Optimal. Leaf size=223 \[ \frac{\left (-3 a^2 b \left (c^2-d^2\right )+2 a^3 c d-6 a b^2 c d+b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f \left (c^2+d^2\right )^2}+\frac{x \left (6 a^2 b c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (c^2+d^2\right )^2}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}+\frac{\left (2 a c d+b \left (c^2+3 d^2\right )\right ) (b c-a d)^2 \log (c+d \tan (e+f x))}{d^2 f \left (c^2+d^2\right )^2} \]
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Rubi [A] time = 0.363817, antiderivative size = 223, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3565, 3626, 3617, 31, 3475} \[ \frac{\left (-3 a^2 b \left (c^2-d^2\right )+2 a^3 c d-6 a b^2 c d+b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{f \left (c^2+d^2\right )^2}+\frac{x \left (6 a^2 b c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (c^2+d^2\right )^2}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d f \left (c^2+d^2\right ) (c+d \tan (e+f x))}+\frac{\left (2 a c d+b \left (c^2+3 d^2\right )\right ) (b c-a d)^2 \log (c+d \tan (e+f x))}{d^2 f \left (c^2+d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3626
Rule 3617
Rule 31
Rule 3475
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^2} \, dx &=-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\int \frac{b^3 c^2+a^3 c d-3 a b^2 c d+3 a^2 b d^2+d \left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \tan (e+f x)+b^3 \left (c^2+d^2\right ) \tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d \left (c^2+d^2\right )}\\ &=\frac{\left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (c^2+d^2\right )^2}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}-\frac{\left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \int \tan (e+f x) \, dx}{\left (c^2+d^2\right )^2}+\frac{\left ((b c-a d)^2 \left (2 a c d+b \left (c^2+3 d^2\right )\right )\right ) \int \frac{1+\tan ^2(e+f x)}{c+d \tan (e+f x)} \, dx}{d \left (c^2+d^2\right )^2}\\ &=\frac{\left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (c^2+d^2\right )^2}+\frac{\left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^2 f}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}+\frac{\left ((b c-a d)^2 \left (2 a c d+b \left (c^2+3 d^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{c+x} \, dx,x,d \tan (e+f x)\right )}{d^2 \left (c^2+d^2\right )^2 f}\\ &=\frac{\left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (c^2+d^2\right )^2}+\frac{\left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \log (\cos (e+f x))}{\left (c^2+d^2\right )^2 f}+\frac{(b c-a d)^2 \left (2 a c d+b \left (c^2+3 d^2\right )\right ) \log (c+d \tan (e+f x))}{d^2 \left (c^2+d^2\right )^2 f}-\frac{(b c-a d)^2 (a+b \tan (e+f x))}{d \left (c^2+d^2\right ) f (c+d \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 4.53075, size = 538, normalized size = 2.41 \[ \frac{\cos (e+f x) (a+b \tan (e+f x))^3 (c \cos (e+f x)+d \sin (e+f x)) \left (c^2 \cos (e+f x) \left (2 (c+i d)^2 (e+f x) \left (-3 i a^2 b d^2+a^3 d^2-3 a b^2 d^2+b^3 c (2 d+i c)\right )+(b c-a d)^2 \left (2 a c d+b \left (c^2+3 d^2\right )\right ) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )-2 b^3 \left (c^2+d^2\right )^2 \log (\cos (e+f x))\right )+d \sin (e+f x) \left (2 (c+i d) \left (3 a^2 b c d^2 (d (e+f x+i)-i c (e+f x-i))+a^3 d^2 \left (c^2 (e+f x)+c d (i e+i f x+1)-i d^2\right )+3 a b^2 c d \left (c^2-c d (e+f x+i)-i d^2 (e+f x)\right )+b^3 c^2 \left (i c^2 (e+f x+i)+c d (e+f x+i)+2 i d^2 (e+f x)\right )\right )+c (b c-a d)^2 \left (2 a c d+b \left (c^2+3 d^2\right )\right ) \log \left ((c \cos (e+f x)+d \sin (e+f x))^2\right )-2 b^3 c \left (c^2+d^2\right )^2 \log (\cos (e+f x))\right )-2 i c (b c-a d)^2 \left (2 a c d+b \left (c^2+3 d^2\right )\right ) \tan ^{-1}(\tan (e+f x)) (c \cos (e+f x)+d \sin (e+f x))\right )}{2 c d^2 f (c-i d)^2 (c+i d)^2 (c+d \tan (e+f x))^2 (a \cos (e+f x)+b \sin (e+f x))^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.038, size = 671, normalized size = 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 = 1.79262, size = 419, normalized size = 1.88 \begin{align*} \frac{\frac{2 \,{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c d -{\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )}{\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac{2 \,{\left (b^{3} c^{4} + 3 \, a^{2} b d^{4} - 3 \,{\left (a^{2} b - b^{3}\right )} c^{2} d^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{4} d^{2} + 2 \, c^{2} d^{4} + d^{6}} + \frac{{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} - 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c d -{\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac{2 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )}}{c^{3} d^{2} + c d^{4} +{\left (c^{2} d^{3} + d^{5}\right )} \tan \left (f x + e\right )}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.00547, size = 1025, normalized size = 4.6 \begin{align*} \frac{2 \, b^{3} c^{3} d^{2} - 6 \, a b^{2} c^{2} d^{3} + 6 \, a^{2} b c d^{4} - 2 \, a^{3} d^{5} + 2 \,{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{3} d^{2} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c^{2} d^{3} -{\left (a^{3} - 3 \, a b^{2}\right )} c d^{4}\right )} f x +{\left (b^{3} c^{5} + 3 \, a^{2} b c d^{4} - 3 \,{\left (a^{2} b - b^{3}\right )} c^{3} d^{2} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{3} +{\left (b^{3} c^{4} d + 3 \, a^{2} b d^{5} - 3 \,{\left (a^{2} b - b^{3}\right )} c^{2} d^{3} + 2 \,{\left (a^{3} - 3 \, a b^{2}\right )} c d^{4}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac{d^{2} \tan \left (f x + e\right )^{2} + 2 \, c d \tan \left (f x + e\right ) + c^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) -{\left (b^{3} c^{5} + 2 \, b^{3} c^{3} d^{2} + b^{3} c d^{4} +{\left (b^{3} c^{4} d + 2 \, b^{3} c^{2} d^{3} + b^{3} d^{5}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac{1}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \,{\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4} -{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} d^{3} + 2 \,{\left (3 \, a^{2} b - b^{3}\right )} c d^{4} -{\left (a^{3} - 3 \, a b^{2}\right )} d^{5}\right )} f x\right )} \tan \left (f x + e\right )}{2 \,{\left ({\left (c^{4} d^{3} + 2 \, c^{2} d^{5} + d^{7}\right )} f \tan \left (f x + e\right ) +{\left (c^{5} d^{2} + 2 \, c^{3} d^{4} + c d^{6}\right )} f\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 [B] time = 1.89217, size = 603, normalized size = 2.7 \begin{align*} \frac{\frac{2 \,{\left (a^{3} c^{2} - 3 \, a b^{2} c^{2} + 6 \, a^{2} b c d - 2 \, b^{3} c d - a^{3} d^{2} + 3 \, a b^{2} d^{2}\right )}{\left (f x + e\right )}}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac{{\left (3 \, a^{2} b c^{2} - b^{3} c^{2} - 2 \, a^{3} c d + 6 \, a b^{2} c d - 3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{4} + 2 \, c^{2} d^{2} + d^{4}} + \frac{2 \,{\left (b^{3} c^{4} - 3 \, a^{2} b c^{2} d^{2} + 3 \, b^{3} c^{2} d^{2} + 2 \, a^{3} c d^{3} - 6 \, a b^{2} c d^{3} + 3 \, a^{2} b d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{4} d^{2} + 2 \, c^{2} d^{4} + d^{6}} - \frac{2 \,{\left (b^{3} c^{4} \tan \left (f x + e\right ) - 3 \, a^{2} b c^{2} d^{2} \tan \left (f x + e\right ) + 3 \, b^{3} c^{2} d^{2} \tan \left (f x + e\right ) + 2 \, a^{3} c d^{3} \tan \left (f x + e\right ) - 6 \, a b^{2} c d^{3} \tan \left (f x + e\right ) + 3 \, a^{2} b d^{4} \tan \left (f x + e\right ) + 3 \, a b^{2} c^{4} - 6 \, a^{2} b c^{3} d + 2 \, b^{3} c^{3} d + 3 \, a^{3} c^{2} d^{2} - 3 \, a b^{2} c^{2} d^{2} + a^{3} d^{4}\right )}}{{\left (c^{4} d + 2 \, c^{2} d^{3} + d^{5}\right )}{\left (d \tan \left (f x + e\right ) + c\right )}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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