*b^
7 - 1366875/4*b^8 + 27/4*(256000*a^6*b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b
^6 - 371250*a*b^7 + 50625*b^8)*cos(d*x + c)^2 + 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 65
0*a^13*b^4 - 361*a^12*b^5 + 112*a^11*b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a
^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/(
(a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 4
5*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4))*cos(d*x + c)*sin(d*x + c) - (12800*a^10*b - 54080*a^9*b^2 + 98420*
a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(40
0*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a
^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4
*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*
a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*
d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^1
3*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x
+ c)^2 - (400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^
6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4
*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*
a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*
d^4)))*sin(d*x + c) - 3*((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 - 4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*
d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x + c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a
^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3 + a^3*b^4)*d)*sqrt(-(400*a^4*b - 1
044*a^3*b^2 + 1085*a^2*b^3 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^
6*b^5)*d^2*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389
980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 +
210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^
11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2))*log(-1728000*a^6*b^2 + 7369920*a^5*b^3 -
13507020*a^4*b^4 + 13573305*a^3*b^5 - 31519503/4*a^2*b^6 + 5011875/2*a*b^7 - 1366875/4*b^8 + 27/4*(256000*a^6*
b^2 - 1091840*a^5*b^3 + 2001040*a^4*b^4 - 2010860*a^3*b^5 + 1167389*a^2*b^6 - 371250*a*b^7 + 50625*b^8)*cos(d*
x + c)^2 - 27/2*((26*a^17 - 167*a^16*b + 460*a^15*b^2 - 705*a^14*b^3 + 650*a^13*b^4 - 361*a^12*b^5 + 112*a^11*
b^6 - 15*a^10*b^7)*d^3*sqrt((409600*a^8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^
4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120
*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)
*d^4))*cos(d*x + c)*sin(d*x + c) - (12800*a^10*b - 54080*a^9*b^2 + 98420*a^8*b^3 - 98415*a^7*b^4 + 56973*a^6*b
^5 - 18109*a^5*b^6 + 2475*a^4*b^7)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-(400*a^4*b - 1044*a^3*b^2 + 1085*a^2*b^3
 - 530*a*b^4 + 105*b^5 + (a^11 - 5*a^10*b + 10*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((409600*a^
8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*
b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^
5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))/((a^11 - 5*a^10*b + 10*a^9*b^2
 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2)) - 27/4*(2*(400*a^14 - 2556*a^13*b + 7005*a^12*b^2 - 10685*a^11*b^3
+ 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2*cos(d*x + c)^2 - (400*a^14 - 2556*a^13*b + 70
05*a^12*b^2 - 10685*a^11*b^3 + 9810*a^10*b^4 - 5430*a^9*b^5 + 1681*a^8*b^6 - 225*a^7*b^7)*d^2)*sqrt((409600*a^
8*b^3 - 2355200*a^7*b^4 + 6054400*a^6*b^5 - 9073120*a^5*b^6 + 8661145*a^4*b^7 - 5389980*a^3*b^8 + 2135086*a^2*
b^9 - 492300*a*b^10 + 50625*b^11)/((a^23 - 10*a^22*b + 45*a^21*b^2 - 120*a^20*b^3 + 210*a^19*b^4 - 252*a^18*b^
5 + 210*a^17*b^6 - 120*a^16*b^7 + 45*a^15*b^8 - 10*a^14*b^9 + a^13*b^10)*d^4)))*sin(d*x + c) + 8*(32*a^4 - 110
*a^3*b + 189*a^2*b^2 - 156*a*b^3 + 45*b^4)*cos(d*x + c))/(((a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^8 -
4*(a^5*b^2 - 2*a^4*b^3 + a^3*b^4)*d*cos(d*x + c)^6 - 2*(a^6*b - 5*a^5*b^2 + 7*a^4*b^3 - 3*a^3*b^4)*d*cos(d*x +
 c)^4 + 4*(a^6*b - 3*a^5*b^2 + 3*a^4*b^3 - a^3*b^4)*d*cos(d*x + c)^2 + (a^7 - 4*a^6*b + 6*a^5*b^2 - 4*a^4*b^3
+ a^3*b^4)*d)*sin(d*x + c))

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giac [B]  time = 2.27, size = 2203, normalized size = 6.17 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="giac")

[Out]

