Optimal. Leaf size=127 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 a d (a \sec (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.0865185, antiderivative size = 127, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3887, 302, 203} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a \sec (c+d x)+a}}+\frac{2 \tan ^5(c+d x)}{5 d (a \sec (c+d x)+a)^{5/2}}-\frac{2 \tan ^3(c+d x)}{3 a d (a \sec (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 302
Rule 203
Rubi steps
\begin{align*} \int \frac{\tan ^6(c+d x)}{(a+a \sec (c+d x))^{5/2}} \, dx &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^6}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=-\frac{(2 a) \operatorname{Subst}\left (\int \left (\frac{1}{a^3}-\frac{x^2}{a^2}+\frac{x^4}{a}-\frac{1}{a^3 \left (1+a x^2\right )}\right ) \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{d}\\ &=\frac{2 \tan (c+d x)}{a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{2 \tan ^3(c+d x)}{3 a d (a+a \sec (c+d x))^{3/2}}+\frac{2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+a x^2} \, dx,x,-\frac{\tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^2 d}\\ &=-\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{5/2} d}+\frac{2 \tan (c+d x)}{a^2 d \sqrt{a+a \sec (c+d x)}}-\frac{2 \tan ^3(c+d x)}{3 a d (a+a \sec (c+d x))^{3/2}}+\frac{2 \tan ^5(c+d x)}{5 d (a+a \sec (c+d x))^{5/2}}\\ \end{align*}
Mathematica [B] time = 6.06771, size = 447, normalized size = 3.52 \[ \frac{\sqrt{2} \sqrt{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1} \left (\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}-1\right )^3 \tan ^7(c+d x) \cot ^8\left (\frac{1}{2} (c+d x)\right ) \left (\frac{1}{\sec (c+d x)+1}\right )^{9/2} \left (\frac{8 \tan ^6\left (\frac{1}{2} (c+d x)\right )}{5 \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^3 \left (\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}-1\right )^3}+\frac{4 \tan ^4\left (\frac{1}{2} (c+d x)\right )}{3 \left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right )^2 \left (\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}-1\right )^2}+\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\left (\tan ^2\left (\frac{1}{2} (c+d x)\right )+1\right ) \left (\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}-1\right )}+\frac{\sqrt{2} \tan \left (\frac{1}{2} (c+d x)\right ) \sin ^{-1}\left (\frac{\sqrt{2} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}\right )}{\sqrt{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1} \sqrt{1-\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}}}\right )}{d \left (1-\frac{2 \tan ^2\left (\frac{1}{2} (c+d x)\right )}{\tan ^2\left (\frac{1}{2} (c+d x)\right )+1}\right )^{5/2} (a (\sec (c+d x)+1))^{5/2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.214, size = 302, normalized size = 2.4 \begin{align*}{\frac{1}{60\,d{a}^{3}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}}\sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \right ) }}} \left ( 15\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\sqrt{2}\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}+30\,{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\sqrt{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) +15\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}} \right ) \left ( -2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}} \right ) ^{5/2}\sin \left ( dx+c \right ) -184\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+272\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-112\,\cos \left ( dx+c \right ) +24 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A] time = 2.03001, size = 849, normalized size = 6.69 \begin{align*} \left [-\frac{15 \,{\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \sqrt{-a} \log \left (\frac{2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) - 2 \,{\left (23 \, \cos \left (d x + c\right )^{2} - 11 \, \cos \left (d x + c\right ) + 3\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\right )}}, \frac{2 \,{\left (15 \,{\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}\right )} \sqrt{a} \arctan \left (\frac{\sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt{a} \sin \left (d x + c\right )}\right ) +{\left (23 \, \cos \left (d x + c\right )^{2} - 11 \, \cos \left (d x + c\right ) + 3\right )} \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )\right )}}{15 \,{\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2}\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{\tan ^{6}{\left (c + d x \right )}}{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 16.5845, size = 394, normalized size = 3.1 \begin{align*} -\frac{15 \, \sqrt{-a}{\left (\frac{\log \left ({\left |{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} - a{\left (2 \, \sqrt{2} + 3\right )} \right |}\right )}{a^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{\log \left ({\left |{\left (\sqrt{-a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}\right )}^{2} + a{\left (2 \, \sqrt{2} - 3\right )} \right |}\right )}{a^{3} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )} + \frac{2 \,{\left ({\left (\frac{37 \, \sqrt{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )} - \frac{40 \, \sqrt{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + \frac{15 \, \sqrt{2}}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - a\right )}^{2} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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