3.3.35 \(\int \frac {\csc ^2(c+d x)}{(a-b \sin ^4(c+d x))^3} \, dx\) [235]

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

[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 = 0.87, 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 = {3296, 1348, 1683, 1678, 1180, 211} \begin {gather*} \frac {3 \sqrt {b} \left (-34 \sqrt {a} \sqrt {b}+20 a+15 b\right ) \text {ArcTan}\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 ) \text {ArcTan}\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 )} \end {gather*}

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 211

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

Rule 1180

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 1348

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 1678

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 1683

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 3296

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 {\text {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 {\text {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 {\text {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 {\text {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) \text {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 ) \text {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 ) \text {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 = 3.51, size = 357, normalized size = 1.00 \begin {gather*} -\frac {\frac {3 \sqrt {b} \left (20 a+34 \sqrt {a} \sqrt {b}+15 b\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}+\sqrt {b}\right )^2 \sqrt {a+\sqrt {a} \sqrt {b}}}+\frac {3 \sqrt {b} \left (20 a-34 \sqrt {a} \sqrt {b}+15 b\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {-a+\sqrt {a} \sqrt {b}}}\right )}{\left (\sqrt {a}-\sqrt {b}\right )^2 \sqrt {-a+\sqrt {a} \sqrt {b}}}+64 \cot (c+d x)+\frac {4 b \left (28 a^2+3 a b-13 b^2+b (-19 a+13 b) \cos (2 (c+d x))\right ) \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)))}+\frac {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}}{64 a^3 d} \end {gather*}

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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Maple [A]
time = 1.78, size = 437, normalized size = 1.22

method	result	size
derivativedivides	\(\frac {-\frac {1}{a^{3} \tan \left (d x +c \right )}+\frac {b \left (\frac {-\frac {\left (18 a^{2}+15 a b -13 b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 \left (a -b \right )}-\frac {a \left (27 a^{2}-2 a b -13 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{16 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (54 a^{2}-13 a b -17 b^{2}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a -2 b \right ) \tan \left (d x +c \right )}{16 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )^{2}}+\frac {3 \left (a -b \right ) \left (\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}+20 a^{3}-27 a^{2} b +11 a \,b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}-20 a^{3}+27 a^{2} b -11 a \,b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}}{d}\)	\(437\)
default	\(\frac {-\frac {1}{a^{3} \tan \left (d x +c \right )}+\frac {b \left (\frac {-\frac {\left (18 a^{2}+15 a b -13 b^{2}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{32 \left (a -b \right )}-\frac {a \left (27 a^{2}-2 a b -13 b^{2}\right ) \left (\tan ^{5}\left (d x +c \right )\right )}{16 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (54 a^{2}-13 a b -17 b^{2}\right ) a \left (\tan ^{3}\left (d x +c \right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}-\frac {3 a^{2} \left (3 a -2 b \right ) \tan \left (d x +c \right )}{16 \left (a^{2}-2 a b +b^{2}\right )}}{\left (\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right )^{2}}+\frac {3 \left (a -b \right ) \left (\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}+20 a^{3}-27 a^{2} b +11 a \,b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {\left (26 a^{2} \sqrt {a b}-37 a b \sqrt {a b}+15 b^{2} \sqrt {a b}-20 a^{3}+27 a^{2} b -11 a \,b^{2}\right ) \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{a^{3}}}{d}\)	\(437\)
risch	\(\text {Expression too large to display}\)	\(2739\)

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,method=_RETURNVERBOSE)

[Out]

