Optimal. Leaf size=161 \[ -\frac{\left (a^2+b^2\right )^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac{\left (3 a^2-b^2\right ) \left (a^2+b^2\right )}{a^2 b^4 d (a \cot (c+d x)+b)}+\frac{2 \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac{2 \left (3 a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d} \]
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Rubi [A] time = 0.167825, antiderivative size = 161, 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} \[ -\frac{\left (a^2+b^2\right )^2}{2 a^2 b^3 d (a \cot (c+d x)+b)^2}-\frac{\left (3 a^2-b^2\right ) \left (a^2+b^2\right )}{a^2 b^4 d (a \cot (c+d x)+b)}+\frac{2 \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}+\frac{2 \left (3 a^2+b^2\right ) \log (a \cot (c+d x)+b)}{b^5 d}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d} \]
Antiderivative was successfully verified.
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Rule 3088
Rule 894
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^3} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2}{x^3 (b+a x)^3} \, dx,x,\cot (c+d x)\right )}{d}\\ &=-\frac{\operatorname{Subst}\left (\int \left (\frac{1}{b^3 x^3}-\frac{3 a}{b^4 x^2}+\frac{2 \left (3 a^2+b^2\right )}{b^5 x}-\frac{\left (a^2+b^2\right )^2}{a b^3 (b+a x)^3}+\frac{-3 a^4-2 a^2 b^2+b^4}{a b^4 (b+a x)^2}-\frac{2 a \left (3 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}{2 a^2 b^3 d (b+a \cot (c+d x))^2}-\frac{\left (3 a^2-b^2\right ) \left (a^2+b^2\right )}{a^2 b^4 d (b+a \cot (c+d x))}+\frac{2 \left (3 a^2+b^2\right ) \log (b+a \cot (c+d x))}{b^5 d}+\frac{2 \left (3 a^2+b^2\right ) \log (\tan (c+d x))}{b^5 d}-\frac{3 a \tan (c+d x)}{b^4 d}+\frac{\tan ^2(c+d x)}{2 b^3 d}\\ \end{align*}
Mathematica [A] time = 3.10517, size = 140, normalized size = 0.87 \[ \frac{-2 a \left (-\frac{a^2+b^2}{a+b \tan (c+d x)}-2 a \log (a+b \tan (c+d x))+b \tan (c+d x)\right )+2 \left (a^2+b^2\right ) \left (\frac{3 a^2+4 a b \tan (c+d x)-b^2}{2 (a+b \tan (c+d x))^2}+\log (a+b \tan (c+d x))\right )+\frac{b^4 \sec ^4(c+d x)}{2 (a+b \tan (c+d x))^2}}{b^5 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.267, size = 184, normalized size = 1.1 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{2\,{b}^{3}d}}-3\,{\frac{a\tan \left ( dx+c \right ) }{{b}^{4}d}}+6\,{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{5}}}+2\,{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{{b}^{3}d}}-{\frac{{a}^{4}}{2\,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 ) ^{2}}}-{\frac{1}{2\,db \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+4\,{\frac{{a}^{3}}{d{b}^{5} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+4\,{\frac{a}{{b}^{3}d \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 [B] time = 1.32987, size = 880, normalized size = 5.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.598297, size = 814, normalized size = 5.06 \begin{align*} \frac{24 \, a^{2} b^{2} \cos \left (d x + c\right )^{4} + b^{4} - 2 \,{\left (9 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \,{\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) +{\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\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 ) - 2 \,{\left ({\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) +{\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\cos \left (d x + c\right )^{2}\right ) - 4 \,{\left (a b^{3} \cos \left (d x + c\right ) + 3 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{2 \,{\left (2 \, a b^{6} d \cos \left (d x + c\right )^{3} \sin \left (d x + c\right ) + b^{7} d \cos \left (d x + c\right )^{2} +{\left (a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.343, size = 189, normalized size = 1.17 \begin{align*} \frac{\frac{4 \,{\left (3 \, a^{2} + b^{2}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{b^{5}} + \frac{b^{3} \tan \left (d x + c\right )^{2} - 6 \, a b^{2} \tan \left (d x + c\right )}{b^{6}} - \frac{18 \, a^{2} b^{2} \tan \left (d x + c\right )^{2} + 6 \, b^{4} \tan \left (d x + c\right )^{2} + 28 \, a^{3} b \tan \left (d x + c\right ) + 4 \, a b^{3} \tan \left (d x + c\right ) + 11 \, a^{4} + b^{4}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{2} b^{5}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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