Optimal. Leaf size=122 \[ \frac{\left (3 a^2 A-2 a b B-A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left (\left (3 a^2 A-2 a b B+A b^2\right ) \sin (c+d x)+2 b (2 a A-b B)\right )}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2 (A \sin (c+d x)+B)}{4 d} \]
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Rubi [A] time = 0.155705, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {2837, 821, 778, 206} \[ \frac{\left (3 a^2 A-2 a b B-A b^2\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^2(c+d x) \left (\left (3 a^2 A-2 a b B+A b^2\right ) \sin (c+d x)+2 b (2 a A-b B)\right )}{8 d}+\frac{\sec ^4(c+d x) (a+b \sin (c+d x))^2 (A \sin (c+d x)+B)}{4 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 821
Rule 778
Rule 206
Rubi steps
\begin{align*} \int \sec ^5(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac{b^5 \operatorname{Subst}\left (\int \frac{(a+x)^2 \left (A+\frac{B x}{b}\right )}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^4(c+d x) (B+A \sin (c+d x)) (a+b \sin (c+d x))^2}{4 d}-\frac{b^3 \operatorname{Subst}\left (\int \frac{(a+x) (-3 a A+2 b B-A x)}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=\frac{\sec ^4(c+d x) (B+A \sin (c+d x)) (a+b \sin (c+d x))^2}{4 d}+\frac{\sec ^2(c+d x) \left (2 b (2 a A-b B)+\left (3 a^2 A+A b^2-2 a b B\right ) \sin (c+d x)\right )}{8 d}+\frac{\left (b \left (3 a^2 A-A b^2-2 a b B\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{\left (3 a^2 A-A b^2-2 a b B\right ) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac{\sec ^4(c+d x) (B+A \sin (c+d x)) (a+b \sin (c+d x))^2}{4 d}+\frac{\sec ^2(c+d x) \left (2 b (2 a A-b B)+\left (3 a^2 A+A b^2-2 a b B\right ) \sin (c+d x)\right )}{8 d}\\ \end{align*}
Mathematica [A] time = 1.72069, size = 186, normalized size = 1.52 \[ \frac{4 \left (b^2-a^2\right ) \sec ^4(c+d x) (a+b \sin (c+d x))^3 ((b B-a A) \sin (c+d x)-a B+A b)+\left (-3 a^2 A+2 a b B+A b^2\right ) \left (\left (4 a b^3-6 a^3 b\right ) \tan ^2(c+d x)+\left (a^2-b^2\right )^2 (\log (1-\sin (c+d x))-\log (\sin (c+d x)+1))-2 \left (a^4-b^4\right ) \tan (c+d x) \sec (c+d x)+2 a^3 b \sec ^2(c+d x)\right )}{16 d \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.097, size = 299, normalized size = 2.5 \begin{align*}{\frac{{a}^{2}A\tan \left ( dx+c \right ) \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{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{B{a}^{2}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{Aab}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{Bab \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{2\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{Bab \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{Bab\sin \left ( dx+c \right ) }{4\,d}}-{\frac{Bab\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{4\,d}}+{\frac{A{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{A{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{3}}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{A{b}^{2}\sin \left ( dx+c \right ) }{8\,d}}-{\frac{A{b}^{2}\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{8\,d}}+{\frac{B{b}^{2} \left ( \sin \left ( dx+c \right ) \right ) ^{4}}{4\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99136, size = 231, normalized size = 1.89 \begin{align*} \frac{{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) + \frac{2 \,{\left (4 \, B b^{2} \sin \left (d x + c\right )^{2} -{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \sin \left (d x + c\right )^{3} + 2 \, B a^{2} + 4 \, A a b - 2 \, B b^{2} +{\left (5 \, A a^{2} + 2 \, B a b + A b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.52496, size = 414, normalized size = 3.39 \begin{align*} \frac{{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 8 \, B b^{2} \cos \left (d x + c\right )^{2} + 4 \, B a^{2} + 8 \, A a b + 4 \, B b^{2} + 2 \,{\left (2 \, A a^{2} + 4 \, B a b + 2 \, A b^{2} +{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, 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 [A] time = 1.40568, size = 252, normalized size = 2.07 \begin{align*} \frac{{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) -{\left (3 \, A a^{2} - 2 \, B a b - A b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (3 \, A a^{2} \sin \left (d x + c\right )^{3} - 2 \, B a b \sin \left (d x + c\right )^{3} - A b^{2} \sin \left (d x + c\right )^{3} - 4 \, B b^{2} \sin \left (d x + c\right )^{2} - 5 \, A a^{2} \sin \left (d x + c\right ) - 2 \, B a b \sin \left (d x + c\right ) - A b^{2} \sin \left (d x + c\right ) - 2 \, B a^{2} - 4 \, A a b + 2 \, B b^{2}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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