1/d*(-1/a^3/tan(d*x+c)+b/a^3*((-1/32*(18*a^2+15*a*b-13*b^2)/(a-b)*tan(d*x+c)^7-1/16*a*(27*a^2-2*a*b-13*b^2)/(a
^2-2*a*b+b^2)*tan(d*x+c)^5-1/32*(54*a^2-13*a*b-17*b^2)*a/(a^2-2*a*b+b^2)*tan(d*x+c)^3-3/16*a^2*(3*a-2*b)/(a^2-
2*a*b+b^2)*tan(d*x+c))/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)^2+3/32/(a^2-2*a*b+b^2)*(a-b)*(1/2*(2
6*a^2*(a*b)^(1/2)-37*a*b*(a*b)^(1/2)+15*b^2*(a*b)^(1/2)+20*a^3-27*a^2*b+11*a*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))+1/2*(26*a^2*(a*b)^(1/2)-37*a*b*(a*b
)^(1/2)+15*b^2*(a*b)^(1/2)-20*a^3+27*a^2*b-11*a*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)))))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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]

1/16*(12*(160*a^3*b^3 - 57*a^2*b^4 - 195*a*b^5 + 135*b^6)*cos(4*d*x + 4*c)*sin(2*d*x + 2*c) + (3*(20*a^2*b^4 -
33*a*b^5 + 15*b^6)*sin(16*d*x + 16*c) - 12*(43*a^2*b^4 - 68*a*b^5 + 30*b^6)*sin(14*d*x + 14*c) - 4*(400*a^3*b
^3 - 1137*a^2*b^4 + 1031*a*b^5 - 315*b^6)*sin(12*d*x + 12*c) + 12*(592*a^3*b^3 - 1237*a^2*b^4 + 886*a*b^5 - 21
0*b^6)*sin(10*d*x + 10*c) + 2*(4096*a^4*b^2 - 12192*a^3*b^3 + 13634*a^2*b^4 - 7113*a*b^5 + 1575*b^6)*sin(8*d*x
+ 8*c) + 4*(880*a^3*b^3 - 2855*a^2*b^4 + 2512*a*b^5 - 630*b^6)*sin(6*d*x + 6*c) - 4*(256*a^3*b^3 - 823*a^2*b^
4 + 903*a*b^5 - 315*b^6)*sin(4*d*x + 4*c) - 12*(19*a^2*b^4 - 54*a*b^5 + 30*b^6)*sin(2*d*x + 2*c))*cos(18*d*x +
18*c) + 3*(4*(160*a^3*b^3 - 57*a^2*b^4 - 195*a*b^5 + 135*b^6)*sin(14*d*x + 14*c) + 4*(400*a^3*b^3 - 1671*a^2*
b^4 + 1800*a*b^5 - 630*b^6)*sin(12*d*x + 12*c) - 2*(2560*a^4*b^2 + 3232*a^3*b^3 - 13806*a^2*b^4 + 11469*a*b^5
- 2835*b^6)*sin(10*d*x + 10*c) - 4*(4864*a^4*b^2 - 14576*a^3*b^3 + 16221*a^2*b^4 - 8430*a*b^5 + 1890*b^6)*sin(
8*d*x + 8*c) - 4*(1840*a^3*b^3 - 6825*a^2*b^4 + 6243*a*b^5 - 1575*b^6)*sin(6*d*x + 6*c) + 4*(608*a^3*b^3 - 202
5*a^2*b^4 + 2292*a*b^5 - 810*b^6)*sin(4*d*x + 4*c) + 9*(56*a^2*b^4 - 183*a*b^5 + 105*b^6)*sin(2*d*x + 2*c))*co
s(16*d*x + 16*c) + 4*(4*(3200*a^4*b^2 - 7536*a^3*b^3 + 7612*a^2*b^4 - 3915*a*b^5 + 945*b^6)*sin(12*d*x + 12*c)
- 6*(3968*a^4*b^2 - 14864*a^3*b^3 + 19013*a^2*b^4 - 10224*a*b^5 + 1890*b^6)*sin(10*d*x + 10*c) - 2*(32768*a^5
