Optimal. Leaf size=88 \[ \frac{8 \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}+\frac{4 \tan (c+d x)}{15 a d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{\tan (c+d x)}{5 d \left (a \sec ^2(c+d x)\right )^{5/2}} \]
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Rubi [A] time = 0.0436301, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 192, 191} \[ \frac{8 \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}+\frac{4 \tan (c+d x)}{15 a d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{\tan (c+d x)}{5 d \left (a \sec ^2(c+d x)\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3657
Rule 4122
Rule 192
Rule 191
Rubi steps
\begin{align*} \int \frac{1}{\left (a+a \tan ^2(c+d x)\right )^{5/2}} \, dx &=\int \frac{1}{\left (a \sec ^2(c+d x)\right )^{5/2}} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{7/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\tan (c+d x)}{5 d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{4 \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\tan (c+d x)\right )}{5 d}\\ &=\frac{\tan (c+d x)}{5 d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{4 \tan (c+d x)}{15 a d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{8 \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{15 a d}\\ &=\frac{\tan (c+d x)}{5 d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{4 \tan (c+d x)}{15 a d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{8 \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0999582, size = 52, normalized size = 0.59 \[ \frac{\left (3 \sin ^4(c+d x)-10 \sin ^2(c+d x)+15\right ) \tan (c+d x)}{15 a^2 d \sqrt{a \sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.022, size = 88, normalized size = 1. \begin{align*}{\frac{a}{d} \left ({\frac{\tan \left ( dx+c \right ) }{5\,a} \left ( a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{5}{2}}}}+{\frac{4}{5\,a} \left ({\frac{\tan \left ( dx+c \right ) }{3\,a} \left ( a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{3}{2}}}}+{\frac{2\,\tan \left ( dx+c \right ) }{3\,{a}^{2}}{\frac{1}{\sqrt{a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.81389, size = 53, normalized size = 0.6 \begin{align*} \frac{3 \, \sin \left (5 \, d x + 5 \, c\right ) + 25 \, \sin \left (3 \, d x + 3 \, c\right ) + 150 \, \sin \left (d x + c\right )}{240 \, a^{\frac{5}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57921, size = 231, normalized size = 2.62 \begin{align*} \frac{{\left (8 \, \tan \left (d x + c\right )^{5} + 20 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} \sqrt{a \tan \left (d x + c\right )^{2} + a}}{15 \,{\left (a^{3} d \tan \left (d x + c\right )^{6} + 3 \, a^{3} d \tan \left (d x + c\right )^{4} + 3 \, a^{3} d \tan \left (d x + c\right )^{2} + a^{3} d\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{1}{\left (a \tan ^{2}{\left (c + d x \right )} + a\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.87904, size = 147, normalized size = 1.67 \begin{align*} -\frac{2 \,{\left (15 \, \sqrt{a}{\left (\frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{4} - 40 \, \sqrt{a}{\left (\frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{2} + 48 \, \sqrt{a}\right )}}{15 \, a^{3} d{\left (\frac{1}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}^{5} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 1\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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