Optimal. Leaf size=74 \[ \frac{5 a^2 \tan (c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d} \]
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Rubi [A] time = 0.0808184, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {3788, 3768, 3770, 4046, 3767, 8} \[ \frac{5 a^2 \tan (c+d x)}{3 d}+\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x)}{3 d}+\frac{a^2 \tan (c+d x) \sec (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 3788
Rule 3768
Rule 3770
Rule 4046
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \sec ^2(c+d x) (a+a \sec (c+d x))^2 \, dx &=\left (2 a^2\right ) \int \sec ^3(c+d x) \, dx+\int \sec ^2(c+d x) \left (a^2+a^2 \sec ^2(c+d x)\right ) \, dx\\ &=\frac{a^2 \sec (c+d x) \tan (c+d x)}{d}+\frac{a^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}+a^2 \int \sec (c+d x) \, dx+\frac{1}{3} \left (5 a^2\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{a^2 \sec (c+d x) \tan (c+d x)}{d}+\frac{a^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{\left (5 a^2\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac{5 a^2 \tan (c+d x)}{3 d}+\frac{a^2 \sec (c+d x) \tan (c+d x)}{d}+\frac{a^2 \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [B] time = 0.630396, size = 318, normalized size = 4.3 \[ -\frac{a^2 \sec (c) \sec ^3(c+d x) \left (6 \sin (2 c+d x)-6 \sin (c+2 d x)-6 \sin (3 c+2 d x)-10 \sin (2 c+3 d x)+3 \cos (2 c+3 d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+3 \cos (4 c+3 d x) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )+9 \cos (d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )+9 \cos (2 c+d x) \left (\log \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )-\log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )\right )-3 \cos (2 c+3 d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-3 \cos (4 c+3 d x) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )-24 \sin (d x)\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 78, normalized size = 1.1 \begin{align*}{\frac{5\,{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{2}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{{a}^{2} \left ( \sec \left ( dx+c \right ) \right ) ^{2}\tan \left ( dx+c \right ) }{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.12788, size = 115, normalized size = 1.55 \begin{align*} \frac{2 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{2} - 3 \, a^{2}{\left (\frac{2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, a^{2} \tan \left (d x + c\right )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.74291, size = 246, normalized size = 3.32 \begin{align*} \frac{3 \, a^{2} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \, a^{2} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (5 \, a^{2} \cos \left (d x + c\right )^{2} + 3 \, a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sin \left (d x + c\right )}{6 \, d \cos \left (d x + c\right )^{3}} \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*} a^{2} \left (\int \sec ^{2}{\left (c + d x \right )}\, dx + \int 2 \sec ^{3}{\left (c + d x \right )}\, dx + \int \sec ^{4}{\left (c + d x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.4357, size = 143, normalized size = 1.93 \begin{align*} \frac{3 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \, a^{2} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 8 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 9 \, a^{2} \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}}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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