Optimal. Leaf size=156 \[ \frac{\left (2 a^2 B+4 a A b+3 b^2 B\right ) \tan (c+d x)}{3 d}+\frac{\left (3 a^2 A+8 a b B+4 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left (3 a^2 A+8 a b B+4 A b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rubi [A] time = 0.293472, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {2988, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac{\left (2 a^2 B+4 a A b+3 b^2 B\right ) \tan (c+d x)}{3 d}+\frac{\left (3 a^2 A+8 a b B+4 A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left (3 a^2 A+8 a b B+4 A b^2\right ) \tan (c+d x) \sec (c+d x)}{8 d}+\frac{a^2 A \tan (c+d x) \sec ^3(c+d x)}{4 d}+\frac{a (a B+2 A b) \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 2988
Rule 3021
Rule 2748
Rule 3768
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int (a+b \cos (c+d x))^2 (A+B \cos (c+d x)) \sec ^5(c+d x) \, dx &=\frac{a^2 A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{4} \int \left (-4 a (2 A b+a B)-\left (3 a^2 A+4 A b^2+8 a b B\right ) \cos (c+d x)-4 b^2 B \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx\\ &=\frac{a (2 A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a^2 A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{12} \int \left (-3 \left (3 a^2 A+4 A b^2+8 a b B\right )-4 \left (4 a A b+2 a^2 B+3 b^2 B\right ) \cos (c+d x)\right ) \sec ^3(c+d x) \, dx\\ &=\frac{a (2 A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a^2 A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{4} \left (-3 a^2 A-4 A b^2-8 a b B\right ) \int \sec ^3(c+d x) \, dx-\frac{1}{3} \left (-4 a A b-2 a^2 B-3 b^2 B\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac{\left (3 a^2 A+4 A b^2+8 a b B\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (2 A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a^2 A \sec ^3(c+d x) \tan (c+d x)}{4 d}-\frac{1}{8} \left (-3 a^2 A-4 A b^2-8 a b B\right ) \int \sec (c+d x) \, dx-\frac{\left (4 a A b+2 a^2 B+3 b^2 B\right ) \operatorname{Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac{\left (3 a^2 A+4 A b^2+8 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\left (4 a A b+2 a^2 B+3 b^2 B\right ) \tan (c+d x)}{3 d}+\frac{\left (3 a^2 A+4 A b^2+8 a b B\right ) \sec (c+d x) \tan (c+d x)}{8 d}+\frac{a (2 A b+a B) \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac{a^2 A \sec ^3(c+d x) \tan (c+d x)}{4 d}\\ \end{align*}
Mathematica [A] time = 0.693543, size = 120, normalized size = 0.77 \[ \frac{3 \left (3 a^2 A+8 a b B+4 A b^2\right ) \tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \left (3 \left (3 a^2 A+8 a b B+4 A b^2\right ) \sec (c+d x)+24 \left (a^2 B+2 a A b+b^2 B\right )+6 a^2 A \sec ^3(c+d x)+8 a (a B+2 A b) \tan ^2(c+d x)\right )}{24 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.084, size = 241, normalized size = 1.5 \begin{align*}{\frac{{a}^{2}A \left ( \sec \left ( dx+c \right ) \right ) ^{3}\tan \left ( dx+c \right ) }{4\,d}}+{\frac{3\,{a}^{2}A\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{8\,d}}+{\frac{3\,{a}^{2}A\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{2\,B{a}^{2}\tan \left ( dx+c \right ) }{3\,d}}+{\frac{B{a}^{2}\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{4\,Aab\tan \left ( dx+c \right ) }{3\,d}}+{\frac{2\,Aab\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{2}}{3\,d}}+{\frac{Bab\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{d}}+{\frac{Bab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{d}}+{\frac{A{b}^{2}\tan \left ( dx+c \right ) \sec \left ( dx+c \right ) }{2\,d}}+{\frac{A{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{2\,d}}+{\frac{{b}^{2}B\tan \left ( dx+c \right ) }{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10652, size = 308, normalized size = 1.97 \begin{align*} \frac{16 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} B a^{2} + 32 \,{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a b - 3 \, A a^{2}{\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 )} - 24 \, B a 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 )} - 12 \, A b^{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 )} + 48 \, B b^{2} \tan \left (d x + c\right )}{48 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47, size = 443, normalized size = 2.84 \begin{align*} \frac{3 \,{\left (3 \, A a^{2} + 8 \, B a b + 4 \, A b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (3 \, A a^{2} + 8 \, B a b + 4 \, A b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (8 \,{\left (2 \, B a^{2} + 4 \, A a b + 3 \, B b^{2}\right )} \cos \left (d x + c\right )^{3} + 6 \, A a^{2} + 3 \,{\left (3 \, A a^{2} + 8 \, B a b + 4 \, A b^{2}\right )} \cos \left (d x + c\right )^{2} + 8 \,{\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\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(-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 = 1.49832, size = 645, normalized size = 4.13 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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