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

Optimal. Leaf size=617 \[ -\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac{b \cos (c+d x) \left (-(5 a+b) \cos ^2(c+d x)+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}-\frac{\sqrt [4]{b} \left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

[Out]

-((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a]
- Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] -
Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqr
t[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(
8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*
a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a]
+ Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*a*(a - b)*d*(
a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*d*(a
- b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) - (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a^2
*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

________________________________________________________________________________________

Rubi [A]  time = 0.836883, antiderivative size = 617, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 7, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.318, Rules used = {3215, 1238, 207, 1178, 1166, 205, 208} \[ -\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac{b \cos (c+d x) \left (-(5 a+b) \cos ^2(c+d x)+11 a+b\right )}{32 a^2 d (a-b)^2 \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}-\sqrt{b}}}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{3/2}}-\frac{\sqrt [4]{b} \left (5 \sqrt{a}-2 \sqrt{b}\right ) \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}-\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \left (5 \sqrt{a}+2 \sqrt{b}\right ) \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{5/2}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 d \sqrt{\sqrt{a}+\sqrt{b}}}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{5/2} d \left (\sqrt{a}+\sqrt{b}\right )^{3/2}}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a d (a-b) \left (a-b \cos ^4(c+d x)+2 b \cos ^2(c+d x)-b\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((5*Sqrt[a] - 2*Sqrt[b])*b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a]
- Sqrt[b])^(5/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(8*a^(5/2)*(Sqrt[a] -
Sqrt[b])^(3/2)*d) - (b^(1/4)*ArcTan[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] - Sqrt[b]]])/(2*a^3*Sqrt[Sqrt[a] - Sqr
t[b]]*d) - ArcTanh[Cos[c + d*x]]/(a^3*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(
8*a^(5/2)*(Sqrt[a] + Sqrt[b])^(3/2)*d) + (b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a] + Sqrt[b]]])/(2*
a^3*Sqrt[Sqrt[a] + Sqrt[b]]*d) + ((5*Sqrt[a] + 2*Sqrt[b])*b^(1/4)*ArcTanh[(b^(1/4)*Cos[c + d*x])/Sqrt[Sqrt[a]
+ Sqrt[b]]])/(64*a^(5/2)*(Sqrt[a] + Sqrt[b])^(5/2)*d) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(8*a*(a - b)*d*(
a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)^2) - (b*Cos[c + d*x]*(2 - Cos[c + d*x]^2))/(4*a^2*(a - b)*d*(a
- b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4)) - (b*Cos[c + d*x]*(11*a + b - (5*a + b)*Cos[c + d*x]^2))/(32*a^2
*(a - b)^2*d*(a - b + 2*b*Cos[c + d*x]^2 - b*Cos[c + d*x]^4))

Rule 3215

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

Rule 1238

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 207

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 1178

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(x*(a*b*e - d*(b^2 - 2*
a*c) - c*(b*d - 2*a*e)*x^2)*(a + b*x^2 + c*x^4)^(p + 1))/(2*a*(p + 1)*(b^2 - 4*a*c)), x] + Dist[1/(2*a*(p + 1)
*(b^2 - 4*a*c)), Int[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 7)*(d*b - 2*a*e)*c*x^2, x]*(a +
b*x^2 + c*x^4)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0] && LtQ[p, -1] && IntegerQ[2*p]

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]

Rule 208

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

Rubi steps

\begin{align*} \int \frac{\csc (c+d x)}{\left (a-b \sin ^4(c+d x)\right )^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{a^3 \left (-1+x^2\right )}+\frac{b-b x^2}{a \left (a-b+2 b x^2-b x^4\right )^3}+\frac{b-b x^2}{a^2 \left (a-b+2 b x^2-b x^4\right )^2}+\frac{b-b x^2}{a^3 \left (a-b+2 b x^2-b x^4\right )}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{a^3 d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{a^2 d}-\frac{\operatorname{Subst}\left (\int \frac{b-b x^2}{\left (a-b+2 b x^2-b x^4\right )^3} \, dx,x,\cos (c+d x)\right )}{a d}\\ &=-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{-4 a b^2+2 a b^2 x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{8 a^3 (a-b) b d}+\frac{\operatorname{Subst}\left (\int \frac{-12 a b^2+10 a b^2 x^2}{\left (a-b+2 b x^2-b x^4\right )^2} \, dx,x,\cos (c+d x)\right )}{16 a^2 (a-b) b d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^3 d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{2 a^3 d}\\ &=-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 \sqrt{\sqrt{a}-\sqrt{b}} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 \sqrt{\sqrt{a}+\sqrt{b}} d}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac{b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{4 a (13 a-b) b^3-4 a b^3 (5 a+b) x^2}{a-b+2 b x^2-b x^4} \, dx,x,\cos (c+d x)\right )}{128 a^3 (a-b)^2 b^2 d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right ) d}+\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{8 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right ) d}\\ &=-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right )^{3/2} d}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 \sqrt{\sqrt{a}-\sqrt{b}} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right )^{3/2} d}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 \sqrt{\sqrt{a}+\sqrt{b}} d}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac{b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}+\frac{\left (\left (5 \sqrt{a}-2 \sqrt{b}\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{-\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right )^2 d}+\frac{\left (\left (5 \sqrt{a}+2 \sqrt{b}\right ) b\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a} \sqrt{b}+b-b x^2} \, dx,x,\cos (c+d x)\right )}{64 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right )^2 d}\\ &=-\frac{\left (5 \sqrt{a}-2 \sqrt{b}\right ) \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{64 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right )^{5/2} d}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{8 a^{5/2} \left (\sqrt{a}-\sqrt{b}\right )^{3/2} d}-\frac{\sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}-\sqrt{b}}}\right )}{2 a^3 \sqrt{\sqrt{a}-\sqrt{b}} d}-\frac{\tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{8 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right )^{3/2} d}+\frac{\sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{2 a^3 \sqrt{\sqrt{a}+\sqrt{b}} d}+\frac{\left (5 \sqrt{a}+2 \sqrt{b}\right ) \sqrt [4]{b} \tanh ^{-1}\left (\frac{\sqrt [4]{b} \cos (c+d x)}{\sqrt{\sqrt{a}+\sqrt{b}}}\right )}{64 a^{5/2} \left (\sqrt{a}+\sqrt{b}\right )^{5/2} d}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{8 a (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )^2}-\frac{b \cos (c+d x) \left (2-\cos ^2(c+d x)\right )}{4 a^2 (a-b) d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}-\frac{b \cos (c+d x) \left (11 a+b-(5 a+b) \cos ^2(c+d x)\right )}{32 a^2 (a-b)^2 d \left (a-b+2 b \cos ^2(c+d x)-b \cos ^4(c+d x)\right )}\\ \end{align*}

Mathematica [C]  time = 4.27989, size = 920, normalized size = 1.49 \[ \frac{\frac{512 b (\cos (3 (c+d x))-5 \cos (c+d x)) a^2}{(a-b) (-8 a+3 b-4 b \cos (2 (c+d x))+b \cos (4 (c+d x)))^2}+\frac{32 b \cos (c+d x) (-41 a+23 b+(13 a-7 b) \cos (2 (c+d x))) a}{(a-b)^2 (8 a-3 b+4 b \cos (2 (c+d x))-b \cos (4 (c+d x)))}-256 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+256 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-\frac{i b \text{RootSum}\left [b \text{$\#$1}^8-4 b \text{$\#$1}^6-16 a \text{$\#$1}^4+6 b \text{$\#$1}^4-4 b \text{$\#$1}^2+b\& ,\frac{90 a^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^6+64 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^6-142 a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^6-45 i a^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^6-32 i b^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^6+71 i a b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^6-398 a^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^4-192 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^4+506 a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^4+199 i a^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^4+96 i b^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^4-253 i a b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^4+398 a^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^2+192 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^2-506 a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right ) \text{$\#$1}^2-199 i a^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^2-96 i b^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^2+253 i a b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right ) \text{$\#$1}^2-90 a^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )-64 b^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+142 a b \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+45 i a^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right )+32 i b^2 \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right )-71 i a b \log \left (\text{$\#$1}^2-2 \cos (c+d x) \text{$\#$1}+1\right )}{b \text{$\#$1}^7-3 b \text{$\#$1}^5-8 a \text{$\#$1}^3+3 b \text{$\#$1}^3-b \text{$\#$1}}\& \right ]}{(a-b)^2}}{256 a^3 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((32*a*b*Cos[c + d*x]*(-41*a + 23*b + (13*a - 7*b)*Cos[2*(c + d*x)]))/((a - b)^2*(8*a - 3*b + 4*b*Cos[2*(c + d
*x)] - b*Cos[4*(c + d*x)])) + (512*a^2*b*(-5*Cos[c + d*x] + Cos[3*(c + d*x)]))/((a - b)*(-8*a + 3*b - 4*b*Cos[
2*(c + d*x)] + b*Cos[4*(c + d*x)])^2) - 256*Log[Cos[(c + d*x)/2]] + 256*Log[Sin[(c + d*x)/2]] - (I*b*RootSum[b
- 4*b*#1^2 - 16*a*#1^4 + 6*b*#1^4 - 4*b*#1^6 + b*#1^8 & , (-90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] +
142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] - 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)] + (45*I)*a
^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] - (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + (32*I)*b^2*Log[1 - 2*Cos
[c + d*x]*#1 + #1^2] + 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - 506*a*b*ArcTan[Sin[c + d*x]/(Co
s[c + d*x] - #1)]*#1^2 + 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - (199*I)*a^2*Log[1 - 2*Cos[c +
d*x]*#1 + #1^2]*#1^2 + (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^2 - (96*I)*b^2*Log[1 - 2*Cos[c + d*x]
*#1 + #1^2]*#1^2 - 398*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + 506*a*b*ArcTan[Sin[c + d*x]/(Cos[c
+ d*x] - #1)]*#1^4 - 192*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^4 + (199*I)*a^2*Log[1 - 2*Cos[c + d*x
]*#1 + #1^2]*#1^4 - (253*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^4 + (96*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1
+ #1^2]*#1^4 + 90*a^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - 142*a*b*ArcTan[Sin[c + d*x]/(Cos[c + d*x
] - #1)]*#1^6 + 64*b^2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^6 - (45*I)*a^2*Log[1 - 2*Cos[c + d*x]*#1 +
#1^2]*#1^6 + (71*I)*a*b*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*#1^6 - (32*I)*b^2*Log[1 - 2*Cos[c + d*x]*#1 + #1^2]*
#1^6)/(-(b*#1) - 8*a*#1^3 + 3*b*#1^3 - 3*b*#1^5 + b*#1^7) & ])/(a - b)^2)/(256*a^3*d)

