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

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

[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))

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Rubi [A]  time = 1.29425, antiderivative size = 357, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 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]

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 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 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 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 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]

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*}

Mathematica [A]  time = 5.27675, size = 357, normalized size = 1. \[ -\frac{\frac{4 b \sin (2 (c+d x)) \left (28 a^2+b (13 b-19 a) \cos (2 (c+d x))+3 a b-13 b^2\right )}{(a-b)^2 (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac{3 \sqrt{b} \left (34 \sqrt{a} \sqrt{b}+20 a+15 b\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\left (\sqrt{a}+\sqrt{b}\right )^2 \sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{128 a b \sin (2 (c+d x)) (2 a-b \cos (2 (c+d x))+b)}{(a-b) (-8 a-4 b \cos (2 (c+d x))+b \cos (4 (c+d x))+3 b)^2}+\frac{3 \sqrt{b} \left (-34 \sqrt{a} \sqrt{b}+20 a+15 b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\left (\sqrt{a}-\sqrt{b}\right )^2 \sqrt{\sqrt{a} \sqrt{b}-a}}+64 \cot (c+d x)}{64 a^3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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]*Sqr
t[b]]])/((Sqrt[a] + Sqrt[b])^2*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (3*Sqrt[b]*(20*a - 34*Sqrt[a]*Sqrt[b] + 15*b)*ArcT
anh[((Sqrt[a] - Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])^2*Sqrt[-a + Sqrt[a]*S
qrt[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))/(64*a^3*d)

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Maple [B]  time = 0.206, size = 1959, normalized size = 5.5 \begin{align*} \text{result too large to display} \end{align*}

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]

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))-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)^3-1/d/a^3/tan(d*x+c)-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)^5-15/32/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+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+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^3-57/32/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+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)^5-9/16/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*tan(d*x+c)^7-
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+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^4-189/64/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))-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))+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))+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))-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))+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))+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+39/16/d/a^2/(a^2-2*a*b+b^2)/(a-b)/(((a*b)^(1/2)-a)*(a-b))^(1/2)*arc
tanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))*b^3-45/64/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))+1/8/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)^5*b^2-9/16/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+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)-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))+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+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

________________________________________________________________________________________

Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

Fricas [B]  time = 26.8953, size = 16077, normalized size = 45.03 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError