Optimal. Leaf size=732 \[ -\frac {6 b f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^4 \sqrt {a^2-b^2}}+\frac {6 b f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a d^4 \sqrt {a^2-b^2}}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^3 \sqrt {a^2-b^2}}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a d^3 \sqrt {a^2-b^2}}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^2 \sqrt {a^2-b^2}}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a d^2 \sqrt {a^2-b^2}}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d \sqrt {a^2-b^2}}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.12, antiderivative size = 732, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 9, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {4535, 4183, 2531, 6609, 2282, 6589, 3323, 2264, 2190} \[ \frac {6 i b f^2 (e+f x) \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^3 \sqrt {a^2-b^2}}-\frac {6 i b f^2 (e+f x) \text {PolyLog}\left (3,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d^3 \sqrt {a^2-b^2}}+\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^2 \sqrt {a^2-b^2}}-\frac {3 b f (e+f x)^2 \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d^2 \sqrt {a^2-b^2}}-\frac {6 b f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d^4 \sqrt {a^2-b^2}}+\frac {6 b f^3 \text {PolyLog}\left (4,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d^4 \sqrt {a^2-b^2}}-\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {PolyLog}\left (3,e^{i (c+d x)}\right )}{a d^3}+\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {6 i f^3 \text {PolyLog}\left (4,-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {PolyLog}\left (4,e^{i (c+d x)}\right )}{a d^4}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a d \sqrt {a^2-b^2}}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a d \sqrt {a^2-b^2}}-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2190
Rule 2264
Rule 2282
Rule 2531
Rule 3323
Rule 4183
Rule 4535
Rule 6589
Rule 6609
Rubi steps
\begin {align*} \int \frac {(e+f x)^3 \csc (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x)^3 \csc (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x)^3}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {(2 b) \int \frac {e^{i (c+d x)} (e+f x)^3}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a}-\frac {(3 f) \int (e+f x)^2 \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {(3 f) \int (e+f x)^2 \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {\left (2 i b^2\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a \sqrt {a^2-b^2}}-\frac {\left (2 i b^2\right ) \int \frac {e^{i (c+d x)} (e+f x)^3}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a \sqrt {a^2-b^2}}-\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (-e^{i (c+d x)}\right ) \, dx}{a d^2}+\frac {\left (6 i f^2\right ) \int (e+f x) \text {Li}_2\left (e^{i (c+d x)}\right ) \, dx}{a d^2}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {(3 i b f) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a \sqrt {a^2-b^2} d}+\frac {(3 i b f) \int (e+f x)^2 \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a \sqrt {a^2-b^2} d}+\frac {\left (6 f^3\right ) \int \text {Li}_3\left (-e^{i (c+d x)}\right ) \, dx}{a d^3}-\frac {\left (6 f^3\right ) \int \text {Li}_3\left (e^{i (c+d x)}\right ) \, dx}{a d^3}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}-\frac {\left (6 b f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a \sqrt {a^2-b^2} d^2}+\frac {\left (6 b f^2\right ) \int (e+f x) \text {Li}_2\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a \sqrt {a^2-b^2} d^2}-\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}+\frac {\left (6 i f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3(x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^4}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {\left (6 i b f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a \sqrt {a^2-b^2} d^3}+\frac {\left (6 i b f^3\right ) \int \text {Li}_3\left (\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a \sqrt {a^2-b^2} d^3}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {\left (6 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a \sqrt {a^2-b^2} d^4}+\frac {\left (6 b f^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_3\left (\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a \sqrt {a^2-b^2} d^4}\\ &=-\frac {2 (e+f x)^3 \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}-\frac {i b (e+f x)^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d}+\frac {3 i f (e+f x)^2 \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {3 i f (e+f x)^2 \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {3 b f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^2}-\frac {6 f^2 (e+f x) \text {Li}_3\left (-e^{i (c+d x)}\right )}{a d^3}+\frac {6 f^2 (e+f x) \text {Li}_3\left (e^{i (c+d x)}\right )}{a d^3}+\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i b f^2 (e+f x) \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^3}-\frac {6 i f^3 \text {Li}_4\left (-e^{i (c+d x)}\right )}{a d^4}+\frac {6 i f^3 \text {Li}_4\left (e^{i (c+d x)}\right )}{a d^4}-\frac {6 b f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^4}+\frac {6 b f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a \sqrt {a^2-b^2} d^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.64, size = 894, normalized size = 1.22 \[ \frac {-2 d^3 \tanh ^{-1}(\cos (c+d x)+i \sin (c+d x)) (e+f x)^3+\frac {b \left (3 d^2 f \text {Li}_2\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right ) (e+f x)^2+i \left (2 i e^3 \tan ^{-1}\left (\frac {i a+b e^{i (c+d x)}}{\sqrt {a^2-b^2}}\right ) d^3+f^3 x^3 \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right ) d^3+3 e f^2 x^2 \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right ) d^3+3 e^2 f x \log \left (\frac {i e^{i (c+d x)} b}{\sqrt {a^2-b^2}-a}+1\right ) d^3-f^3 x^3 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^3-3 e f^2 x^2 \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^3-3 e^2 f x \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^3+3 i f (e+f x)^2 \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d^2+6 f^2 (e+f x) \text {Li}_3\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right ) d-6 e f^2 \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d-6 f^3 x \text {Li}_3\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) d+6 i f^3 \text {Li}_4\left (-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}-a}\right )-6 i f^3 \text {Li}_4\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )\right )\right )}{\sqrt {a^2-b^2}}+3 i f \left (-2 \text {Li}_4(-\cos (c+d x)-i \sin (c+d x)) f^2+2 i d (e+f x) \text {Li}_3(-\cos (c+d x)-i \sin (c+d x)) f+d^2 (e+f x)^2 \text {Li}_2(-\cos (c+d x)-i \sin (c+d x))\right )-3 i f \left (-2 \text {Li}_4(\cos (c+d x)+i \sin (c+d x)) f^2+2 i d (e+f x) \text {Li}_3(\cos (c+d x)+i \sin (c+d x)) f+d^2 (e+f x)^2 \text {Li}_2(\cos (c+d x)+i \sin (c+d x))\right )}{a d^4} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [C] time = 0.77, size = 3608, normalized size = 4.93 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.58, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{3} \csc \left (d x +c \right )}{a +b \sin \left (d x +c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (e + f x\right )^{3} \csc {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________