\(\int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx\) [188]

Optimal result

Integrand size = 23, antiderivative size = 344 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]

[Out]

-3/8*arctanh(cos(d*x+c))/a/d+b*cot(d*x+c)/a^2/d-3/8*cot(d*x+c)*csc(d*x+c)/a/d-1/4*cot(d*x+c)*csc(d*x+c)^3/a/d-
2/3*b^(5/3)*arctan((b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)-b^(2/3))^(1/2))/a^(7/3)/d/(a^(2/3)-b^(2/3))^(
1/2)+2/3*(-1)^(1/3)*b^(5/3)*arctan(((-1)^(2/3)*b^(1/3)+a^(1/3)*tan(1/2*d*x+1/2*c))/(a^(2/3)+(-1)^(1/3)*b^(2/3)
)^(1/2))/a^(7/3)/d/(a^(2/3)+(-1)^(1/3)*b^(2/3))^(1/2)+2/3*(-1)^(2/3)*b^(5/3)*arctan(((-1)^(1/3)*b^(1/3)-a^(1/3
)*tan(1/2*d*x+1/2*c))/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2))/a^(7/3)/d/(a^(2/3)-(-1)^(2/3)*b^(2/3))^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3299, 3852, 8, 3853, 3855, 2739, 632, 210} \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d} \]

[In]

Int[Csc[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]

[Out]

(2*(-1)^(2/3)*b^(5/3)*ArcTan[((-1)^(1/3)*b^(1/3) - a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)
]])/(3*a^(7/3)*Sqrt[a^(2/3) - (-1)^(2/3)*b^(2/3)]*d) - (2*b^(5/3)*ArcTan[(b^(1/3) + a^(1/3)*Tan[(c + d*x)/2])/
Sqrt[a^(2/3) - b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) - b^(2/3)]*d) + (2*(-1)^(1/3)*b^(5/3)*ArcTan[((-1)^(2/3)*b^(
1/3) + a^(1/3)*Tan[(c + d*x)/2])/Sqrt[a^(2/3) + (-1)^(1/3)*b^(2/3)]])/(3*a^(7/3)*Sqrt[a^(2/3) + (-1)^(1/3)*b^(
2/3)]*d) - (3*ArcTanh[Cos[c + d*x]])/(8*a*d) + (b*Cot[c + d*x])/(a^2*d) - (3*Cot[c + d*x]*Csc[c + d*x])/(8*a*d
) - (Cot[c + d*x]*Csc[c + d*x]^3)/(4*a*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 210

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

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 3299

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.), x_Symbol] :> Int[ExpandTr
ig[sin[e + f*x]^m*(a + b*sin[e + f*x]^n)^p, x], x] /; FreeQ[{a, b, e, f}, x] && IntegersQ[m, p] && (EqQ[n, 4]
|| GtQ[p, 0] || (EqQ[p, -1] && IntegerQ[n]))

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {b \csc ^2(c+d x)}{a^2}+\frac {\csc ^5(c+d x)}{a}+\frac {b^2 \sin (c+d x)}{a^2 \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx \\ & = \frac {\int \csc ^5(c+d x) \, dx}{a}-\frac {b \int \csc ^2(c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {\sin (c+d x)}{a+b \sin ^3(c+d x)} \, dx}{a^2} \\ & = -\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc ^3(c+d x) \, dx}{4 a}+\frac {b^2 \int \left (-\frac {1}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)\right )}-\frac {(-1)^{2/3}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}+\frac {\sqrt [3]{-1}}{3 \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{a^2}+\frac {b \text {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^2 d} \\ & = \frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {3 \int \csc (c+d x) \, dx}{8 a}-\frac {b^{5/3} \int \frac {1}{\sqrt [3]{a}+\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{7/3}}+\frac {\left (\sqrt [3]{-1} b^{5/3}\right ) \int \frac {1}{\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{7/3}}-\frac {\left ((-1)^{2/3} b^{5/3}\right ) \int \frac {1}{\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{7/3}} \\ & = -\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {\left (2 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}+\frac {\left (2 \sqrt [3]{-1} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}+2 (-1)^{2/3} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}-\frac {\left (2 (-1)^{2/3} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [3]{a}-2 \sqrt [3]{-1} \sqrt [3]{b} x+\sqrt [3]{a} x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d} \\ & = -\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}+\frac {\left (4 b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-b^{2/3}\right )-x^2} \, dx,x,2 \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}-\frac {\left (4 \sqrt [3]{-1} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}+\sqrt [3]{-1} b^{2/3}\right )-x^2} \, dx,x,2 (-1)^{2/3} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d}+\frac {\left (4 (-1)^{2/3} b^{5/3}\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^{2/3}-(-1)^{2/3} b^{2/3}\right )-x^2} \, dx,x,-2 \sqrt [3]{-1} \sqrt [3]{b}+2 \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{3 a^{7/3} d} \\ & = \frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 1.23 (sec) , antiderivative size = 290, normalized size of antiderivative = 0.84 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {-64 b^2 \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (32 b \cot \left (\frac {1}{2} (c+d x)\right )-6 a \csc ^2\left (\frac {1}{2} (c+d x)\right )-a \csc ^4\left (\frac {1}{2} (c+d x)\right )-24 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \sec ^4\left (\frac {1}{2} (c+d x)\right )-32 b \tan \left (\frac {1}{2} (c+d x)\right )\right )}{192 a^2 d} \]

[In]

Integrate[Csc[c + d*x]^5/(a + b*Sin[c + d*x]^3),x]

[Out]

(-64*b^2*RootSum[-b + 3*b*#1^2 - (8*I)*a*#1^3 - 3*b*#1^4 + b*#1^6 & , (-2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] -
#1)] + I*Log[1 - 2*Cos[c + d*x]*#1 + #1^2] + 2*ArcTan[Sin[c + d*x]/(Cos[c + d*x] - #1)]*#1^2 - I*Log[1 - 2*Cos
[c + d*x]*#1 + #1^2]*#1^2)/(b - (4*I)*a*#1 - 2*b*#1^2 + b*#1^4) & ] + 3*(32*b*Cot[(c + d*x)/2] - 6*a*Csc[(c +
d*x)/2]^2 - a*Csc[(c + d*x)/2]^4 - 24*a*Log[Cos[(c + d*x)/2]] + 24*a*Log[Sin[(c + d*x)/2]] + 6*a*Sec[(c + d*x)
/2]^2 + a*Sec[(c + d*x)/2]^4 - 32*b*Tan[(c + d*x)/2]))/(192*a^2*d)

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 2.04 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.57

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{4}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -8 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{2}}+\frac {2 b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(196\)
default	\(\frac {\frac {\frac {\left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a}{4}+2 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -8 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 a^{2}}+\frac {2 b^{2} \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{3}+\textit {\_R} \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a^{2}}-\frac {1}{64 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {3 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}+\frac {b}{2 a^{2} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{d}\)	\(196\)
risch	\(\frac {3 a \,{\mathrm e}^{7 i \left (d x +c \right )}-11 a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i b \,{\mathrm e}^{6 i \left (d x +c \right )}-11 a \,{\mathrm e}^{3 i \left (d x +c \right )}-24 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a \,{\mathrm e}^{i \left (d x +c \right )}+24 i b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i b}{4 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+32 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (782757789696 a^{16} d^{6}-782757789696 a^{14} b^{2} d^{6}\right ) \textit {\_Z}^{6}-254803968 a^{10} b^{4} d^{4} \textit {\_Z}^{4}-b^{10}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {8153726976 i d^{5} a^{15}}{a^{2} b^{8}+b^{10}}-\frac {8153726976 i d^{5} a^{13} b^{2}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{5}+\left (\frac {84934656 i d^{4} a^{13} b}{a^{2} b^{8}+b^{10}}-\frac {84934656 i d^{4} a^{11} b^{3}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{4}+\left (-\frac {1769472 i d^{3} a^{9} b^{4}}{a^{2} b^{8}+b^{10}}-\frac {884736 i d^{3} a^{7} b^{6}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{3}+\left (-\frac {18432 i d^{2} a^{7} b^{5}}{a^{2} b^{8}+b^{10}}-\frac {9216 i d^{2} a^{5} b^{7}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{2}+\left (-\frac {96 i d \,a^{5} b^{6}}{a^{2} b^{8}+b^{10}}-\frac {192 i d \,a^{3} b^{8}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R} -\frac {i b^{9} a}{a^{2} b^{8}+b^{10}}\right )\right )-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}\)	\(503\)

[In]

int(csc(d*x+c)^5/(a+b*sin(d*x+c)^3),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/16/a^2*(1/4*tan(1/2*d*x+1/2*c)^4*a+2*tan(1/2*d*x+1/2*c)^2*a-8*b*tan(1/2*d*x+1/2*c))+2/3*b^2/a^2*sum((_R
^3+_R)/(_R^5*a+2*_R^3*a+4*_R^2*b+_R*a)*ln(tan(1/2*d*x+1/2*c)-_R),_R=RootOf(_Z^6*a+3*_Z^4*a+8*_Z^3*b+3*_Z^2*a+a
))-1/64/a/tan(1/2*d*x+1/2*c)^4-1/8/a/tan(1/2*d*x+1/2*c)^2+3/8/a*ln(tan(1/2*d*x+1/2*c))+1/2/a^2*b/tan(1/2*d*x+1
/2*c))

