Integrand size = 22, antiderivative size = 115 \[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2} \]
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Time = 0.19 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {4494, 4269, 3800, 2221, 2317, 2438} \[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}+\frac {3 i d (c+d x)^2}{2 b^2} \]
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Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4269
Rule 4494
Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {(3 d) \int (c+d x)^2 \sec ^2(a+b x) \, dx}{2 b} \\ & = \frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {\left (3 d^2\right ) \int (c+d x) \tan (a+b x) \, dx}{b^2} \\ & = \frac {3 i d (c+d x)^2}{2 b^2}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}-\frac {\left (6 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)}{1+e^{2 i (a+b x)}} \, dx}{b^2} \\ & = \frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}+\frac {\left (3 d^3\right ) \int \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = \frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2}-\frac {\left (3 i d^3\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{2 b^4} \\ & = \frac {3 i d (c+d x)^2}{2 b^2}-\frac {3 d^2 (c+d x) \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {3 i d^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{2 b^4}+\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 d (c+d x)^2 \tan (a+b x)}{2 b^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(286\) vs. \(2(115)=230\).
Time = 6.40 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.49 \[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\frac {(c+d x)^3 \sec ^2(a+b x)}{2 b}-\frac {3 c d^2 \sec (a) (\cos (a) \log (\cos (a) \cos (b x)-\sin (a) \sin (b x))+b x \sin (a))}{b^3 \left (\cos ^2(a)+\sin ^2(a)\right )}-\frac {3 d^3 \csc (a) \left (b^2 e^{-i \arctan (\cot (a))} x^2-\frac {\cot (a) \left (i b x (-\pi -2 \arctan (\cot (a)))-\pi \log \left (1+e^{-2 i b x}\right )-2 (b x-\arctan (\cot (a))) \log \left (1-e^{2 i (b x-\arctan (\cot (a)))}\right )+\pi \log (\cos (b x))-2 \arctan (\cot (a)) \log (\sin (b x-\arctan (\cot (a))))+i \operatorname {PolyLog}\left (2,e^{2 i (b x-\arctan (\cot (a)))}\right )\right )}{\sqrt {1+\cot ^2(a)}}\right ) \sec (a)}{2 b^4 \sqrt {\csc ^2(a) \left (\cos ^2(a)+\sin ^2(a)\right )}}-\frac {3 \sec (a) \sec (a+b x) \left (c^2 d \sin (b x)+2 c d^2 x \sin (b x)+d^3 x^2 \sin (b x)\right )}{2 b^2} \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 300 vs. \(2 (101 ) = 202\).
Time = 3.18 (sec) , antiderivative size = 301, normalized size of antiderivative = 2.62
method | result | size |
risch | \(\frac {2 b \,d^{3} x^{3} {\mathrm e}^{2 i \left (x b +a \right )}-3 i d^{3} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}+6 b c \,d^{2} x^{2} {\mathrm e}^{2 i \left (x b +a \right )}-6 i c \,d^{2} x \,{\mathrm e}^{2 i \left (x b +a \right )}+6 b \,c^{2} d x \,{\mathrm e}^{2 i \left (x b +a \right )}-3 i c^{2} d \,{\mathrm e}^{2 i \left (x b +a \right )}-3 i d^{3} x^{2}+2 b \,c^{3} {\mathrm e}^{2 i \left (x b +a \right )}-6 i c \,d^{2} x -3 i c^{2} d}{b^{2} \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )^{2}}-\frac {3 d^{2} c \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right )}{b^{3}}+\frac {6 d^{2} c \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{3}}+\frac {3 i d^{3} x^{2}}{b^{2}}+\frac {6 i d^{3} x a}{b^{3}}+\frac {3 i d^{3} a^{2}}{b^{4}}-\frac {3 d^{3} \ln \left ({\mathrm e}^{2 i \left (x b +a \right )}+1\right ) x}{b^{3}}+\frac {3 i d^{3} \operatorname {polylog}\left (2, -{\mathrm e}^{2 i \left (x b +a \right )}\right )}{2 b^{4}}-\frac {6 d^{3} a \ln \left ({\mathrm e}^{i \left (x b +a \right )}\right )}{b^{4}}\) | \(301\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (98) = 196\).
