Optimal. Leaf size=155 \[ -\frac{a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 d \left (a^2+b^2\right )}-\frac{2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^2}+\frac{2 a b \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]
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Rubi [A] time = 0.297089, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3565, 3647, 3626, 3617, 31, 3475} \[ -\frac{a^2 \tan ^2(c+d x)}{b d \left (a^2+b^2\right ) (a+b \tan (c+d x))}+\frac{\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 d \left (a^2+b^2\right )}-\frac{2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 d \left (a^2+b^2\right )^2}+\frac{2 a b \log (\cos (c+d x))}{d \left (a^2+b^2\right )^2}+\frac{x \left (a^2-b^2\right )}{\left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 3565
Rule 3647
Rule 3626
Rule 3617
Rule 31
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+b \tan (c+d x))^2} \, dx &=-\frac{a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{\int \frac{\tan (c+d x) \left (2 a^2-a b \tan (c+d x)+\left (2 a^2+b^2\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=\frac{\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac{a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}+\frac{\int \frac{-a \left (2 a^2+b^2\right )-b^3 \tan (c+d x)-2 a \left (a^2+b^2\right ) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^2 \left (a^2+b^2\right )}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac{\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac{a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{(2 a b) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^2}-\frac{\left (2 a^3 \left (a^2+2 b^2\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 )^2}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac{2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac{\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac{a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}-\frac{\left (2 a^3 \left (a^2+2 b^2\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 )^2 d}\\ &=\frac{\left (a^2-b^2\right ) x}{\left (a^2+b^2\right )^2}+\frac{2 a b \log (\cos (c+d x))}{\left (a^2+b^2\right )^2 d}-\frac{2 a^3 \left (a^2+2 b^2\right ) \log (a+b \tan (c+d x))}{b^3 \left (a^2+b^2\right )^2 d}+\frac{\left (2 a^2+b^2\right ) \tan (c+d x)}{b^2 \left (a^2+b^2\right ) d}-\frac{a^2 \tan ^2(c+d x)}{b \left (a^2+b^2\right ) d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 2.60801, size = 329, normalized size = 2.12 \[ \frac{2 i a^3 \left (a^2+2 b^2\right ) \tan ^{-1}(\tan (c+d x)) (a+b \tan (c+d x))+b^2 \left (a^2+b^2\right )^2 \tan ^2(c+d x)+a \left ((a+i b)^2 \left (-4 a^2 b-2 i a^3+2 i a b^2+b^3\right ) (c+d x)+2 a \left (a^2+b^2\right )^2 \log (\cos (c+d x))+a^3 \left (-\left (a^2+2 b^2\right )\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )\right )+b \tan (c+d x) \left (2 a \left (a^2+b^2\right )^2 \log (\cos (c+d x))-a^3 \left (a^2+2 b^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right )-4 i a^3 b^2 c+a^2 b^3 c-4 i a^3 b^2 d x+a^2 b^3 d x+3 a^3 b^2-2 i a^5 c-2 i a^5 d x+2 a^5+a b^4-b^5 c-b^5 d x\right )}{b^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.024, size = 183, normalized size = 1.2 \begin{align*}{\frac{\tan \left ( dx+c \right ) }{{b}^{2}d}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{ab\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-{\frac{{a}^{4}}{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) }}-2\,{\frac{{a}^{5}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{b}^{3} \left ({a}^{2}+{b}^{2} \right ) ^{2}}}-4\,{\frac{{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{bd \left ({a}^{2}+{b}^{2} \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.5861, size = 221, normalized size = 1.43 \begin{align*} -\frac{\frac{a^{4}}{a^{3} b^{3} + a b^{5} +{\left (a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )} + \frac{a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (a^{5} + 2 \, a^{3} b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} - \frac{\tan \left (d x + c\right )}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.36519, size = 617, normalized size = 3.98 \begin{align*} -\frac{a^{4} b^{2} -{\left (a^{3} b^{3} - a b^{5}\right )} d x -{\left (a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6}\right )} \tan \left (d x + c\right )^{2} +{\left (a^{6} + 2 \, a^{4} b^{2} +{\left (a^{5} b + 2 \, a^{3} b^{3}\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 (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4} +{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) -{\left (2 \, a^{5} b + 2 \, a^{3} b^{3} + a b^{5} +{\left (a^{2} b^{4} - b^{6}\right )} d x\right )} \tan \left (d x + c\right )}{{\left (a^{4} b^{4} + 2 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right ) +{\left (a^{5} b^{3} + 2 \, a^{3} b^{5} + a b^{7}\right )} d} \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 = 2.43756, size = 271, normalized size = 1.75 \begin{align*} -\frac{\frac{a b \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} - \frac{{\left (a^{2} - b^{2}\right )}{\left (d x + c\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac{2 \,{\left (a^{5} + 2 \, a^{3} b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}} - \frac{2 \, a^{5} b \tan \left (d x + c\right ) + 4 \, a^{3} b^{3} \tan \left (d x + c\right ) + a^{6} + 3 \, a^{4} b^{2}}{{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}} - \frac{\tan \left (d x + c\right )}{b^{2}}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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