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Maple [B]  time = 0.182, size = 1139, normalized size = 1.9 \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)/(a-b*sin(d*x+c)^4)^3,x)

[Out]

1/2/d/a^3*ln(-1+cos(d*x+c))-1/2/d/a^3*ln(1+cos(d*x+c))-13/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b^2/a/(
a^2-2*a*b+b^2)*cos(d*x+c)^7+7/32/d*b^3/a^2/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^
7+53/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^5*b^2-29/32/d*b^3/a^2/(b*cos(d*
x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^5+17/32/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2*b/
(a^2-2*a*b+b^2)*cos(d*x+c)^3-39/16/d/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/a/(a^2-2*a*b+b^2)*cos(d*x+c)^3*b^
2+37/32/d*b^3/a^2/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a^2-2*a*b+b^2)*cos(d*x+c)^3-35/32/d/(b*cos(d*x+c)^4
-2*b*cos(d*x+c)^2-a+b)^2/(a-b)/a*b*cos(d*x+c)+15/32/d*b^2/a^2/(b*cos(d*x+c)^4-2*b*cos(d*x+c)^2-a+b)^2/(a-b)*co
s(d*x+c)-45/64/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b+
71/64/d*b^2/a^2/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/2/d
*b^3/a^3/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))-1/4/d/a/(a^2
-2*a*b+b^2)/(a*b)^(1/2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))*b^2+5/32/d*b^
3/a^2/(a*b)^(1/2)/(a^2-2*a*b+b^2)/(((a*b)^(1/2)-b)*b)^(1/2)*arctan(cos(d*x+c)*b/(((a*b)^(1/2)-b)*b)^(1/2))+45/
64/d/a/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b-71/64/d*b^2
/a^2/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))+1/2/d*b^3/a^3/(
a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))-1/4/d/a/(a^2-2*a*b+b^
2)/(a*b)^(1/2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))*b^2+5/32/d*b^3/a^2/(a
*b)^(1/2)/(a^2-2*a*b+b^2)/(((a*b)^(1/2)+b)*b)^(1/2)*arctanh(cos(d*x+c)*b/(((a*b)^(1/2)+b)*b)^(1/2))

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

[Out]

Timed out

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Fricas [B]  time = 25.2048, size = 12519, normalized size = 20.29 \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)/(a-b*sin(d*x+c)^4)^3,x, algorithm="fricas")

[Out]

