Optimal. Leaf size=221 \[ -\frac{\left (a^2 \left (-\left (3 c^2 d-d^3\right )\right )+2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3}-\frac{x \left (a^2 \left (-\left (c^3-3 c d^2\right )\right )-2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )}{\left (c^2+d^2\right )^3}-\frac{(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}+\frac{2 (a c+b d) (b c-a d)}{f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))} \]
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Rubi [A] time = 0.406876, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3542, 3529, 3531, 3530} \[ -\frac{\left (a^2 \left (-\left (3 c^2 d-d^3\right )\right )+2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{f \left (c^2+d^2\right )^3}-\frac{x \left (a^2 \left (-\left (c^3-3 c d^2\right )\right )-2 a b d \left (3 c^2-d^2\right )+b^2 c \left (c^2-3 d^2\right )\right )}{\left (c^2+d^2\right )^3}-\frac{(b c-a d)^2}{2 d f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}+\frac{2 (a c+b d) (b c-a d)}{f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))} \]
Antiderivative was successfully verified.
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Rule 3542
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{(a+b \tan (e+f x))^2}{(c+d \tan (e+f x))^3} \, dx &=-\frac{(b c-a d)^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{\int \frac{a^2 c-b^2 c+2 a b d+\left (2 a b c-a^2 d+b^2 d\right ) \tan (e+f x)}{(c+d \tan (e+f x))^2} \, dx}{c^2+d^2}\\ &=-\frac{(b c-a d)^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{2 (b c-a d) (a c+b d)}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac{\int \frac{(a c+b c-a d+b d) (a c-b c+a d+b d)+2 (b c-a d) (a c+b d) \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{\left (c^2+d^2\right )^2}\\ &=-\frac{\left (b^2 c \left (c^2-3 d^2\right )-2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac{(b c-a d)^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{2 (b c-a d) (a c+b d)}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac{\left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \int \frac{d-c \tan (e+f x)}{c+d \tan (e+f x)} \, dx}{\left (c^2+d^2\right )^3}\\ &=-\frac{\left (b^2 c \left (c^2-3 d^2\right )-2 a b d \left (3 c^2-d^2\right )-a^2 \left (c^3-3 c d^2\right )\right ) x}{\left (c^2+d^2\right )^3}-\frac{\left (2 a b c \left (c^2-3 d^2\right )+b^2 d \left (3 c^2-d^2\right )-a^2 \left (3 c^2 d-d^3\right )\right ) \log (c \cos (e+f x)+d \sin (e+f x))}{\left (c^2+d^2\right )^3 f}-\frac{(b c-a d)^2}{2 d \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac{2 (b c-a d) (a c+b d)}{\left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 3.8701, size = 292, normalized size = 1.32 \[ -\frac{(b c-a d) \left (\frac{2 \left (a^2 \left (3 c^2 d-d^3\right )-2 a b c \left (c^2-3 d^2\right )+b^2 d \left (d^2-3 c^2\right )\right ) \log (c+d \tan (e+f x))}{\left (c^2+d^2\right )^2}-\frac{2 (a d-b c) \left (2 a c d+b \left (d^2-c^2\right )\right )}{d \left (c^2+d^2\right ) (c+d \tan (e+f x))}+\frac{(a+i b)^2 (d+i c)^3 \log (-\tan (e+f x)+i)}{\left (c^2+d^2\right )^2}+\frac{i (a-i b)^2 (c+i d) \log (\tan (e+f x)+i)}{(c-i d)^2}\right )+\frac{d^2 (a+b \tan (e+f x))^3}{(c+d \tan (e+f x))^2}-\frac{b d (a+b \tan (e+f x))^2}{c+d \tan (e+f x)}}{2 f \left (c^2+d^2\right ) (a d-b c)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.04, size = 753, normalized size = 3.4 \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 = 1.74511, size = 562, normalized size = 2.54 \begin{align*} \frac{\frac{2 \,{\left (6 \, a b c^{2} d - 2 \, a b d^{3} +{\left (a^{2} - b^{2}\right )} c^{3} - 3 \,{\left (a^{2} - b^{2}\right )} c d^{2}\right )}{\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac{2 \,{\left (2 \, a b c^{3} - 6 \, a b c d^{2} - 3 \,{\left (a^{2} - b^{2}\right )} c^{2} d +{\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (d \tan \left (f x + e\right ) + c\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac{{\left (2 \, a b c^{3} - 6 \, a b c d^{2} - 3 \,{\left (a^{2} - b^{2}\right )} c^{2} d +{\left (a^{2} - b^{2}\right )} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac{b^{2} c^{4} - 6 \, a b c^{3} d + 2 \, a b c d^{3} + a^{2} d^{4} +{\left (5 \, a^{2} - 3 \, b^{2}\right )} c^{2} d^{2} - 4 \,{\left (a b c^{2} d^{2} - a b d^{4} -{\left (a^{2} - b^{2}\right )} c d^{3}\right )} \tan \left (f x + e\right )}{c^{6} d + 2 \, c^{4} d^{3} + c^{2} d^{5} +{\left (c^{4} d^{3} + 2 \, c^{2} d^{5} + d^{7}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (c^{5} d^{2} + 2 \, c^{3} d^{4} + c d^{6}\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 = 1.62974, size = 1389, normalized size = 6.29 \begin{align*} -\frac{3 \, b^{2} c^{4} d - 10 \, a b c^{3} d^{2} + 2 \, a b c d^{4} + a^{2} d^{5} +{\left (7 \, a^{2} - 3 \, b^{2}\right )} c^{2} d^{3} - 2 \,{\left (6 \, a b c^{4} d - 2 \, a b c^{2} d^{3} +{\left (a^{2} - b^{2}\right )} c^{5} - 3 \,{\left (a^{2} - b^{2}\right )} c^{3} d^{2}\right )} f x -{\left (b^{2} c^{4} d - 6 \, a b c^{3} d^{2} + 6 \, a b c d^{4} - a^{2} d^{5} + 5 \,{\left (a^{2} - b^{2}\right )} c^{2} d^{3} + 2 \,{\left (6 \, a b c^{2} d^{3} - 2 \, a b d^{5} +{\left (a^{2} - b^{2}\right )} c^{3} d^{2} - 3 \,{\left (a^{2} - b^{2}\right )} c d^{4}\right )} f x\right )} \tan \left (f x + e\right )^{2} +{\left (2 \, a b c^{5} - 6 \, a b c^{3} d^{2} - 3 \,{\left (a^{2} - b^{2}\right )} c^{4} d +{\left (a^{2} - b^{2}\right )} c^{2} d^{3} +{\left (2 \, a b c^{3} d^{2} - 6 \, a b c d^{4} - 3 \,{\left (a^{2} - b^{2}\right )} c^{2} d^{3} +{\left (a^{2} - b^{2}\right )} d^{5}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (2 \, a b c^{4} d - 6 \, a b c^{2} d^{3} - 3 \,{\left (a^{2} - b^{2}\right )} c^{3} d^{2} +{\left (a^{2} - 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 ) - 2 \,{\left (b^{2} c^{5} - 4 \, a b c^{4} d + 6 \, a b c^{2} d^{3} - 2 \, a b d^{5} + 3 \,{\left (a^{2} - b^{2}\right )} c^{3} d^{2} -{\left (3 \, a^{2} - 2 \, b^{2}\right )} c d^{4} + 2 \,{\left (6 \, a b c^{3} d^{2} - 2 \, a b c d^{4} +{\left (a^{2} - b^{2}\right )} c^{4} d - 3 \,{\left (a^{2} - b^{2}\right )} c^{2} d^{3}\right )} f x\right )} \tan \left (f x + e\right )}{2 \,{\left ({\left (c^{6} d^{2} + 3 \, c^{4} d^{4} + 3 \, c^{2} d^{6} + d^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \,{\left (c^{7} d + 3 \, c^{5} d^{3} + 3 \, c^{3} d^{5} + c d^{7}\right )} f \tan \left (f x + e\right ) +{\left (c^{8} + 3 \, c^{6} d^{2} + 3 \, c^{4} d^{4} + c^{2} 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 = 1.52094, size = 829, normalized size = 3.75 \begin{align*} \frac{\frac{2 \,{\left (a^{2} c^{3} - b^{2} c^{3} + 6 \, a b c^{2} d - 3 \, a^{2} c d^{2} + 3 \, b^{2} c d^{2} - 2 \, a b d^{3}\right )}{\left (f x + e\right )}}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} + \frac{{\left (2 \, a b c^{3} - 3 \, a^{2} c^{2} d + 3 \, b^{2} c^{2} d - 6 \, a b c d^{2} + a^{2} d^{3} - b^{2} d^{3}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{c^{6} + 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} + d^{6}} - \frac{2 \,{\left (2 \, a b c^{3} d - 3 \, a^{2} c^{2} d^{2} + 3 \, b^{2} c^{2} d^{2} - 6 \, a b c d^{3} + a^{2} d^{4} - b^{2} d^{4}\right )} \log \left ({\left | d \tan \left (f x + e\right ) + c \right |}\right )}{c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}} + \frac{6 \, a b c^{3} d^{3} \tan \left (f x + e\right )^{2} - 9 \, a^{2} c^{2} d^{4} \tan \left (f x + e\right )^{2} + 9 \, b^{2} c^{2} d^{4} \tan \left (f x + e\right )^{2} - 18 \, a b c d^{5} \tan \left (f x + e\right )^{2} + 3 \, a^{2} d^{6} \tan \left (f x + e\right )^{2} - 3 \, b^{2} d^{6} \tan \left (f x + e\right )^{2} + 16 \, a b c^{4} d^{2} \tan \left (f x + e\right ) - 22 \, a^{2} c^{3} d^{3} \tan \left (f x + e\right ) + 22 \, b^{2} c^{3} d^{3} \tan \left (f x + e\right ) - 36 \, a b c^{2} d^{4} \tan \left (f x + e\right ) + 2 \, a^{2} c d^{5} \tan \left (f x + e\right ) - 2 \, b^{2} c d^{5} \tan \left (f x + e\right ) - 4 \, a b d^{6} \tan \left (f x + e\right ) - b^{2} c^{6} + 12 \, a b c^{5} d - 14 \, a^{2} c^{4} d^{2} + 11 \, b^{2} c^{4} d^{2} - 14 \, a b c^{3} d^{3} - 3 \, a^{2} c^{2} d^{4} - 2 \, a b c d^{5} - a^{2} d^{6}}{{\left (c^{6} d + 3 \, c^{4} d^{3} + 3 \, c^{2} d^{5} + d^{7}\right )}{\left (d \tan \left (f x + e\right ) + c\right )}^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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