Optimal. Leaf size=250 \[ \frac{a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{a \left (a^2 b^3 B+3 a^3 b^2 C+a^5 C+6 a b^4 C-3 b^5 B\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right )}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.581399, antiderivative size = 250, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.175, Rules used = {3632, 3605, 3635, 3626, 3617, 31, 3475} \[ \frac{a (b B-a C) \tan ^2(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac{a^2 \left (a^3 (-C)-3 a b^2 C+2 b^3 B\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac{a \left (a^2 b^3 B+3 a^3 b^2 C+a^5 C+6 a b^4 C-3 b^5 B\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}-\frac{x \left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3632
Rule 3605
Rule 3635
Rule 3626
Rule 3617
Rule 31
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^2(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx &=\int \frac{\tan ^3(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=\frac{a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{\tan (c+d x) \left (-2 a (b B-a C)+2 b (b B-a C) \tan (c+d x)+2 \left (a^2+b^2\right ) C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac{a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{-2 a \left (2 b^3 B-a^3 C-3 a b^2 C\right )-2 b^2 \left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 C \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac{a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac{\left (a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right )\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )^3}\\ &=-\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\left (a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^3 \left (a^2+b^2\right )^3 d}\\ &=-\frac{\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (a^3 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a \left (a^2 b^3 B-3 b^5 B+a^5 C+3 a^3 b^2 C+6 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^3 d}+\frac{a (b B-a C) \tan ^2(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac{a^2 \left (2 b^3 B-a^3 C-3 a b^2 C\right )}{b^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 4.5326, size = 462, normalized size = 1.85 \[ \frac{\sec ^2(c+d x) (B+C \tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x)) \left (2 i a (c+d x) \left (a^2 b^3 B+3 a^3 b^2 C+a^5 C+6 a b^4 C-3 b^5 B\right ) (a \cos (c+d x)+b \sin (c+d x))^2-2 a b \left (a^2+b^2\right ) \left (a^3 C+4 a b^2 C-3 b^3 B\right ) \sin (c+d x) (a \cos (c+d x)+b \sin (c+d x))+2 b^3 (c+d x) \left (-3 a^2 b B+a^3 C-3 a b^2 C+b^3 B\right ) (a \cos (c+d x)+b \sin (c+d x))^2+a \left (a^2 b^3 B+3 a^3 b^2 C+a^5 C+6 a b^4 C-3 b^5 B\right ) (a \cos (c+d x)+b \sin (c+d x))^2 \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )-2 i a \left (a^2 b^3 B+3 a^3 b^2 C+a^5 C+6 a b^4 C-3 b^5 B\right ) \tan ^{-1}(\tan (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2+a^3 b^2 \left (a^2+b^2\right ) (b B-a C)-2 C \left (a^2+b^2\right )^3 \log (\cos (c+d x)) (a \cos (c+d x)+b \sin (c+d x))^2\right )}{2 b^3 d \left (a^2+b^2\right )^3 (a+b \tan (c+d x))^3 (B \cos (c+d x)+C \sin (c+d x))} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.054, size = 566, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.89238, size = 494, normalized size = 1.98 \begin{align*} \frac{\frac{2 \,{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (C a^{6} + 3 \, C a^{4} b^{2} + B a^{3} b^{3} + 6 \, C a^{2} b^{4} - 3 \, B a b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac{{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{3 \, C a^{6} - B a^{5} b + 7 \, C a^{4} b^{2} - 5 \, B a^{3} b^{3} + 2 \,{\left (2 \, C a^{5} b - B a^{4} b^{2} + 4 \, C a^{3} b^{3} - 3 \, B a^{2} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{3} + 2 \, a^{4} b^{5} + a^{2} b^{7} +{\left (a^{4} b^{5} + 2 \, a^{2} b^{7} + b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b^{4} + 2 \, a^{3} b^{6} + a b^{8}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.60578, size = 1432, normalized size = 5.73 \begin{align*} \frac{C a^{6} b^{2} + B a^{5} b^{3} + 7 \, C a^{4} b^{4} - 5 \, B a^{3} b^{5} + 2 \,{\left (C a^{5} b^{3} - 3 \, B a^{4} b^{4} - 3 \, C a^{3} b^{5} + B a^{2} b^{6}\right )} d x -{\left (3 \, C a^{6} b^{2} - B a^{5} b^{3} + 9 \, C a^{4} b^{4} - 7 \, B a^{3} b^{5} - 2 \,{\left (C a^{3} b^{5} - 3 \, B a^{2} b^{6} - 3 \, C a b^{7} + B b^{8}\right )} d x\right )} \tan \left (d x + c\right )^{2} +{\left (C a^{8} + 3 \, C a^{6} b^{2} + B a^{5} b^{3} + 6 \, C a^{4} b^{4} - 3 \, B a^{3} b^{5} +{\left (C a^{6} b^{2} + 3 \, C a^{4} b^{4} + B a^{3} b^{5} + 6 \, C a^{2} b^{6} - 3 \, B a b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (C a^{7} b + 3 \, C a^{5} b^{3} + B a^{4} b^{4} + 6 \, C a^{3} b^{5} - 3 \, B a^{2} b^{6}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) -{\left (C a^{8} + 3 \, C a^{6} b^{2} + 3 \, C a^{4} b^{4} + C a^{2} b^{6} +{\left (C a^{6} b^{2} + 3 \, C a^{4} b^{4} + 3 \, C a^{2} b^{6} + C b^{8}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (C a^{7} b + 3 \, C a^{5} b^{3} + 3 \, C a^{3} b^{5} + C a b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \,{\left (C a^{7} b + 3 \, C a^{5} b^{3} - 3 \, B a^{4} b^{4} - 4 \, C a^{3} b^{5} + 3 \, B a^{2} b^{6} - 2 \,{\left (C a^{4} b^{4} - 3 \, B a^{3} b^{5} - 3 \, C a^{2} b^{6} + B a b^{7}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{6} b^{5} + 3 \, a^{4} b^{7} + 3 \, a^{2} b^{9} + b^{11}\right )} d \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b^{4} + 3 \, a^{5} b^{6} + 3 \, a^{3} b^{8} + a b^{10}\right )} d \tan \left (d x + c\right ) +{\left (a^{8} b^{3} + 3 \, a^{6} b^{5} + 3 \, a^{4} b^{7} + a^{2} b^{9}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.93239, size = 618, normalized size = 2.47 \begin{align*} \frac{\frac{2 \,{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )}{\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (C a^{6} + 3 \, C a^{4} b^{2} + B a^{3} b^{3} + 6 \, C a^{2} b^{4} - 3 \, B a b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{3} + 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} + b^{9}} - \frac{3 \, C a^{6} b \tan \left (d x + c\right )^{2} + 9 \, C a^{4} b^{3} \tan \left (d x + c\right )^{2} + 3 \, B a^{3} b^{4} \tan \left (d x + c\right )^{2} + 18 \, C a^{2} b^{5} \tan \left (d x + c\right )^{2} - 9 \, B a b^{6} \tan \left (d x + c\right )^{2} + 2 \, C a^{7} \tan \left (d x + c\right ) + 2 \, B a^{6} b \tan \left (d x + c\right ) + 6 \, C a^{5} b^{2} \tan \left (d x + c\right ) + 14 \, B a^{4} b^{3} \tan \left (d x + c\right ) + 28 \, C a^{3} b^{4} \tan \left (d x + c\right ) - 12 \, B a^{2} b^{5} \tan \left (d x + c\right ) + B a^{7} - C a^{6} b + 9 \, B a^{5} b^{2} + 11 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}}{{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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