Optimal. Leaf size=118 \[ \frac{16 \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}+\frac{8 \tan (c+d x)}{35 a^2 d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{6 \tan (c+d x)}{35 a d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{\tan (c+d x)}{7 d \left (a \sec ^2(c+d x)\right )^{7/2}} \]
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Rubi [A] time = 0.0532144, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3657, 4122, 192, 191} \[ \frac{16 \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}+\frac{8 \tan (c+d x)}{35 a^2 d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{6 \tan (c+d x)}{35 a d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{\tan (c+d x)}{7 d \left (a \sec ^2(c+d x)\right )^{7/2}} \]
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 )^{7/2}} \, dx &=\int \frac{1}{\left (a \sec ^2(c+d x)\right )^{7/2}} \, dx\\ &=\frac{a \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{9/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\tan (c+d x)}{7 d \left (a \sec ^2(c+d x)\right )^{7/2}}+\frac{6 \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{7/2}} \, dx,x,\tan (c+d x)\right )}{7 d}\\ &=\frac{\tan (c+d x)}{7 d \left (a \sec ^2(c+d x)\right )^{7/2}}+\frac{6 \tan (c+d x)}{35 a d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{24 \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{5/2}} \, dx,x,\tan (c+d x)\right )}{35 a d}\\ &=\frac{\tan (c+d x)}{7 d \left (a \sec ^2(c+d x)\right )^{7/2}}+\frac{6 \tan (c+d x)}{35 a d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{8 \tan (c+d x)}{35 a^2 d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{16 \operatorname{Subst}\left (\int \frac{1}{\left (a+a x^2\right )^{3/2}} \, dx,x,\tan (c+d x)\right )}{35 a^2 d}\\ &=\frac{\tan (c+d x)}{7 d \left (a \sec ^2(c+d x)\right )^{7/2}}+\frac{6 \tan (c+d x)}{35 a d \left (a \sec ^2(c+d x)\right )^{5/2}}+\frac{8 \tan (c+d x)}{35 a^2 d \left (a \sec ^2(c+d x)\right )^{3/2}}+\frac{16 \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.184002, size = 62, normalized size = 0.53 \[ \frac{\left (-5 \sin ^6(c+d x)+21 \sin ^4(c+d x)-35 \sin ^2(c+d x)+35\right ) \tan (c+d x)}{35 a^3 d \sqrt{a \sec ^2(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 119, normalized size = 1. \begin{align*}{\frac{a}{d} \left ({\frac{\tan \left ( dx+c \right ) }{7\,a} \left ( a+a \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) ^{-{\frac{7}{2}}}}+{\frac{6}{7\,a} \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 ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.82782, size = 68, normalized size = 0.58 \begin{align*} \frac{5 \, \sin \left (7 \, d x + 7 \, c\right ) + 49 \, \sin \left (5 \, d x + 5 \, c\right ) + 245 \, \sin \left (3 \, d x + 3 \, c\right ) + 1225 \, \sin \left (d x + c\right )}{2240 \, a^{\frac{7}{2}} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.53594, size = 293, normalized size = 2.48 \begin{align*} \frac{{\left (16 \, \tan \left (d x + c\right )^{7} + 56 \, \tan \left (d x + c\right )^{5} + 70 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} \sqrt{a \tan \left (d x + c\right )^{2} + a}}{35 \,{\left (a^{4} d \tan \left (d x + c\right )^{8} + 4 \, a^{4} d \tan \left (d x + c\right )^{6} + 6 \, a^{4} d \tan \left (d x + c\right )^{4} + 4 \, a^{4} d \tan \left (d x + c\right )^{2} + a^{4} 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{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.11193, size = 185, normalized size = 1.57 \begin{align*} -\frac{2 \,{\left (35 \, \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 )}^{6} - 140 \, \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} + 336 \, \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} - 320 \, \sqrt{a}\right )}}{35 \, a^{4} 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 )}^{7} \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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