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 131.01 (sec) , antiderivative size = 21564, normalized size of antiderivative = 62.69 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]

[In]

integrate(csc(d*x+c)**5/(a+b*sin(d*x+c)**3),x)

[Out]

Integral(csc(c + d*x)**5/(a + b*sin(c + d*x)**3), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [F]

\[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]

[In]

integrate(csc(d*x+c)^5/(a+b*sin(d*x+c)^3),x, algorithm="giac")

[Out]

integrate(csc(d*x + c)^5/(b*sin(d*x + c)^3 + a), x)

Mupad [B] (verification not implemented)

Time = 15.22 (sec) , antiderivative size = 1560, normalized size of antiderivative = 4.53 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(sin(c + d*x)^5*(a + b*sin(c + d*x)^3)),x)

[Out]

symsum(log((262144*b^14*tan(c/2 + (d*x)/2) - 3072*a^3*b^11 + 155648*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243
*a^10*b^4*z^4 - b^10, z, k)*a^4*b^11 - 393216*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10,
z, k)^2*a^5*b^11 + 774144*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^2*a^7*b^9 - 20
64384*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^8*b^9 + 2073600*root(729*a^14*
b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^10*b^7 - 9510912*root(729*a^14*b^2*z^6 - 729*a^16*
z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^4*a^11*b^7 + 2737152*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*
z^4 - b^10, z, k)^4*a^13*b^5 + 10616832*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^
5*a^12*b^7 - 10285056*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^5*a^14*b^5 + 37324
80*root(729*a^14*b^2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^5*a^16*b^3 + 7962624*root(729*a^14*b^
2*z^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^6*a^15*b^5 - 9953280*root(729*a^14*b^2*z^6 - 729*a^16*z^
6 - 243*a^10*b^4*z^4 - b^10, z, k)^6*a^17*b^3 + 98304*a^2*b^12*tan(c/2 + (d*x)/2) - 262144*root(729*a^14*b^2*z
^6 - 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)*a^3*b^12*tan(c/2 + (d*x)/2) + 165888*root(729*a^14*b^2*z^6
- 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)*a^5*b^10*tan(c/2 + (d*x)/2) - 1327104*root(729*a^14*b^2*z^6 -
729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^2*a^6*b^10*tan(c/2 + (d*x)/2) + 165888*root(729*a^14*b^2*z^6 - 7
29*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^2*a^8*b^8*tan(c/2 + (d*x)/2) + 2359296*root(729*a^14*b^2*z^6 - 72
9*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^7*b^10*tan(c/2 + (d*x)/2) - 7077888*root(729*a^14*b^2*z^6 - 72
9*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^9*b^8*tan(c/2 + (d*x)/2) + 82944*root(729*a^14*b^2*z^6 - 729*a
^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^3*a^11*b^6*tan(c/2 + (d*x)/2) + 81395712*root(729*a^14*b^2*z^6 - 729*
a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^4*a^10*b^8*tan(c/2 + (d*x)/2) - 1714176*root(729*a^14*b^2*z^6 - 729*
a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^4*a^12*b^6*tan(c/2 + (d*x)/2) + 27869184*root(729*a^14*b^2*z^6 - 729
*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^5*a^13*b^6*tan(c/2 + (d*x)/2) - 23141376*root(729*a^14*b^2*z^6 - 72
9*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^5*a^15*b^4*tan(c/2 + (d*x)/2) - 254803968*root(729*a^14*b^2*z^6 -
729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^6*a^14*b^6*tan(c/2 + (d*x)/2) + 252813312*root(729*a^14*b^2*z^6
- 729*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k)^6*a^16*b^4*tan(c/2 + (d*x)/2))/a^9)*root(729*a^14*b^2*z^6 - 72
9*a^16*z^6 - 243*a^10*b^4*z^4 - b^10, z, k), k, 1, 6)/d - cot(c/2 + (d*x)/2)^2/(8*a*d) - cot(c/2 + (d*x)/2)^4/
(64*a*d) + tan(c/2 + (d*x)/2)^2/(8*a*d) + tan(c/2 + (d*x)/2)^4/(64*a*d) + (3*log(tan(c/2 + (d*x)/2)))/(8*a*d)
+ (b*cot(c/2 + (d*x)/2))/(2*a^2*d) - (b*tan(c/2 + (d*x)/2))/(2*a^2*d)