-1/64*(3*((78*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 267*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sq
rt(a*b)*a^3*b^2 + 241*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 53*sqrt(a^2 - a*b + sqrt(a*b)*(a
 - b))*sqrt(a*b)*a*b^4 - 15*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^5)*(a^5 - 2*a^4*b + a^3*b^2)^2*abs
(-a + b) - 2*(9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^10*b - 51*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^9*b^2 +
108*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^8*b^3 - 106*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^7*b^4 + 45*sqrt(a^
2 - a*b + sqrt(a*b)*(a - b))*a^6*b^5 - 3*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^6 - 2*sqrt(a^2 - a*b + sqrt
(a*b)*(a - b))*a^4*b^7)*abs(a^5 - 2*a^4*b + a^3*b^2)*abs(-a + b) - (60*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqr
t(a*b)*a^15 - 441*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b + 1339*sqrt(a^2 - a*b + sqrt(a*b)*(a -
b))*sqrt(a*b)*a^13*b^2 - 2185*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 2059*sqrt(a^2 - a*b + s
qrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 1091*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5 + 265*sqrt(
a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^8*b^7 - 1
1*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(t
an(d*x + c)/sqrt((a^6 - 2*a^5*b + a^4*b^2 + sqrt((a^6 - 2*a^5*b + a^4*b^2)^2 - (a^6 - 2*a^5*b + a^4*b^2)*(a^6
- 3*a^5*b + 3*a^4*b^2 - a^3*b^3)))/(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3))))/((3*a^16 - 27*a^15*b + 104*a^14*b^
2 - 224*a^13*b^3 + 294*a^12*b^4 - 238*a^11*b^5 + 112*a^10*b^6 - 24*a^9*b^7 - a^8*b^8 + a^7*b^9)*abs(a^5 - 2*a^
4*b + a^3*b^2)) - 3*((78*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b - 267*sqrt(a^2 - a*b - sqrt(a*b)*
(a - b))*sqrt(a*b)*a^3*b^2 + 241*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^3 - 53*sqrt(a^2 - a*b - s
qrt(a*b)*(a - b))*sqrt(a*b)*a*b^4 - 15*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^5)*(a^5 - 2*a^4*b + a^3
*b^2)^2*abs(-a + b) + 2*(9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^10*b - 51*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))
*a^9*b^2 + 108*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^8*b^3 - 106*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^7*b^4 +
 45*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^6*b^5 - 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^6 - 2*sqrt(a^2 -
 a*b - sqrt(a*b)*(a - b))*a^4*b^7)*abs(a^5 - 2*a^4*b + a^3*b^2)*abs(-a + b) - (60*sqrt(a^2 - a*b - sqrt(a*b)*(
a - b))*sqrt(a*b)*a^15 - 441*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^14*b + 1339*sqrt(a^2 - a*b - sqrt
(a*b)*(a - b))*sqrt(a*b)*a^13*b^2 - 2185*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^12*b^3 + 2059*sqrt(a^
2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^11*b^4 - 1091*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^10*b^5
+ 265*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^9*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*
a^8*b^7 - 11*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^7*b^8)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2)
 + arctan(tan(d*x + c)/sqrt((a^6 - 2*a^5*b + a^4*b^2 - sqrt((a^6 - 2*a^5*b + a^4*b^2)^2 - (a^6 - 2*a^5*b + a^4
*b^2)*(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3)))/(a^6 - 3*a^5*b + 3*a^4*b^2 - a^3*b^3))))/((3*a^16 - 27*a^15*b +
104*a^14*b^2 - 224*a^13*b^3 + 294*a^12*b^4 - 238*a^11*b^5 + 112*a^10*b^6 - 24*a^9*b^7 - a^8*b^8 + a^7*b^9)*abs
(a^5 - 2*a^4*b + a^3*b^2)) + 2*(18*a^3*b*tan(d*x + c)^7 - 3*a^2*b^2*tan(d*x + c)^7 - 28*a*b^3*tan(d*x + c)^7 +
 13*b^4*tan(d*x + c)^7 + 54*a^3*b*tan(d*x + c)^5 - 4*a^2*b^2*tan(d*x + c)^5 - 26*a*b^3*tan(d*x + c)^5 + 54*a^3
*b*tan(d*x + c)^3 - 13*a^2*b^2*tan(d*x + c)^3 - 17*a*b^3*tan(d*x + c)^3 + 18*a^3*b*tan(d*x + c) - 12*a^2*b^2*t
an(d*x + c))/((a^5 - 2*a^4*b + a^3*b^2)*(a*tan(d*x + c)^4 - b*tan(d*x + c)^4 + 2*a*tan(d*x + c)^2 + a)^2) + 64
/(a^3*tan(d*x + c)))/d

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maple [B]  time = 0.61, size = 1959, normalized size = 5.49 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x)

[Out]

-57/32/d/a/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(
1/2)-a)*(a-b))^(1/2))*b^3+15/16/d*b/(a^2-2*a*b+b^2)*a/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((
a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-15/16/d*b/(a^2-2*a*b+b^2)*a/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*
(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+57/32/d/a/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a
-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*b^3+141/64/d*b^2/(a^2
-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^
(1/2))+33/64/d/a^2/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(
((a*b)^(1/2)-a)*(a-b))^(1/2))*b^4-33/64/d/a^2/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*
arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))*b^4-9/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c
)^2+a)^2/(a^2-2*a*b+b^2)*tan(d*x+c)*b-9/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)/a*b*ta
n(d*x+c)^7+13/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/a/(a^2-2*a*b+b^2)*tan(d*x+c)^3*b^2-27/
16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^2-2*a*b+b^2)*tan(d*x+c)^3-189/64/d/a*b^2/(a^2-2
*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-189/64/
d/a*b^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1
/2))+39/16/d/a^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a
)*(a-b))^(1/2))*b^3+39/16/d/a^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/((
(a*b)^(1/2)+a)*(a-b))^(1/2))*b^3-1/d/a^3/tan(d*x+c)-141/64/d*b^2/(a^2-2*a*b+b^2)/(a*b)^(1/2)/(a-b)/(((a*b)^(1/
2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-45/64/d*b^4/a^3/(a^2-2*a*b+b^2)/(a-b
)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-45/64/d*b^4/a^3/(a^2-
2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+1/8/d/(t
an(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/a/(a^2-2*a*b+b^2)*tan(d*x+c)^5*b^2+17/32/d*b^3/a^2/(tan(d*x
+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a^2-2*a*b+b^2)*tan(d*x+c)^3+13/32/d*b^3/a^3/(tan(d*x+c)^4*a-tan(
d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/(a-b)*tan(d*x+c)^7-15/32/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)
^2/a^2/(a-b)*tan(d*x+c)^7*b^2-27/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b/(a^2-2*a*b+b^2)*t
an(d*x+c)^5+13/16/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2/a^2*b^3/(a^2-2*a*b+b^2)*tan(d*x+c)^5+
39/32/d*b/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b)
)^(1/2))+39/32/d*b/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)+a)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a
)*(a-b))^(1/2))+3/8/d/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2*b^2/a/(a^2-2*a*b+b^2)*tan(d*x+c)

________________________________________________________________________________________

maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)^2/(a-b*sin(d*x+c)^4)^3,x, algorithm="maxima")

[Out]

Timed out

________________________________________________________________________________________

mupad [B]  time = 20.80, size = 7364, normalized size = 20.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)^2*(a - b*sin(c + d*x)^4)^3),x)

[Out]

(atan((((-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1
085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b
^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2315255808*a
^15*b^12 - 201326592*a^14*b^13 - 12079595520*a^16*b^11 + 37748736000*a^17*b^10 - 78517370880*a^18*b^9 + 114152
177664*a^19*b^8 - 118380036096*a^20*b^7 + 87577067520*a^21*b^6 - 45298483200*a^22*b^5 + 15602810880*a^23*b^4 -
 3221225472*a^24*b^3 + 301989888*a^25*b^2 + tan(c + d*x)*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(
1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) -
 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 +
 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2147483648*a^29*b + 2147483648*a^17*b^13 - 25769803776*a^18*b^12 + 141733
920768*a^19*b^11 - 472446402560*a^20*b^10 + 1063004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^
23*b^7 - 1700807049216*a^24*b^6 + 1063004405760*a^25*b^5 - 472446402560*a^26*b^4 + 141733920768*a^27*b^3 - 257
69803776*a^28*b^2)) + tan(c + d*x)*(3024617472*a^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520
940032*a^13*b^11 - 94402510848*a^14*b^10 + 125505110016*a^15*b^9 - 108421447680*a^16*b^8 + 51536461824*a^17*b^
7 + 484835328*a^18*b^6 - 18454413312*a^19*b^5 + 12354453504*a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^
2))*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*
a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^
(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*1i + ((-(9*(640*a
^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*
a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*
(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(201326592*a^14*b^13 - 231525580
8*a^15*b^12 + 12079595520*a^16*b^11 - 37748736000*a^17*b^10 + 78517370880*a^18*b^9 - 114152177664*a^19*b^8 + 1
18380036096*a^20*b^7 - 87577067520*a^21*b^6 + 45298483200*a^22*b^5 - 15602810880*a^23*b^4 + 3221225472*a^24*b^
3 - 301989888*a^25*b^2 + tan(c + d*x)*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b -
 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b
^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a
^16*b^2)))^(1/2)*(2147483648*a^29*b + 2147483648*a^17*b^13 - 25769803776*a^18*b^12 + 141733920768*a^19*b^11 -
472446402560*a^20*b^10 + 1063004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 1700807049
216*a^24*b^6 + 1063004405760*a^25*b^5 - 472446402560*a^26*b^4 + 141733920768*a^27*b^3 - 25769803776*a^28*b^2))
 + tan(c + d*x)*(3024617472*a^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13*b^11 -
94402510848*a^14*b^10 + 125505110016*a^15*b^9 - 108421447680*a^16*b^8 + 51536461824*a^17*b^7 + 484835328*a^18*
b^6 - 18454413312*a^19*b^5 + 12354453504*a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^2))*(-(9*(640*a^4*(
a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10
*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a
^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*1i)/(((-(9*(640*a^4*(a^13*b^3)^(1/2)
 + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2
*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 +
a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2315255808*a^15*b^12 - 201326592*a^14*b^13 - 12079
595520*a^16*b^11 + 37748736000*a^17*b^10 - 78517370880*a^18*b^9 + 114152177664*a^19*b^8 - 118380036096*a^20*b^
7 + 87577067520*a^21*b^6 - 45298483200*a^22*b^5 + 15602810880*a^23*b^4 - 3221225472*a^24*b^3 + 301989888*a^25*
b^2 + tan(c + d*x)*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*
a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^
3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2
147483648*a^29*b + 2147483648*a^17*b^13 - 25769803776*a^18*b^12 + 141733920768*a^19*b^11 - 472446402560*a^20*b
^10 + 1063004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 1700807049216*a^24*b^6 + 1063
004405760*a^25*b^5 - 472446402560*a^26*b^4 + 141733920768*a^27*b^3 - 25769803776*a^28*b^2)) + tan(c + d*x)*(30
24617472*a^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13*b^11 - 94402510848*a^14*b^
10 + 125505110016*a^15*b^9 - 108421447680*a^16*b^8 + 51536461824*a^17*b^7 + 484835328*a^18*b^6 - 18454413312*a
^19*b^5 + 12354453504*a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^2))*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 2
25*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2
*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13
*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2) - ((-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^
(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2)
- 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4
+ 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(201326592*a^14*b^13 - 2315255808*a^15*b^12 + 12079595520*a^16*b^11 - 377
48736000*a^17*b^10 + 78517370880*a^18*b^9 - 114152177664*a^19*b^8 + 118380036096*a^20*b^7 - 87577067520*a^21*b
^6 + 45298483200*a^22*b^5 - 15602810880*a^23*b^4 + 3221225472*a^24*b^3 - 301989888*a^25*b^2 + tan(c + d*x)*(-(
9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3
 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))
/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2147483648*a^29*b + 214
7483648*a^17*b^13 - 25769803776*a^18*b^12 + 141733920768*a^19*b^11 - 472446402560*a^20*b^10 + 1063004405760*a^
21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 1700807049216*a^24*b^6 + 1063004405760*a^25*b^5 - 4
72446402560*a^26*b^4 + 141733920768*a^27*b^3 - 25769803776*a^28*b^2)) + tan(c + d*x)*(3024617472*a^11*b^13 - 2
65420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13*b^11 - 94402510848*a^14*b^10 + 125505110016*a^15
*b^9 - 108421447680*a^16*b^8 + 51536461824*a^17*b^7 + 484835328*a^18*b^6 - 18454413312*a^19*b^5 + 12354453504*
a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^2))*(-(9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2
) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 10
94*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10
*a^15*b^3 - 10*a^16*b^2)))^(1/2) + 186624000*a^7*b^14 - 2227875840*a^8*b^13 + 12162465792*a^9*b^12 - 400508928
00*a^10*b^11 + 88332816384*a^11*b^10 - 136918204416*a^12*b^9 + 152103813120*a^13*b^8 - 121034760192*a^14*b^7 +
 67571435520*a^15*b^6 - 25193631744*a^16*b^5 + 5643288576*a^17*b^4 - 575078400*a^18*b^3))*(-(9*(640*a^4*(a^13*
b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) - 400*a^11*b - 105*a^7*b^5 + 530*a^8*b^4 - 1085*a^9*b^3 + 1044*a^10*b^2
+ 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b
 - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*2i)/d - (1/a + (tan(c + d*x)^2*(64*a^2 -
119*a*b + 58*b^2))/(16*a*(a^2 - 2*a*b + b^2)) + (tan(c + d*x)^8*(111*a*b^2 - 78*a^2*b + 32*a^3 - 45*b^3))/(32*
a^3*(a - b)) + (tan(c + d*x)^6*(190*a*b^2 - 165*a^2*b + 64*a^3 - 77*b^3))/(16*a^2*(a - b)^2) + (tan(c + d*x)^4
*(307*a*b^2 - 394*a^2*b + 192*a^3 - 81*b^3))/(32*a^2*(a^2 - 2*a*b + b^2)))/(d*(tan(c + d*x)^9*(a^2 - 2*a*b + b
^2) + a^2*tan(c + d*x) - tan(c + d*x)^5*(2*a*b - 6*a^2) - tan(c + d*x)^7*(4*a*b - 4*a^2) + 4*a^2*tan(c + d*x)^
3)) + (atan(((((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^
4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a
^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2315255
808*a^15*b^12 - 201326592*a^14*b^13 - 12079595520*a^16*b^11 + 37748736000*a^17*b^10 - 78517370880*a^18*b^9 + 1
14152177664*a^19*b^8 - 118380036096*a^20*b^7 + 87577067520*a^21*b^6 - 45298483200*a^22*b^5 + 15602810880*a^23*
b^4 - 3221225472*a^24*b^3 + 301989888*a^25*b^2 + tan(c + d*x)*((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^
3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/
2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b
^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2147483648*a^29*b + 2147483648*a^17*b^13 - 25769803776*a^18*b^12 + 14
1733920768*a^19*b^11 - 472446402560*a^20*b^10 + 1063004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 198427489075
2*a^23*b^7 - 1700807049216*a^24*b^6 + 1063004405760*a^25*b^5 - 472446402560*a^26*b^4 + 141733920768*a^27*b^3 -
 25769803776*a^28*b^2)) + tan(c + d*x)*(3024617472*a^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 4
7520940032*a^13*b^11 - 94402510848*a^14*b^10 + 125505110016*a^15*b^9 - 108421447680*a^16*b^8 + 51536461824*a^1
7*b^7 + 484835328*a^18*b^6 - 18454413312*a^19*b^5 + 12354453504*a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^2
2*b^2))*((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 10
85*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^
3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*1i + (((9*(640
*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 104
4*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(1638
4*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(201326592*a^14*b^13 - 2315255
808*a^15*b^12 + 12079595520*a^16*b^11 - 37748736000*a^17*b^10 + 78517370880*a^18*b^9 - 114152177664*a^19*b^8 +
 118380036096*a^20*b^7 - 87577067520*a^21*b^6 + 45298483200*a^22*b^5 - 15602810880*a^23*b^4 + 3221225472*a^24*
b^3 - 301989888*a^25*b^2 + tan(c + d*x)*((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b
+ 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*
b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*
a^16*b^2)))^(1/2)*(2147483648*a^29*b + 2147483648*a^17*b^13 - 25769803776*a^18*b^12 + 141733920768*a^19*b^11 -
 472446402560*a^20*b^10 + 1063004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 170080704
9216*a^24*b^6 + 1063004405760*a^25*b^5 - 472446402560*a^26*b^4 + 141733920768*a^27*b^3 - 25769803776*a^28*b^2)
) + tan(c + d*x)*(3024617472*a^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13*b^11 -
 94402510848*a^14*b^10 + 125505110016*a^15*b^9 - 108421447680*a^16*b^8 + 51536461824*a^17*b^7 + 484835328*a^18
*b^6 - 18454413312*a^19*b^5 + 12354453504*a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^2))*((9*(640*a^4*(
a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10
*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a
^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*1i)/((((9*(640*a^4*(a^13*b^3)^(1/2)
+ 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*
b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a
^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2315255808*a^15*b^12 - 201326592*a^14*b^13 - 120795
95520*a^16*b^11 + 37748736000*a^17*b^10 - 78517370880*a^18*b^9 + 114152177664*a^19*b^8 - 118380036096*a^20*b^7
 + 87577067520*a^21*b^6 - 45298483200*a^22*b^5 + 15602810880*a^23*b^4 - 3221225472*a^24*b^3 + 301989888*a^25*b
^2 + tan(c + d*x)*((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^
8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*
b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(214
7483648*a^29*b + 2147483648*a^17*b^13 - 25769803776*a^18*b^12 + 141733920768*a^19*b^11 - 472446402560*a^20*b^1
0 + 1063004405760*a^21*b^9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 1700807049216*a^24*b^6 + 106300
4405760*a^25*b^5 - 472446402560*a^26*b^4 + 141733920768*a^27*b^3 - 25769803776*a^28*b^2)) + tan(c + d*x)*(3024
617472*a^11*b^13 - 265420800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13*b^11 - 94402510848*a^14*b^10
 + 125505110016*a^15*b^9 - 108421447680*a^16*b^8 + 51536461824*a^17*b^7 + 484835328*a^18*b^6 - 18454413312*a^1
9*b^5 + 12354453504*a^20*b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^2))*((9*(640*a^4*(a^13*b^3)^(1/2) + 225*
b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a
^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^
5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2) - (((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2
) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 10
94*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10
*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(201326592*a^14*b^13 - 2315255808*a^15*b^12 + 12079595520*a^16*b^11 - 3774873
6000*a^17*b^10 + 78517370880*a^18*b^9 - 114152177664*a^19*b^8 + 118380036096*a^20*b^7 - 87577067520*a^21*b^6 +
 45298483200*a^22*b^5 - 15602810880*a^23*b^4 + 3221225472*a^24*b^3 - 301989888*a^25*b^2 + tan(c + d*x)*((9*(64
0*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 10
44*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(163
84*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*(2147483648*a^29*b + 21474836
48*a^17*b^13 - 25769803776*a^18*b^12 + 141733920768*a^19*b^11 - 472446402560*a^20*b^10 + 1063004405760*a^21*b^
9 - 1700807049216*a^22*b^8 + 1984274890752*a^23*b^7 - 1700807049216*a^24*b^6 + 1063004405760*a^25*b^5 - 472446
402560*a^26*b^4 + 141733920768*a^27*b^3 - 25769803776*a^28*b^2)) + tan(c + d*x)*(3024617472*a^11*b^13 - 265420
800*a^10*b^14 - 15574892544*a^12*b^12 + 47520940032*a^13*b^11 - 94402510848*a^14*b^10 + 125505110016*a^15*b^9
- 108421447680*a^16*b^8 + 51536461824*a^17*b^7 + 484835328*a^18*b^6 - 18454413312*a^19*b^5 + 12354453504*a^20*
b^4 - 3779592192*a^21*b^3 + 471859200*a^22*b^2))*((9*(640*a^4*(a^13*b^3)^(1/2) + 225*b^4*(a^13*b^3)^(1/2) + 40
0*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b
^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*
b^3 - 10*a^16*b^2)))^(1/2) + 186624000*a^7*b^14 - 2227875840*a^8*b^13 + 12162465792*a^9*b^12 - 40050892800*a^1
0*b^11 + 88332816384*a^11*b^10 - 136918204416*a^12*b^9 + 152103813120*a^13*b^8 - 121034760192*a^14*b^7 + 67571
435520*a^15*b^6 - 25193631744*a^16*b^5 + 5643288576*a^17*b^4 - 575078400*a^18*b^3))*((9*(640*a^4*(a^13*b^3)^(1
/2) + 225*b^4*(a^13*b^3)^(1/2) + 400*a^11*b + 105*a^7*b^5 - 530*a^8*b^4 + 1085*a^9*b^3 - 1044*a^10*b^2 + 2085*
a^2*b^2*(a^13*b^3)^(1/2) - 1094*a*b^3*(a^13*b^3)^(1/2) - 1840*a^3*b*(a^13*b^3)^(1/2)))/(16384*(5*a^17*b - a^18
 + a^13*b^5 - 5*a^14*b^4 + 10*a^15*b^3 - 10*a^16*b^2)))^(1/2)*2i)/d

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2/(a-b*sin(d*x+c)**4)**3,x)

[Out]

Timed out

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