Optimal. Leaf size=175 \[ -\frac{b c-a d}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{a^2 (-d)+2 a b c+b^2 d}{f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}+\frac{\left (3 a^2 b c+a^3 (-d)+3 a b^2 d-b^3 c\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 b d+a^3 c-3 a b^2 c-b^3 d\right )}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.270096, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3531, 3530} \[ -\frac{b c-a d}{2 f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac{a^2 (-d)+2 a b c+b^2 d}{f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}+\frac{\left (3 a^2 b c+a^3 (-d)+3 a b^2 d-b^3 c\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 b d+a^3 c-3 a b^2 c-b^3 d\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{c+d \tan (e+f x)}{(a+b \tan (e+f x))^3} \, dx &=-\frac{b c-a d}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac{\int \frac{a c+b d-(b c-a d) \tan (e+f x)}{(a+b \tan (e+f x))^2} \, dx}{a^2+b^2}\\ &=-\frac{b c-a d}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{2 a b c-a^2 d+b^2 d}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\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)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac{\left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right ) x}{\left (a^2+b^2\right )^3}-\frac{b c-a d}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{2 a b c-a^2 d+b^2 d}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac{\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \int \frac{b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac{\left (a^3 c-3 a b^2 c+3 a^2 b d-b^3 d\right ) x}{\left (a^2+b^2\right )^3}+\frac{\left (3 a^2 b c-b^3 c-a^3 d+3 a b^2 d\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac{b c-a d}{2 \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac{2 a b c-a^2 d+b^2 d}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end{align*}
Mathematica [C] time = 3.95667, size = 243, normalized size = 1.39 \[ -\frac{(b c-a d) \left (\frac{b \left (\frac{\left (a^2+b^2\right ) \left (5 a^2+4 a b \tan (e+f x)+b^2\right )}{(a+b \tan (e+f x))^2}+\left (2 b^2-6 a^2\right ) \log (a+b \tan (e+f x))\right )}{\left (a^2+b^2\right )^3}+\frac{i \log (-\tan (e+f x)+i)}{(a+i b)^3}-\frac{\log (\tan (e+f x)+i)}{(b+i a)^3}\right )+d \left (\frac{2 b \left (\frac{a^2+b^2}{a+b \tan (e+f x)}-2 a \log (a+b \tan (e+f x))\right )}{\left (a^2+b^2\right )^2}+\frac{i \log (-\tan (e+f x)+i)}{(a+i b)^2}-\frac{i \log (\tan (e+f x)+i)}{(a-i b)^2}\right )}{2 b f} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.035, size = 483, normalized size = 2.8 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){a}^{3}d}{2\,f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){a}^{2}bc}{2\,f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ) a{b}^{2}d}{2\,f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( fx+e \right ) \right ) ^{2} \right ){b}^{3}c}{2\,f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){a}^{3}c}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){a}^{2}bd}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ) a{b}^{2}c}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\arctan \left ( \tan \left ( fx+e \right ) \right ){b}^{3}d}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{ad}{2\,f \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}}-{\frac{bc}{2\,f \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( fx+e \right ) \right ) ^{2}}}+{\frac{{a}^{2}d}{f \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( fx+e \right ) \right ) }}-2\,{\frac{abc}{f \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( fx+e \right ) \right ) }}-{\frac{{b}^{2}d}{f \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( fx+e \right ) \right ) }}-{\frac{\ln \left ( a+b\tan \left ( fx+e \right ) \right ){a}^{3}d}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{\ln \left ( a+b\tan \left ( fx+e \right ) \right ){a}^{2}bc}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{\ln \left ( a+b\tan \left ( fx+e \right ) \right ) a{b}^{2}d}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\ln \left ( a+b\tan \left ( fx+e \right ) \right ){b}^{3}c}{f \left ({a}^{2}+{b}^{2} \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.85684, size = 450, normalized size = 2.57 \begin{align*} \frac{\frac{2 \,{\left ({\left (a^{3} - 3 \, a b^{2}\right )} c +{\left (3 \, a^{2} b - b^{3}\right )} d\right )}{\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left ({\left (3 \, a^{2} b - b^{3}\right )} c -{\left (a^{3} - 3 \, a b^{2}\right )} d\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left ({\left (3 \, a^{2} b - b^{3}\right )} c -{\left (a^{3} - 3 \, a b^{2}\right )} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (5 \, a^{2} b + b^{3}\right )} c -{\left (3 \, a^{3} - a b^{2}\right )} d + 2 \,{\left (2 \, a b^{2} c -{\left (a^{2} b - b^{3}\right )} d\right )} \tan \left (f x + e\right )}{a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} +{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (f x + e\right )^{2} + 2 \,{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{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 = 1.64349, size = 1030, normalized size = 5.89 \begin{align*} \frac{2 \,{\left ({\left (a^{5} - 3 \, a^{3} b^{2}\right )} c +{\left (3 \, a^{4} b - a^{2} b^{3}\right )} d\right )} f x +{\left (2 \,{\left ({\left (a^{3} b^{2} - 3 \, a b^{4}\right )} c +{\left (3 \, a^{2} b^{3} - b^{5}\right )} d\right )} f x +{\left (5 \, a^{2} b^{3} - b^{5}\right )} c - 3 \,{\left (a^{3} b^{2} - a b^{4}\right )} d\right )} \tan \left (f x + e\right )^{2} -{\left (7 \, a^{2} b^{3} + b^{5}\right )} c +{\left (5 \, a^{3} b^{2} - a b^{4}\right )} d +{\left ({\left ({\left (3 \, a^{2} b^{3} - b^{5}\right )} c -{\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d\right )} \tan \left (f x + e\right )^{2} +{\left (3 \, a^{4} b - a^{2} b^{3}\right )} c -{\left (a^{5} - 3 \, a^{3} b^{2}\right )} d + 2 \,{\left ({\left (3 \, a^{3} b^{2} - a b^{4}\right )} c -{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d\right )} \tan \left (f x + e\right )\right )} \log \left (\frac{b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \,{\left (2 \,{\left ({\left (a^{4} b - 3 \, a^{2} b^{3}\right )} c +{\left (3 \, a^{3} b^{2} - a b^{4}\right )} d\right )} f x + 3 \,{\left (a^{3} b^{2} - a b^{4}\right )} c -{\left (2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}\right )} d\right )} \tan \left (f x + e\right )}{2 \,{\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} f \tan \left (f x + e\right ) +{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{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 = 1.42732, size = 575, normalized size = 3.29 \begin{align*} \frac{\frac{2 \,{\left (a^{3} c - 3 \, a b^{2} c + 3 \, a^{2} b d - b^{3} d\right )}{\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{2} b c - b^{3} c - a^{3} d + 3 \, a b^{2} d\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (3 \, a^{2} b^{2} c - b^{4} c - a^{3} b d + 3 \, a b^{3} d\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac{9 \, a^{2} b^{3} c \tan \left (f x + e\right )^{2} - 3 \, b^{5} c \tan \left (f x + e\right )^{2} - 3 \, a^{3} b^{2} d \tan \left (f x + e\right )^{2} + 9 \, a b^{4} d \tan \left (f x + e\right )^{2} + 22 \, a^{3} b^{2} c \tan \left (f x + e\right ) - 2 \, a b^{4} c \tan \left (f x + e\right ) - 8 \, a^{4} b d \tan \left (f x + e\right ) + 18 \, a^{2} b^{3} d \tan \left (f x + e\right ) + 2 \, b^{5} d \tan \left (f x + e\right ) + 14 \, a^{4} b c + 3 \, a^{2} b^{3} c + b^{5} c - 6 \, a^{5} d + 7 \, a^{3} b^{2} d + a b^{4} d}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (b \tan \left (f x + e\right ) + a\right )}^{2}}}{2 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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