*b - 117888*a^4*b^2 + 172048*a^3*b^3 - 127323*a^2*b^4 + 49365*a*b^5 - 8505*b^6)*sin(8*d*x + 8*c) - 8*(3520*a^4
*b^2 - 12800*a^3*b^3 + 17461*a^2*b^4 - 9882*a*b^5 + 1890*b^6)*sin(6*d*x + 6*c) + 4*(2048*a^4*b^2 - 7856*a^3*b^
3 + 11838*a^2*b^4 - 8091*a*b^5 + 2025*b^6)*sin(4*d*x + 4*c) + 3*(608*a^3*b^3 - 2025*a^2*b^4 + 2292*a*b^5 - 810
*b^6)*sin(2*d*x + 2*c))*cos(14*d*x + 14*c) + 4*(2*(51200*a^5*b - 67456*a^4*b^2 - 32384*a^3*b^3 + 91591*a^2*b^4
- 46683*a*b^5 + 6615*b^6)*sin(10*d*x + 10*c) + 2*(112640*a^5*b - 364160*a^4*b^2 + 462304*a^3*b^3 - 293923*a^2
*b^4 + 97020*a*b^5 - 13230*b^6)*sin(8*d*x + 8*c) + 4*(19200*a^4*b^2 - 78800*a^3*b^3 + 95318*a^2*b^4 - 43701*a*
b^5 + 6615*b^6)*sin(6*d*x + 6*c) - 8*(3520*a^4*b^2 - 12800*a^3*b^3 + 17461*a^2*b^4 - 9882*a*b^5 + 1890*b^6)*si
n(4*d*x + 4*c) - 3*(1840*a^3*b^3 - 6825*a^2*b^4 + 6243*a*b^5 - 1575*b^6)*sin(2*d*x + 2*c))*cos(12*d*x + 12*c)
+ 4*((524288*a^6 - 1761280*a^5*b + 2435584*a^4*b^2 - 1768256*a^3*b^3 + 719196*a^2*b^4 - 163611*a*b^5 + 19845*b
^6)*sin(8*d*x + 8*c) + 2*(112640*a^5*b - 364160*a^4*b^2 + 462304*a^3*b^3 - 293923*a^2*b^4 + 97020*a*b^5 - 1323
0*b^6)*sin(6*d*x + 6*c) - 2*(32768*a^5*b - 117888*a^4*b^2 + 172048*a^3*b^3 - 127323*a^2*b^4 + 49365*a*b^5 - 85
05*b^6)*sin(4*d*x + 4*c) - 3*(4864*a^4*b^2 - 14576*a^3*b^3 + 16221*a^2*b^4 - 8430*a*b^5 + 1890*b^6)*sin(2*d*x
+ 2*c))*cos(10*d*x + 10*c) + 2*(4*(51200*a^5*b - 67456*a^4*b^2 - 32384*a^3*b^3 + 91591*a^2*b^4 - 46683*a*b^5 +
6615*b^6)*sin(6*d*x + 6*c) - 12*(3968*a^4*b^2 - 14864*a^3*b^3 + 19013*a^2*b^4 - 10224*a*b^5 + 1890*b^6)*sin(4
*d*x + 4*c) - 3*(2560*a^4*b^2 + 3232*a^3*b^3 - 13806*a^2*b^4 + 11469*a*b^5 - 2835*b^6)*sin(2*d*x + 2*c))*cos(8
*d*x + 8*c) + 4*(4*(3200*a^4*b^2 - 7536*a^3*b^3 + 7612*a^2*b^4 - 3915*a*b^5 + 945*b^6)*sin(4*d*x + 4*c) + 3*(4
00*a^3*b^3 - 1671*a^2*b^4 + 1800*a*b^5 - 630*b^6)*sin(2*d*x + 2*c))*cos(6*d*x + 6*c) + 16*((a^5*b^4 - 2*a^4*b^
5 + a^3*b^6)*d*cos(18*d*x + 18*c)^2 + 81*(a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*cos(16*d*x + 16*c)^2 + 16*(64*a^7*b
^2 - 272*a^6*b^3 + 433*a^5*b^4 - 306*a^4*b^5 + 81*a^3*b^6)*d*cos(14*d*x + 14*c)^2 + 16*(1600*a^7*b^2 - 4880*a^
6*b^3 + 5401*a^5*b^4 - 2562*a^4*b^5 + 441*a^3*b^6)*d*cos(12*d*x + 12*c)^2 + 4*(16384*a^9 - 73728*a^8*b + 14003
