Integrand size = 23, antiderivative size = 287 \[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {2 b \arctan \left (\frac {\sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}-b^{2/3}} d}-\frac {2 b \arctan \left (\frac {(-1)^{2/3} \sqrt [3]{b}+\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}+\frac {2 b \arctan \left (\frac {\sqrt [3]{-1} \left (\sqrt [3]{b}+(-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Time = 0.26 (sec) , antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3299, 3853, 3855, 3292, 2739, 632, 210} \[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=-\frac {2 b \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}}{\sqrt {a^{2/3}-b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {a^{2/3}-b^{2/3}}}-\frac {2 b \arctan \left (\frac {\sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+(-1)^{2/3} \sqrt [3]{b}}{\sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}}}+\frac {2 b \arctan \left (\frac {\sqrt [3]{-1} \left ((-1)^{2/3} \sqrt [3]{a} \tan \left (\frac {1}{2} (c+d x)\right )+\sqrt [3]{b}\right )}{\sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}\right )}{3 a^{5/3} d \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}}}-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \]
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Rule 210
Rule 632
Rule 2739
Rule 3292
Rule 3299
Rule 3853
Rule 3855
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\csc ^3(c+d x)}{a}-\frac {b}{a \left (a+b \sin ^3(c+d x)\right )}\right ) \, dx \\ & = \frac {\int \csc ^3(c+d x) \, dx}{a}-\frac {b \int \frac {1}{a+b \sin ^3(c+d x)} \, dx}{a} \\ & = -\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {\int \csc (c+d x) \, dx}{2 a}-\frac {b \int \left (-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)\right )}-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)\right )}-\frac {1}{3 a^{2/3} \left (-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)\right )}\right ) \, dx}{a} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {b \int \frac {1}{-\sqrt [3]{a}-\sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{5/3}}+\frac {b \int \frac {1}{-\sqrt [3]{a}+\sqrt [3]{-1} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{5/3}}+\frac {b \int \frac {1}{-\sqrt [3]{a}-(-1)^{2/3} \sqrt [3]{b} \sin (c+d x)} \, dx}{3 a^{5/3}} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}+\frac {(2 b) \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^{5/3} d}+\frac {(2 b) \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^{5/3} d}+\frac {(2 b) \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^{5/3} d} \\ & = -\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d}-\frac {(4 b) \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^{5/3} d}-\frac {(4 b) \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^{5/3} d}-\frac {(4 b) \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^{5/3} d} \\ & = \frac {2 b \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^{5/3} \sqrt {a^{2/3}-(-1)^{2/3} b^{2/3}} d}-\frac {2 b \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^{5/3} \sqrt {a^{2/3}-b^{2/3}} d}-\frac {2 b \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^{5/3} \sqrt {a^{2/3}+\sqrt [3]{-1} b^{2/3}} d}-\frac {\text {arctanh}(\cos (c+d x))}{2 a d}-\frac {\cot (c+d x) \csc (c+d x)}{2 a d} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.40 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.63 \[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\frac {16 i b \text {RootSum}\left [-b+3 b \text {$\#$1}^2-8 i a \text {$\#$1}^3-3 b \text {$\#$1}^4+b \text {$\#$1}^6\&,\frac {2 \arctan \left (\frac {\sin (c+d x)}{\cos (c+d x)-\text {$\#$1}}\right ) \text {$\#$1}-i \log \left (1-2 \cos (c+d x) \text {$\#$1}+\text {$\#$1}^2\right ) \text {$\#$1}}{b-4 i a \text {$\#$1}-2 b \text {$\#$1}^2+b \text {$\#$1}^4}\&\right ]-3 \left (\csc ^2\left (\frac {1}{2} (c+d x)\right )+4 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-4 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-\sec ^2\left (\frac {1}{2} (c+d x)\right )\right )}{24 a d} \]
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Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.49 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.47
method | result | size |
derivativedivides | \(\frac {\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(136\) |
default | \(\frac {\frac {\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}-\frac {b \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a \,\textit {\_Z}^{6}+3 a \,\textit {\_Z}^{4}+8 b \,\textit {\_Z}^{3}+3 a \,\textit {\_Z}^{2}+a \right )}{\sum }\frac {\left (\textit {\_R}^{4}+2 \textit {\_R}^{2}+1\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\textit {\_R} \right )}{\textit {\_R}^{5} a +2 \textit {\_R}^{3} a +4 \textit {\_R}^{2} b +\textit {\_R} a}\right )}{3 a}-\frac {1}{8 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 a}}{d}\) | \(136\) |
risch | \(\frac {{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{i \left (d x +c \right )}}{d a \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{2}}-8 i \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (191102976 a^{12} d^{6}-191102976 a^{10} b^{2} d^{6}\right ) \textit {\_Z}^{6}-995328 a^{8} b^{2} d^{4} \textit {\_Z}^{4}+1728 a^{4} b^{4} d^{2} \textit {\_Z}^{2}-b^{6}\right )}{\sum }\textit {\_R} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\left (-\frac {15925248 i d^{5} a^{11}}{b^{6}}+\frac {15925248 i d^{5} a^{9}}{b^{4}}\right ) \textit {\_R}^{5}+\left (-\frac {331776 i d^{4} a^{9}}{b^{5}}+\frac {331776 i d^{4} a^{7}}{b^{3}}\right ) \textit {\_R}^{4}+\left (\frac {69120 i d^{3} a^{7}}{b^{4}}+\frac {13824 i d^{3} a^{5}}{b^{2}}\right ) \textit {\_R}^{3}+\frac {1728 i d^{2} a^{5} \textit {\_R}^{2}}{b^{3}}-\frac {72 i d \,a^{3} \textit {\_R}}{b^{2}}-\frac {2 i a}{b}\right )\right )+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{2 d a}\) | \(279\) |
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Result contains complex when optimal does not.
Time = 14.83 (sec) , antiderivative size = 29431, normalized size of antiderivative = 102.55 \[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int \frac {\csc ^{3}{\left (c + d x \right )}}{a + b \sin ^{3}{\left (c + d x \right )}}\, dx \]
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\[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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\[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\int { \frac {\csc \left (d x + c\right )^{3}}{b \sin \left (d x + c\right )^{3} + a} \,d x } \]
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Time = 15.06 (sec) , antiderivative size = 1573, normalized size of antiderivative = 5.48 \[ \int \frac {\csc ^3(c+d x)}{a+b \sin ^3(c+d x)} \, dx=\text {Too large to display} \]
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