Optimal. Leaf size=109 \[ \frac{a \left (2 a^2+9 b^2\right ) \tan (c+d x)}{3 d}+\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{7 a^2 b \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))}{3 d} \]
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Rubi [A] time = 0.182106, antiderivative size = 109, 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 = {2792, 3021, 2748, 3767, 8, 3770} \[ \frac{a \left (2 a^2+9 b^2\right ) \tan (c+d x)}{3 d}+\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{7 a^2 b \tan (c+d x) \sec (c+d x)}{6 d}+\frac{a^2 \tan (c+d x) \sec ^2(c+d x) (a+b \cos (c+d x))}{3 d} \]
Antiderivative was successfully verified.
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Rule 2792
Rule 3021
Rule 2748
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^3 \sec ^4(c+d x) \, dx &=\frac{a^2 (a+b \cos (c+d x)) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{3} \int \left (7 a^2 b+a \left (2 a^2+9 b^2\right ) \cos (c+d x)+b \left (a^2+3 b^2\right ) \cos ^2(c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{7 a^2 b \sec (c+d x) \tan (c+d x)}{6 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{6} \int \left (2 a \left (2 a^2+9 b^2\right )+3 b \left (3 a^2+2 b^2\right ) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac{7 a^2 b \sec (c+d x) \tan (c+d x)}{6 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{1}{2} \left (b \left (3 a^2+2 b^2\right )\right ) \int \sec (c+d x) \, dx+\frac{1}{3} \left (a \left (2 a^2+9 b^2\right )\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{7 a^2 b \sec (c+d x) \tan (c+d x)}{6 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac{\left (a \left (2 a^2+9 b^2\right )\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{b \left (3 a^2+2 b^2\right ) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{a \left (2 a^2+9 b^2\right ) \tan (c+d x)}{3 d}+\frac{7 a^2 b \sec (c+d x) \tan (c+d x)}{6 d}+\frac{a^2 (a+b \cos (c+d x)) \sec ^2(c+d x) \tan (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.268197, size = 70, normalized size = 0.64 \[ \frac{\left (9 a^2 b+6 b^3\right ) \tanh ^{-1}(\sin (c+d x))+a \tan (c+d x) \left (2 a^2 \tan ^2(c+d x)+6 a^2+9 a b \sec (c+d x)+18 b^2\right )}{6 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.065, size = 118, normalized size = 1.1 \begin{align*}{\frac{2\,{a}^{3}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{{a}^{3}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{3\,{a}^{2}b\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{3\,{a}^{2}b\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{\frac{a{b}^{2}\tan \left ( dx+c \right ) }{d}}+{\frac{{b}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.971871, size = 153, normalized size = 1.4 \begin{align*} \frac{4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} a^{3} - 9 \, a^{2} b{\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 \, b^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 36 \, a b^{2} \tan \left (d x + c\right )}{12 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.95837, size = 309, normalized size = 2.83 \begin{align*} \frac{3 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (9 \, a^{2} b \cos \left (d x + c\right ) + 2 \, a^{3} + 2 \,{\left (2 \, a^{3} + 9 \, a b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{12 \, 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(-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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Giac [B] time = 2.57902, size = 277, normalized size = 2.54 \begin{align*} \frac{3 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (3 \, a^{2} b + 2 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 18 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 4 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 9 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 18 \, a b^{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}}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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