Optimal. Leaf size=141 \[ \frac{\left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^4 d}+\frac{\left (a^2+b^2\right )^2}{a b^4 d (a \cot (c+d x)+b)}-\frac{4 a \left (a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}-\frac{4 a \left (a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d} \]
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Rubi [A] time = 0.149899, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {3088, 894} \[ \frac{\left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^4 d}+\frac{\left (a^2+b^2\right )^2}{a b^4 d (a \cot (c+d x)+b)}-\frac{4 a \left (a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}-\frac{4 a \left (a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 894
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^4 (b+a x)^2} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^2 x^4}-\frac{2 a}{b^3 x^3}+\frac{3 a^2+2 b^2}{b^4 x^2}-\frac{4 a \left (a^2+b^2\right )}{b^5 x}+\frac{\left (a^2+b^2\right )^2}{b^4 (b+a x)^2}+\frac{4 a^2 \left (a^2+b^2\right )}{b^5 (b+a x)}\right ) \, dx,x,\cot (c+d x)\right )}{d}\\ &=\frac{\left (a^2+b^2\right )^2}{a b^4 d (b+a \cot (c+d x))}-\frac{4 a \left (a^2+b^2\right ) \log (b+a \cot (c+d x))}{b^5 d}-\frac{4 a \left (a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac{\left (3 a^2+2 b^2\right ) \tan (c+d x)}{b^4 d}-\frac{a \tan ^2(c+d x)}{b^3 d}+\frac{\tan ^3(c+d x)}{3 b^2 d}\\ \end{align*}
Mathematica [A] time = 2.81513, size = 122, normalized size = 0.87 \[ \frac{4 b \left (2 a^2+b^2\right ) \tan (c+d x)+\frac{b^4 \sec ^4(c+d x)-4 \left (a^2+b^2\right ) \left (3 a^2 \log (a+b \tan (c+d x))+a^2+3 a b \tan (c+d x) \log (a+b \tan (c+d x))+b^2\right )}{a+b \tan (c+d x)}-2 a b^2 \tan ^2(c+d x)}{3 b^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.216, size = 174, normalized size = 1.2 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,{b}^{2}d}}-{\frac{a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{{b}^{3}d}}+3\,{\frac{{a}^{2}\tan \left ( dx+c \right ) }{d{b}^{4}}}+2\,{\frac{\tan \left ( dx+c \right ) }{{b}^{2}d}}-4\,{\frac{{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{d{b}^{5}}}-4\,{\frac{a\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{b}^{3}d}}-{\frac{{a}^{4}}{d{b}^{5} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-2\,{\frac{{a}^{2}}{{b}^{3}d \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{1}{db \left ( a+b\tan \left ( dx+c \right ) \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13593, size = 155, normalized size = 1.1 \begin{align*} -\frac{\frac{3 \,{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )}}{b^{6} \tan \left (d x + c\right ) + a b^{5}} - \frac{b^{2} \tan \left (d x + c\right )^{3} - 3 \, a b \tan \left (d x + c\right )^{2} + 3 \,{\left (3 \, a^{2} + 2 \, b^{2}\right )} \tan \left (d x + c\right )}{b^{4}} + \frac{12 \,{\left (a^{3} + a b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{b^{5}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.58217, size = 656, normalized size = 4.65 \begin{align*} -\frac{4 \,{\left (3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{4} - b^{4} - 2 \,{\left (3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (d x + c\right )^{2} + 6 \,{\left ({\left (a^{4} + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )\right )} \log \left (2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) +{\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}\right ) - 6 \,{\left ({\left (a^{4} + a^{2} b^{2}\right )} \cos \left (d x + c\right )^{4} +{\left (a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) + 2 \,{\left (a b^{3} \cos \left (d x + c\right ) - 2 \,{\left (3 \, a^{3} b + 2 \, a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{3 \,{\left (a b^{5} d \cos \left (d x + c\right )^{4} + b^{6} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right )\right )}} \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 (c + d x \right )}}{\left (a \cos{\left (c + d x \right )} + b \sin{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14558, size = 201, normalized size = 1.43 \begin{align*} -\frac{\frac{12 \,{\left (a^{3} + a b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} - \frac{b^{4} \tan \left (d x + c\right )^{3} - 3 \, a b^{3} \tan \left (d x + c\right )^{2} + 9 \, a^{2} b^{2} \tan \left (d x + c\right ) + 6 \, b^{4} \tan \left (d x + c\right )}{b^{6}} - \frac{3 \,{\left (4 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) + 3 \, a^{4} + 2 \, a^{2} b^{2} - b^{4}\right )}}{{\left (b \tan \left (d x + c\right ) + a\right )} b^{5}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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