Optimal. Leaf size=95 \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.0795231, antiderivative size = 95, 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} \[ \frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a \sec (c+d x)+a}}\right )}{a^{3/2} d}+\frac{2 \tan ^3(c+d x)}{3 d (a \sec (c+d x)+a)^{3/2}}-\frac{2 \tan (c+d x)}{a d \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 3887
Rule 302
Rule 203
Rubi steps
\begin{align*} \int \frac{\tan ^4(c+d x)}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^4}{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^2}+\frac{x^2}{a}+\frac{1}{a^2 \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 d \sqrt{a+a \sec (c+d x)}}+\frac{2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}-\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 d}\\ &=\frac{2 \tan ^{-1}\left (\frac{\sqrt{a} \tan (c+d x)}{\sqrt{a+a \sec (c+d x)}}\right )}{a^{3/2} d}-\frac{2 \tan (c+d x)}{a d \sqrt{a+a \sec (c+d x)}}+\frac{2 \tan ^3(c+d x)}{3 d (a+a \sec (c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 4.40266, size = 162, normalized size = 1.71 \[ \frac{64 \cos ^6\left (\frac{1}{2} (c+d x)\right ) \cot ^4\left (\frac{1}{2} (c+d x)\right ) \sec ^5(c+d x) \left (\frac{1}{\sec (c+d x)+1}\right )^{7/2} \left ((\sin (c+d x)-2 \sin (2 (c+d x))) \sqrt{\frac{1}{\cos (c+d x)+1}} \sqrt{\cos (c+d x)}+3 \cos ^2(c+d x) \sin ^{-1}\left (\frac{\tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{\frac{1}{\cos (c+d x)+1}}}\right )\right )}{3 d \left (\cot ^2\left (\frac{1}{2} (c+d x)\right )-1\right )^2 (a (\sec (c+d x)+1))^{3/2}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.202, size = 142, normalized size = 1.5 \begin{align*} -{\frac{1}{3\,d{a}^{2}\sin \left ( dx+c \right ) \cos \left ( dx+c \right ) } \left ( 3\,\sqrt{2}\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}{\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 ) -8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+10\,\cos \left ( dx+c \right ) -2 \right ) \sqrt{{\frac{a \left ( \cos \left ( dx+c \right ) +1 \right ) }{\cos \left ( dx+c \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 = 1.7873, size = 778, normalized size = 8.19 \begin{align*} \left [-\frac{3 \,{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\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 \, \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}{\left (4 \, \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\right )}}, -\frac{2 \,{\left (3 \,{\left (\cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\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 ) + \sqrt{\frac{a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}{\left (4 \, \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right )\right )}}{3 \,{\left (a^{2} d \cos \left (d x + c\right )^{2} + a^{2} d \cos \left (d x + c\right )\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 ^{4}{\left (c + d x \right )}}{\left (a \left (\sec{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 12.6768, size = 348, normalized size = 3.66 \begin{align*} \frac{3 \, \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^{2} \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^{2} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}\right )} + \frac{2 \,{\left (\frac{5 \, \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{3 \, \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 )} \sqrt{-a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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