Optimal. Leaf size=131 \[ -\frac{(3 B-4 C) \tan ^3(c+d x)}{3 a d}-\frac{(3 B-4 C) \tan (c+d x)}{a d}+\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d} \]
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Rubi [A] time = 0.253334, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4072, 4019, 3787, 3768, 3770, 3767} \[ -\frac{(3 B-4 C) \tan ^3(c+d x)}{3 a d}-\frac{(3 B-4 C) \tan (c+d x)}{a d}+\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}+\frac{(B-C) \tan (c+d x) \sec ^3(c+d x)}{d (a \sec (c+d x)+a)}+\frac{3 (B-C) \tan (c+d x) \sec (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
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Rule 4072
Rule 4019
Rule 3787
Rule 3768
Rule 3770
Rule 3767
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right )}{a+a \sec (c+d x)} \, dx &=\int \frac{\sec ^4(c+d x) (B+C \sec (c+d x))}{a+a \sec (c+d x)} \, dx\\ &=\frac{(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac{\int \sec ^3(c+d x) (3 a (B-C)-a (3 B-4 C) \sec (c+d x)) \, dx}{a^2}\\ &=\frac{(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 B-4 C) \int \sec ^4(c+d x) \, dx}{a}+\frac{(3 (B-C)) \int \sec ^3(c+d x) \, dx}{a}\\ &=\frac{3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}+\frac{(3 (B-C)) \int \sec (c+d x) \, dx}{2 a}+\frac{(3 B-4 C) \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{a d}\\ &=\frac{3 (B-C) \tanh ^{-1}(\sin (c+d x))}{2 a d}-\frac{(3 B-4 C) \tan (c+d x)}{a d}+\frac{3 (B-C) \sec (c+d x) \tan (c+d x)}{2 a d}+\frac{(B-C) \sec ^3(c+d x) \tan (c+d x)}{d (a+a \sec (c+d x))}-\frac{(3 B-4 C) \tan ^3(c+d x)}{3 a d}\\ \end{align*}
Mathematica [B] time = 1.11916, size = 550, normalized size = 4.2 \[ -\frac{\cos \left (\frac{1}{2} (c+d x)\right ) \sec ^3(c+d x) \left (27 (B-C) \cos \left (\frac{1}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+27 (B-C) \cos \left (\frac{3}{2} (c+d x)\right ) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-12 B \sin \left (\frac{1}{2} (c+d x)\right )+18 B \sin \left (\frac{3}{2} (c+d x)\right )-6 B \sin \left (\frac{5}{2} (c+d x)\right )+12 B \sin \left (\frac{7}{2} (c+d x)\right )+9 B \cos \left (\frac{5}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 B \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-9 B \cos \left (\frac{5}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-9 B \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-30 C \sin \left (\frac{3}{2} (c+d x)\right )+2 C \sin \left (\frac{5}{2} (c+d x)\right )-16 C \sin \left (\frac{7}{2} (c+d x)\right )-9 C \cos \left (\frac{5}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-9 C \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 C \cos \left (\frac{5}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )+9 C \cos \left (\frac{7}{2} (c+d x)\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )}{24 a d (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 340, normalized size = 2.6 \begin{align*} -{\frac{B}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{C}{ad}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-{\frac{C}{3\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-3}}-{\frac{B}{2\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}+{\frac{C}{ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-2}}-{\frac{3\,C}{2\,ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }+{\frac{3\,B}{2\,ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) }-{\frac{5\,C}{2\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3\,B}{2\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) +1 \right ) ^{-1}}-{\frac{C}{3\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-3}}-{\frac{C}{ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{B}{2\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-2}}+{\frac{3\,C}{2\,ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{3\,B}{2\,ad}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) }-{\frac{5\,C}{2\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3\,B}{2\,ad} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -1 \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.954362, size = 497, normalized size = 3.79 \begin{align*} \frac{C{\left (\frac{2 \,{\left (\frac{9 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{16 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac{15 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}}\right )}}{a - \frac{3 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{3 \, a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac{a \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}} - \frac{9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac{9 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac{6 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )} - 3 \, B{\left (\frac{2 \,{\left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac{3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}\right )}}{a - \frac{2 \, a \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac{a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} - \frac{3 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{a} + \frac{3 \, \log \left (\frac{\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{a} + \frac{2 \, \sin \left (d x + c\right )}{a{\left (\cos \left (d x + c\right ) + 1\right )}}\right )}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.519179, size = 417, normalized size = 3.18 \begin{align*} \frac{9 \,{\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} +{\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \,{\left ({\left (B - C\right )} \cos \left (d x + c\right )^{4} +{\left (B - C\right )} \cos \left (d x + c\right )^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (4 \,{\left (3 \, B - 4 \, C\right )} \cos \left (d x + c\right )^{3} +{\left (3 \, B - 7 \, C\right )} \cos \left (d x + c\right )^{2} -{\left (3 \, B - C\right )} \cos \left (d x + c\right ) - 2 \, C\right )} \sin \left (d x + c\right )}{12 \,{\left (a d \cos \left (d x + c\right )^{4} + a d \cos \left (d x + c\right )^{3}\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*} \frac{\int \frac{B \sec ^{4}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx + \int \frac{C \sec ^{5}{\left (c + d x \right )}}{\sec{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18516, size = 246, normalized size = 1.88 \begin{align*} \frac{\frac{9 \,{\left (B - C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a} - \frac{9 \,{\left (B - C\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a} - \frac{6 \,{\left (B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{a} + \frac{2 \,{\left (9 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 15 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 16 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 3 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{3} a}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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