Optimal. Leaf size=118 \[ \frac{(5 a A-b B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^6(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{6 d}+\frac{(5 a A-b B) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{(5 a A-b B) \tan (c+d x) \sec (c+d x)}{16 d} \]
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Rubi [A] time = 0.105369, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {2837, 778, 199, 206} \[ \frac{(5 a A-b B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^6(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{6 d}+\frac{(5 a A-b B) \tan (c+d x) \sec ^3(c+d x)}{24 d}+\frac{(5 a A-b B) \tan (c+d x) \sec (c+d x)}{16 d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 778
Rule 199
Rule 206
Rubi steps
\begin{align*} \int \sec ^7(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac{b^7 \operatorname{Subst}\left (\int \frac{(a+x) \left (A+\frac{B x}{b}\right )}{\left (b^2-x^2\right )^4} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{\sec ^6(c+d x) (A b+a B+(a A+b B) \sin (c+d x))}{6 d}+\frac{\left (b^5 (5 a A-b B)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{6 d}\\ &=\frac{\sec ^6(c+d x) (A b+a B+(a A+b B) \sin (c+d x))}{6 d}+\frac{(5 a A-b B) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{\left (b^3 (5 a A-b B)\right ) \operatorname{Subst}\left (\int \frac{1}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac{\sec ^6(c+d x) (A b+a B+(a A+b B) \sin (c+d x))}{6 d}+\frac{(5 a A-b B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{(5 a A-b B) \sec ^3(c+d x) \tan (c+d x)}{24 d}+\frac{(b (5 a A-b B)) \operatorname{Subst}\left (\int \frac{1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{16 d}\\ &=\frac{(5 a A-b B) \tanh ^{-1}(\sin (c+d x))}{16 d}+\frac{\sec ^6(c+d x) (A b+a B+(a A+b B) \sin (c+d x))}{6 d}+\frac{(5 a A-b B) \sec (c+d x) \tan (c+d x)}{16 d}+\frac{(5 a A-b B) \sec ^3(c+d x) \tan (c+d x)}{24 d}\\ \end{align*}
Mathematica [A] time = 0.871246, size = 104, normalized size = 0.88 \[ -\frac{\sec ^6(c+d x) \left ((3 b B-15 a A) \sin ^5(c+d x)+8 (5 a A-b B) \sin ^3(c+d x)-3 (11 a A+b B) \sin (c+d x)-3 (5 a A-b B) \cos ^6(c+d x) \tanh ^{-1}(\sin (c+d x))-8 (a B+A b)\right )}{48 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.082, size = 217, normalized size = 1.8 \begin{align*}{\frac{A\tan \left ( dx+c \right ) a \left ( \sec \left ( dx+c \right ) \right ) ^{5}}{6\,d}}+{\frac{5\,A\tan \left ( dx+c \right ) a \left ( \sec \left ( dx+c \right ) \right ) ^{3}}{24\,d}}+{\frac{5\,aA\sec \left ( dx+c \right ) \tan \left ( dx+c \right ) }{16\,d}}+{\frac{5\,aA\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}}+{\frac{aB}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{Ab}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{3}b}{6\,d \left ( \cos \left ( dx+c \right ) \right ) ^{6}}}+{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{3}b}{8\,d \left ( \cos \left ( dx+c \right ) \right ) ^{4}}}+{\frac{B \left ( \sin \left ( dx+c \right ) \right ) ^{3}b}{16\,d \left ( \cos \left ( dx+c \right ) \right ) ^{2}}}+{\frac{Bb\sin \left ( dx+c \right ) }{16\,d}}-{\frac{Bb\ln \left ( \sec \left ( dx+c \right ) +\tan \left ( dx+c \right ) \right ) }{16\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.21774, size = 193, normalized size = 1.64 \begin{align*} \frac{3 \,{\left (5 \, A a - B b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (5 \, A a - B b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac{2 \,{\left (3 \,{\left (5 \, A a - B b\right )} \sin \left (d x + c\right )^{5} - 8 \,{\left (5 \, A a - B b\right )} \sin \left (d x + c\right )^{3} + 8 \, B a + 8 \, A b + 3 \,{\left (11 \, A a + B b\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57303, size = 342, normalized size = 2.9 \begin{align*} \frac{3 \,{\left (5 \, A a - B b\right )} \cos \left (d x + c\right )^{6} \log \left (\sin \left (d x + c\right ) + 1\right ) - 3 \,{\left (5 \, A a - B b\right )} \cos \left (d x + c\right )^{6} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 16 \, B a + 16 \, A b + 2 \,{\left (3 \,{\left (5 \, A a - B b\right )} \cos \left (d x + c\right )^{4} + 2 \,{\left (5 \, A a - B b\right )} \cos \left (d x + c\right )^{2} + 8 \, A a + 8 \, B b\right )} \sin \left (d x + c\right )}{96 \, d \cos \left (d x + c\right )^{6}} \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 = 1.28641, size = 188, normalized size = 1.59 \begin{align*} \frac{3 \,{\left (5 \, A a - B b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - 3 \,{\left (5 \, A a - B b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac{2 \,{\left (15 \, A a \sin \left (d x + c\right )^{5} - 3 \, B b \sin \left (d x + c\right )^{5} - 40 \, A a \sin \left (d x + c\right )^{3} + 8 \, B b \sin \left (d x + c\right )^{3} + 33 \, A a \sin \left (d x + c\right ) + 3 \, B b \sin \left (d x + c\right ) + 8 \, B a + 8 \, A b\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{3}}}{96 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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