Integrand size = 23, antiderivative size = 344 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}-b^{2/3}} d}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d} \]
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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} \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {2 (-1)^{2/3} b^{5/3} \arctan \left (\frac {\sqrt [3]{-1} \sqrt [3]{b}-\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {2 b^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}-b^{2/3}}}+\frac {2 \sqrt [3]{-1} b^{5/3} \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{7/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {b \cot (c+d x)}{a^2 d}-\frac {3 \text {arctanh}(\cos (c+d x))}{8 a d}-\frac {\cot (c+d x) \csc ^3(c+d x)}{4 a d}-\frac {3 \cot (c+d x) \csc (c+d x)}{8 a d} \]
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Rule 8
Rule 210
Rule 632
Rule 2739
Rule 3299
Rule 3852
Rule 3853
Rule 3855
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*}
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 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {-64 b^2 \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {-2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right )+i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right )+2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}^2-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}^2}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]+3 \left (32 b \cot \left (\frac {1}{2} (c+d x)\right )-6 a \csc ^2\left (\frac {1}{2} (c+d x)\right )-a \csc ^4\left (\frac {1}{2} (c+d x)\right )-24 a \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+24 a \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+6 a \sec ^2\left (\frac {1}{2} (c+d x)\right )+a \sec ^4\left (\frac {1}{2} (c+d x)\right )-32 b \tan \left (\frac {1}{2} (c+d x)\right )\right )}{192 a^2 d} \]
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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 | \(\frac {3 a \,{\mathrm e}^{7 i \left (d x +c \right )}-11 a \,{\mathrm e}^{5 i \left (d x +c \right )}+8 i b \,{\mathrm e}^{6 i \left (d x +c \right )}-11 a \,{\mathrm e}^{3 i \left (d x +c \right )}-24 i b \,{\mathrm e}^{4 i \left (d x +c \right )}+3 a \,{\mathrm e}^{i \left (d x +c \right )}+24 i b \,{\mathrm e}^{2 i \left (d x +c \right )}-8 i b}{4 a^{2} d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}+32 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (782757789696 a^{16} d^{6}-782757789696 a^{14} b^{2} d^{6}\right ) \textit {\_Z}^{6}-254803968 a^{10} b^{4} d^{4} \textit {\_Z}^{4}-b^{10}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (\frac {8153726976 i d^{5} a^{15}}{a^{2} b^{8}+b^{10}}-\frac {8153726976 i d^{5} a^{13} b^{2}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{5}+\left (\frac {84934656 i d^{4} a^{13} b}{a^{2} b^{8}+b^{10}}-\frac {84934656 i d^{4} a^{11} b^{3}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{4}+\left (-\frac {1769472 i d^{3} a^{9} b^{4}}{a^{2} b^{8}+b^{10}}-\frac {884736 i d^{3} a^{7} b^{6}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{3}+\left (-\frac {18432 i d^{2} a^{7} b^{5}}{a^{2} b^{8}+b^{10}}-\frac {9216 i d^{2} a^{5} b^{7}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R}^{2}+\left (-\frac {96 i d \,a^{5} b^{6}}{a^{2} b^{8}+b^{10}}-\frac {192 i d \,a^{3} b^{8}}{a^{2} b^{8}+b^{10}}\right ) \textit {\_R} -\frac {i b^{9} a}{a^{2} b^{8}+b^{10}}\right )\right )-\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{8 d a}+\frac {3 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{8 d a}\) | \(503\) |
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Result contains complex when optimal does not.
Time = 131.01 (sec) , antiderivative size = 21564, normalized size of antiderivative = 62.69 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\csc ^{5}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]
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Exception generated. \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\csc \left (d x + c\right )^{5}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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Time = 15.22 (sec) , antiderivative size = 1560, normalized size of antiderivative = 4.53 \[ \int \frac {\csc ^5(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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