Optimal. Leaf size=85 \[ \frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}+\frac{3 C \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 C \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.0716354, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {4047, 3767, 12, 3768, 3770} \[ \frac{B \tan ^3(c+d x)}{3 d}+\frac{B \tan (c+d x)}{d}+\frac{3 C \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 C \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 4047
Rule 3767
Rule 12
Rule 3768
Rule 3770
Rubi steps
\begin{align*} \int \sec ^3(c+d x) \left (B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=B \int \sec ^4(c+d x) \, dx+\int C \sec ^5(c+d x) \, dx\\ &=C \int \sec ^5(c+d x) \, dx-\frac{B \operatorname{Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d}\\ &=\frac{B \tan (c+d x)}{d}+\frac{C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{B \tan ^3(c+d x)}{3 d}+\frac{1}{4} (3 C) \int \sec ^3(c+d x) \, dx\\ &=\frac{B \tan (c+d x)}{d}+\frac{3 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac{C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{B \tan ^3(c+d x)}{3 d}+\frac{1}{8} (3 C) \int \sec (c+d x) \, dx\\ &=\frac{3 C \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{B \tan (c+d x)}{d}+\frac{3 C \sec (c+d x) \tan (c+d x)}{8 d}+\frac{C \sec ^3(c+d x) \tan (c+d x)}{4 d}+\frac{B \tan ^3(c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.229007, size = 76, normalized size = 0.89 \[ \frac{B \left (\frac{1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d}+\frac{C \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{3 C \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{8 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 92, normalized size = 1.1 \begin{align*}{\frac{2\,B\tan \left ( dx+c \right ) }{3\,d}}+{\frac{B\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{C \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934751, size = 128, normalized size = 1.51 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B - 3 \, C{\left (\frac{2 \,{\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )}}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0.505312, size = 266, normalized size = 3.13 \begin{align*} \frac{9 \, C \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 9 \, C \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (16 \, B \cos \left (d x + c\right )^{3} + 9 \, C \cos \left (d x + c\right )^{2} + 8 \, B \cos \left (d x + c\right ) + 6 \, C\right )} \sin \left (d x + c\right )}{48 \, d \cos \left (d x + c\right )^{4}} \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 \left (B + C \sec{\left (c + d x \right )}\right ) \sec ^{4}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.22148, size = 221, normalized size = 2.6 \begin{align*} \frac{9 \, C \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 9 \, C \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (24 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 15 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 40 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 40 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 9 \, C \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 24 \, B \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 15 \, 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 )}^{4}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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