Optimal. Leaf size=73 \[ \frac{\left (a^2+b^2\right ) \tan (x)}{a^3}-\frac{b \left (a^2+b^2\right ) \log (\tan (x))}{a^4}-\frac{b \left (a^2+b^2\right ) \log (a+b \cot (x))}{a^4}-\frac{b \tan ^2(x)}{2 a^2}+\frac{\tan ^3(x)}{3 a} \]
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Rubi [A] time = 0.0980419, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3516, 894} \[ \frac{\left (a^2+b^2\right ) \tan (x)}{a^3}-\frac{b \left (a^2+b^2\right ) \log (\tan (x))}{a^4}-\frac{b \left (a^2+b^2\right ) \log (a+b \cot (x))}{a^4}-\frac{b \tan ^2(x)}{2 a^2}+\frac{\tan ^3(x)}{3 a} \]
Antiderivative was successfully verified.
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Rule 3516
Rule 894
Rubi steps
\begin{align*} \int \frac{\sec ^4(x)}{a+b \cot (x)} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{b^2+x^2}{x^4 (a+x)} \, dx,x,b \cot (x)\right )\right )\\ &=-\left (b \operatorname{Subst}\left (\int \left (\frac{b^2}{a x^4}-\frac{b^2}{a^2 x^3}+\frac{a^2+b^2}{a^3 x^2}+\frac{-a^2-b^2}{a^4 x}+\frac{a^2+b^2}{a^4 (a+x)}\right ) \, dx,x,b \cot (x)\right )\right )\\ &=-\frac{b \left (a^2+b^2\right ) \log (a+b \cot (x))}{a^4}-\frac{b \left (a^2+b^2\right ) \log (\tan (x))}{a^4}+\frac{\left (a^2+b^2\right ) \tan (x)}{a^3}-\frac{b \tan ^2(x)}{2 a^2}+\frac{\tan ^3(x)}{3 a}\\ \end{align*}
Mathematica [A] time = 0.267152, size = 66, normalized size = 0.9 \[ \frac{\left (4 a^3+6 a b^2\right ) \tan (x)+6 b \left (a^2+b^2\right ) (\log (\cos (x))-\log (a \sin (x)+b \cos (x)))+a^2 \sec ^2(x) (2 a \tan (x)-3 b)}{6 a^4} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 64, normalized size = 0.9 \begin{align*}{\frac{ \left ( \tan \left ( x \right ) \right ) ^{3}}{3\,a}}-{\frac{b \left ( \tan \left ( x \right ) \right ) ^{2}}{2\,{a}^{2}}}+{\frac{\tan \left ( x \right ) }{a}}+{\frac{{b}^{2}\tan \left ( x \right ) }{{a}^{3}}}-{\frac{b\ln \left ( a\tan \left ( x \right ) +b \right ) }{{a}^{2}}}-{\frac{{b}^{3}\ln \left ( a\tan \left ( x \right ) +b \right ) }{{a}^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.28219, size = 76, normalized size = 1.04 \begin{align*} \frac{2 \, a^{2} \tan \left (x\right )^{3} - 3 \, a b \tan \left (x\right )^{2} + 6 \,{\left (a^{2} + b^{2}\right )} \tan \left (x\right )}{6 \, a^{3}} - \frac{{\left (a^{2} b + b^{3}\right )} \log \left (a \tan \left (x\right ) + b\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.17184, size = 278, normalized size = 3.81 \begin{align*} -\frac{3 \,{\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{3} \log \left (2 \, a b \cos \left (x\right ) \sin \left (x\right ) -{\left (a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + a^{2}\right ) - 3 \,{\left (a^{2} b + b^{3}\right )} \cos \left (x\right )^{3} \log \left (\cos \left (x\right )^{2}\right ) + 3 \, a^{2} b \cos \left (x\right ) - 2 \,{\left (a^{3} +{\left (2 \, a^{3} + 3 \, a b^{2}\right )} \cos \left (x\right )^{2}\right )} \sin \left (x\right )}{6 \, a^{4} \cos \left (x\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec ^{4}{\left (x \right )}}{a + b \cot{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.2557, size = 81, normalized size = 1.11 \begin{align*} \frac{2 \, a^{2} \tan \left (x\right )^{3} - 3 \, a b \tan \left (x\right )^{2} + 6 \, a^{2} \tan \left (x\right ) + 6 \, b^{2} \tan \left (x\right )}{6 \, a^{3}} - \frac{{\left (a^{2} b + b^{3}\right )} \log \left ({\left | a \tan \left (x\right ) + b \right |}\right )}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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