2*a^7*b^2 - 144576*a^6*b^3 + 86017*a^5*b^4 - 28098*a^4*b^5 + 3969*a^3*b^6)*d*cos(10*d*x + 10*c)^2 + 4*(16384*a
^9 - 73728*a^8*b + 140032*a^7*b^2 - 144576*a^6*b^3 + 86017*a^5*b^4 - 28098*a^4*b^5 + 3969*a^3*b^6)*d*cos(8*d*x
+ 8*c)^2 + 16*(1600*a^7*b^2 - 4880*a^6*b^3 + 5401*a^5*b^4 - 2562*a^4*b^5 + 441*a^3*b^6)*d*cos(6*d*x + 6*c)^2
+ 16*(64*a^7*b^2 - 272*a^6*b^3 + 433*a^5*b^4 - 306*a^4*b^5 + 81*a^3*b^6)*d*cos(4*d*x + 4*c)^2 + 81*(a^5*b^4 -
2*a^4*b^5 + a^3*b^6)*d*cos(2*d*x + 2*c)^2 + (a^5*b^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(18*d*x + 18*c)^2 + 81*(a^5*b
^4 - 2*a^4*b^5 + a^3*b^6)*d*sin(16*d*x + 16*c)^2 + 16*(64*a^7*b^2 - 272*a^6*b^3 + 433*a^5*b^4 - 306*a^4*b^5 +
81*a^3*b^6)*d*sin(14*d*x + 14*c)^2 + 16*(1600*a^7*b^2 - 4880*a^6*b^3 + 5401*a^5*b^4 - 2562*a^4*b^5 + 441*a^3*b
^6)*d*sin(12*d*x + 12*c)^2 + 4*(16384*a^9 - 73728*a^8*b + 140032*a^7*b^2 - 144576*a^6*b^3 + 86017*a^5*b^4 - 28
098*a^4*b^5 + 3969*a^3*b^6)*d*sin(10*d*x + 10*c)^2 + 4*(16384*a^9 - 73728*a^8*b + 140032*a^7*b^2 - 144576*a^6*
b^3 + 86017*a^5*b^4 - 28098*a^4*b^5 + 3969*a^3*b^6)*d*sin(8*d*x + 8*c)^2 + 16*(1600*a^7*b^2 - 4880*a^6*b^3 + 5
401*a^5*b^4 - 2562*a^4*b^5 + 441*a^3*b^6)*d*sin(6*d*x + 6*c)^2 + 16*(64*a^7*b^2 - 272*a^6*b^3 + 433*a^5*b^4 -
306*a^4*b^5 + 81*a^3*b^6)*d*sin(4*d*x + 4*c)^2 + 72*(8*a^6*b^3 - 25*a^5*b^4 + 26*a^4*b^5 - 9*a^3*b^6)*d*sin(4*
d*x + 4*c)*sin(2*d*x + 2*c) + 81*(a^5*b^4 - 2*a...

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6323 vs. \(2 (305) = 610\).
time = 2.75, size = 6323, normalized size = 17.71 \begin {gather*} \text {Too large to display} \end {gather*}

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 - 538...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2203 vs. \(2 (305) = 610\).
time = 1.37, size = 2203, normalized size = 6.17 \begin {gather*} \text {Too large to display} \end {gather*}

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

________________________________________________________________________________________

Mupad [B]
time = 20.80, size = 2500, normalized size = 7.00 \begin {gather*} \text {Too large to display} \end {gather*}

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 - 1557...

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