Optimal. Leaf size=156 \[ \frac{a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A-3 B-5 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 x (3 A+B)-\frac{(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 a d}+\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d} \]
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Rubi [A] time = 0.284327, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {4086, 3917, 3914, 3767, 8, 3770} \[ \frac{a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}-\frac{(6 A-3 B-5 C) \tan (c+d x) \left (a^3 \sec (c+d x)+a^3\right )}{6 d}+a^3 x (3 A+B)-\frac{(3 A-C) \tan (c+d x) \left (a^2 \sec (c+d x)+a^2\right )^2}{3 a d}+\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}+\frac{A \sin (c+d x) (a \sec (c+d x)+a)^3}{d} \]
Antiderivative was successfully verified.
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Rule 4086
Rule 3917
Rule 3914
Rule 3767
Rule 8
Rule 3770
Rubi steps
\begin{align*} \int \cos (c+d x) (a+a \sec (c+d x))^3 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac{A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}+\frac{\int (a+a \sec (c+d x))^3 (a (3 A+B)-a (3 A-C) \sec (c+d x)) \, dx}{a}\\ &=\frac{A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac{(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}+\frac{\int (a+a \sec (c+d x))^2 \left (3 a^2 (3 A+B)-a^2 (6 A-3 B-5 C) \sec (c+d x)\right ) \, dx}{3 a}\\ &=\frac{A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac{(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac{(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac{\int (a+a \sec (c+d x)) \left (6 a^3 (3 A+B)+15 a^3 (B+C) \sec (c+d x)\right ) \, dx}{6 a}\\ &=a^3 (3 A+B) x+\frac{A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac{(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac{(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}+\frac{1}{2} \left (5 a^3 (B+C)\right ) \int \sec ^2(c+d x) \, dx+\frac{1}{2} \left (a^3 (6 A+7 B+5 C)\right ) \int \sec (c+d x) \, dx\\ &=a^3 (3 A+B) x+\frac{a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}-\frac{(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac{(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}-\frac{\left (5 a^3 (B+C)\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{2 d}\\ &=a^3 (3 A+B) x+\frac{a^3 (6 A+7 B+5 C) \tanh ^{-1}(\sin (c+d x))}{2 d}+\frac{A (a+a \sec (c+d x))^3 \sin (c+d x)}{d}+\frac{5 a^3 (B+C) \tan (c+d x)}{2 d}-\frac{(3 A-C) \left (a^2+a^2 \sec (c+d x)\right )^2 \tan (c+d x)}{3 a d}-\frac{(6 A-3 B-5 C) \left (a^3+a^3 \sec (c+d x)\right ) \tan (c+d x)}{6 d}\\ \end{align*}
Mathematica [B] time = 6.45498, size = 1503, normalized size = 9.63 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.107, size = 226, normalized size = 1.5 \begin{align*}{\frac{A{a}^{3}\sin \left ( dx+c \right ) }{d}}+{a}^{3}Bx+{\frac{B{a}^{3}c}{d}}+{\frac{5\,{a}^{3}C\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+3\,{a}^{3}Ax+3\,{\frac{A{a}^{3}c}{d}}+{\frac{7\,B{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{11\,{a}^{3}C\tan \left ( dx+c \right ) }{3\,d}}+3\,{\frac{A{a}^{3}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+3\,{\frac{B{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{3\,{a}^{3}C\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{A{a}^{3}\tan \left ( dx+c \right ) }{d}}+{\frac{B{a}^{3}\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{2\,d}}+{\frac{{a}^{3}C\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.965363, size = 370, normalized size = 2.37 \begin{align*} \frac{36 \,{\left (d x + c\right )} A a^{3} + 12 \,{\left (d x + c\right )} B a^{3} + 4 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{3} - 3 \, B a^{3}{\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 )} - 9 \, C a^{3}{\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 )} + 18 \, A a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 18 \, B a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 6 \, C a^{3}{\left (\log \left (\sin \left (d x + c\right ) + 1\right ) - \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 12 \, A a^{3} \sin \left (d x + c\right ) + 12 \, A a^{3} \tan \left (d x + c\right ) + 36 \, B a^{3} \tan \left (d x + c\right ) + 36 \, C a^{3} \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 = 0.557569, size = 425, normalized size = 2.72 \begin{align*} \frac{12 \,{\left (3 \, A + B\right )} a^{3} d x \cos \left (d x + c\right )^{3} + 3 \,{\left (6 \, A + 7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (6 \, A + 7 \, B + 5 \, C\right )} a^{3} \cos \left (d x + c\right )^{3} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (6 \, A a^{3} \cos \left (d x + c\right )^{3} + 2 \,{\left (3 \, A + 9 \, B + 11 \, C\right )} a^{3} \cos \left (d x + c\right )^{2} + 3 \,{\left (B + 3 \, C\right )} a^{3} \cos \left (d x + c\right ) + 2 \, C a^{3}\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 [A] time = 1.3687, size = 389, normalized size = 2.49 \begin{align*} \frac{\frac{12 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1} + 6 \,{\left (3 \, A a^{3} + B a^{3}\right )}{\left (d x + c\right )} + 3 \,{\left (6 \, A a^{3} + 7 \, B a^{3} + 5 \, C a^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 3 \,{\left (6 \, A a^{3} + 7 \, B a^{3} + 5 \, C a^{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 a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 15 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 12 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 36 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 40 \, C a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 6 \, A a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 21 \, B a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 33 \, C a^{3} \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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