Time = 0.29 (sec) , antiderivative size = 540, normalized size of antiderivative = 4.70 \[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\frac {b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3} - 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) + 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) + 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right )\right ) - 3 i \, d^{3} \cos \left (b x + a\right )^{2} {\rm Li}_2\left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right )\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) + \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b d^{3} x + a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-i \, \cos \left (b x + a\right ) - \sin \left (b x + a\right ) + 1\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) + i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b c d^{2} - a d^{3}\right )} \cos \left (b x + a\right )^{2} \log \left (-\cos \left (b x + a\right ) - i \, \sin \left (b x + a\right ) + i\right ) - 3 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} c^{2} d\right )} \cos \left (b x + a\right ) \sin \left (b x + a\right )}{2 \, b^{4} \cos \left (b x + a\right )^{2}} \]
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\[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\int \left (c + d x\right )^{3} \tan {\left (a + b x \right )} \sec ^{2}{\left (a + b x \right )}\, dx \]
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (98) = 196\).
Time = 0.44 (sec) , antiderivative size = 668, normalized size of antiderivative = 5.81 \[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=-\frac {6 \, b^{2} c^{2} d + 6 \, {\left (b d^{3} x + b c d^{2} + {\left (b d^{3} x + b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right ) + 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (-i \, b d^{3} x - i \, b c d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (-i \, b d^{3} x - i \, b c d^{2}\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \arctan \left (\sin \left (2 \, b x + 2 \, a\right ), \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (b^{2} d^{3} x^{2} + 2 \, b^{2} c d^{2} x\right )} \cos \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (-2 i \, b^{3} d^{3} x^{3} - 2 i \, b^{3} c^{3} - 3 \, b^{2} c^{2} d + 3 \, {\left (-2 i \, b^{3} c d^{2} + b^{2} d^{3}\right )} x^{2} + 6 \, {\left (-i \, b^{3} c^{2} d + b^{2} c d^{2}\right )} x\right )} \cos \left (2 \, b x + 2 \, a\right ) - 3 \, {\left (d^{3} \cos \left (4 \, b x + 4 \, a\right ) + 2 \, d^{3} \cos \left (2 \, b x + 2 \, a\right ) + i \, d^{3} \sin \left (4 \, b x + 4 \, a\right ) + 2 i \, d^{3} \sin \left (2 \, b x + 2 \, a\right ) + d^{3}\right )} {\rm Li}_2\left (-e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 3 \, {\left (i \, b d^{3} x + i \, b c d^{2} + {\left (i \, b d^{3} x + i \, b c d^{2}\right )} \cos \left (4 \, b x + 4 \, a\right ) + 2 \, {\left (i \, b d^{3} x + i \, b c d^{2}\right )} \cos \left (2 \, b x + 2 \, a\right ) - {\left (b d^{3} x + b c d^{2}\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \sin \left (2 \, b x + 2 \, a\right )\right )} \log \left (\cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) - 6 \, {\left (i \, b^{2} d^{3} x^{2} + 2 i \, b^{2} c d^{2} x\right )} \sin \left (4 \, b x + 4 \, a\right ) - 2 \, {\left (2 \, b^{3} d^{3} x^{3} + 2 \, b^{3} c^{3} - 3 i \, b^{2} c^{2} d + 3 \, {\left (2 \, b^{3} c d^{2} + i \, b^{2} d^{3}\right )} x^{2} + 6 \, {\left (b^{3} c^{2} d + i \, b^{2} c d^{2}\right )} x\right )} \sin \left (2 \, b x + 2 \, a\right )}{-2 i \, b^{4} \cos \left (4 \, b x + 4 \, a\right ) - 4 i \, b^{4} \cos \left (2 \, b x + 2 \, a\right ) + 2 \, b^{4} \sin \left (4 \, b x + 4 \, a\right ) + 4 \, b^{4} \sin \left (2 \, b x + 2 \, a\right ) - 2 i \, b^{4}} \]
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\[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \sec \left (b x + a\right )^{2} \tan \left (b x + a\right ) \,d x } \]
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Timed out. \[ \int (c+d x)^3 \sec ^2(a+b x) \tan (a+b x) \, dx=\int \frac {\mathrm {tan}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^3}{{\cos \left (a+b\,x\right )}^2} \,d x \]
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