-1/128*(4*(13*a^2*b^2 - 7*a*b^3)*cos(d*x + c)^7 - 4*(53*a^2*b^2 - 29*a*b^3)*cos(d*x + c)^5 - 4*(17*a^3*b - 78*
a^2*b^2 + 37*a*b^3)*cos(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(-(
3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*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((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 399
47241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*
b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a
^11*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((4100625*a^6*b -
14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d
*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 -
22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b
^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a
^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d
^4)) - (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 122
88*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 + (a^11 - 5*a^10*b + 1
0*a^9*b^2 - 10*a^8*b^3 + 5*a^7*b^4 - a^6*b^5)*d^2*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 -
49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21
- 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13
*b^8 - 10*a^12*b^9 + a^11*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)
)) + ((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(-(3465*a^4*b - 9306*a^3*b^2 + 1004
5*a^2*b^3 - 5084*a*b^4 + 1024*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
((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6
+ 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^
4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*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((4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4
*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b
+ 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a
^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*
a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^1
6*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) + (123525*a^9*b - 450359*a^
8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b
- 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*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((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*
b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*
a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*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))) + ((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(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*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((4100625*a^8*b - 19010700*a^7*b
^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b
^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6
- 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*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(-(4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504
896*a^2*b^5 - 3044864*a*b^6 + 393216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3
+ 995*a^12*b^4 - 540*a^11*b^5 + 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971
086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 14745
6*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14
*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*b^10)*d^4)) - (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957
*a^6*b^4 + 327152*a^5*b^5 - 95104*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3
- 5084*a*b^4 + 1024*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((4100625
*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 740147
2*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a
^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*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))) - ((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(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*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((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 4967945
2*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^
20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 -
10*a^12*b^9 + a^11*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(-
(4100625*a^6*b - 14762250*a^5*b^2 + 23227949*a^4*b^3 - 20354340*a^3*b^4 + 10504896*a^2*b^5 - 3044864*a*b^6 + 3
93216*b^7)*cos(d*x + c) - ((45*a^16 - 280*a^15*b + 747*a^14*b^2 - 1110*a^13*b^3 + 995*a^12*b^4 - 540*a^11*b^5
+ 165*a^10*b^6 - 22*a^9*b^7)*d^3*sqrt((4100625*a^8*b - 19010700*a^7*b^2 + 39971086*a^6*b^3 - 49679452*a^5*b^4
+ 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147456*b^9)/((a^21 - 10*a^20*b + 45*
a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^14*b^7 + 45*a^13*b^8 - 10*a^12*b^
9 + a^11*b^10)*d^4)) + (123525*a^9*b - 450359*a^8*b^2 + 715183*a^7*b^3 - 630957*a^6*b^4 + 327152*a^5*b^5 - 951
04*a^4*b^6 + 12288*a^3*b^7)*d)*sqrt(-(3465*a^4*b - 9306*a^3*b^2 + 10045*a^2*b^3 - 5084*a*b^4 + 1024*b^5 - (a^1
1 - 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((4100625*a^8*b - 19010700*a^7*b^2 + 399
71086*a^6*b^3 - 49679452*a^5*b^4 + 39947241*a^4*b^5 - 21320992*a^3*b^6 + 7401472*a^2*b^7 - 1536000*a*b^8 + 147
456*b^9)/((a^21 - 10*a^20*b + 45*a^19*b^2 - 120*a^18*b^3 + 210*a^17*b^4 - 252*a^16*b^5 + 210*a^15*b^6 - 120*a^
14*b^7 + 45*a^13*b^8 - 10*a^12*b^9 + a^11*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))) + 20*(7*a^3*b - 10*a^2*b^2 + 3*a*b^3)*cos(d*x + c) + 64*((a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x
+ c)^8 - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^6 - 2*(a^3*b - 5*a^2*b^2 + 7*a*b^3 - 3*b^4)*cos(d*x + c)^4 +
a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*(a^3*b - 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(d*x + c)^2)*log(1/2*cos
(d*x + c) + 1/2) - 64*((a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^8 - 4*(a^2*b^2 - 2*a*b^3 + b^4)*cos(d*x + c)^6 -
2*(a^3*b - 5*a^2*b^2 + 7*a*b^3 - 3*b^4)*cos(d*x + c)^4 + a^4 - 4*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4 + 4*(a^3*b
- 3*a^2*b^2 + 3*a*b^3 - b^4)*cos(d*x + c)^2)*log(-1/2*cos(d*x + c) + 1/2))/((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)

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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)/(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: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError