Optimal. Leaf size=88 \[ \frac {(3 a A-b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{4 d}+\frac {(3 a A-b B) \tan (c+d x) \sec (c+d x)}{8 d} \]
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Rubi [A] time = 0.09, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2837, 778, 199, 206} \[ \frac {(3 a A-b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) ((a A+b B) \sin (c+d x)+a B+A b)}{4 d}+\frac {(3 a A-b B) \tan (c+d x) \sec (c+d x)}{8 d} \]
Antiderivative was successfully verified.
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Rule 199
Rule 206
Rule 778
Rule 2837
Rubi steps
\begin {align*} \int \sec ^5(c+d x) (a+b \sin (c+d x)) (A+B \sin (c+d x)) \, dx &=\frac {b^5 \operatorname {Subst}\left (\int \frac {(a+x) \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) (A b+a B+(a A+b B) \sin (c+d x))}{4 d}+\frac {\left (b^3 (3 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 )}{4 d}\\ &=\frac {\sec ^4(c+d x) (A b+a B+(a A+b B) \sin (c+d x))}{4 d}+\frac {(3 a A-b B) \sec (c+d x) \tan (c+d x)}{8 d}+\frac {(b (3 a A-b B)) \operatorname {Subst}\left (\int \frac {1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{8 d}\\ &=\frac {(3 a A-b B) \tanh ^{-1}(\sin (c+d x))}{8 d}+\frac {\sec ^4(c+d x) (A b+a B+(a A+b B) \sin (c+d x))}{4 d}+\frac {(3 a A-b B) \sec (c+d x) \tan (c+d x)}{8 d}\\ \end {align*}
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Mathematica [A] time = 0.59, size = 82, normalized size = 0.93 \[ \frac {\sec ^4(c+d x) \left ((b B-3 a A) \sin ^3(c+d x)+(5 a A+b B) \sin (c+d x)+(3 a A-b B) \cos ^4(c+d x) \tanh ^{-1}(\sin (c+d x))+2 (a B+A b)\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 114, normalized size = 1.30 \[ \frac {{\left (3 \, A a - B b\right )} \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, A a - B b\right )} \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, B a + 4 \, A b + 2 \, {\left ({\left (3 \, A a - B b\right )} \cos \left (d x + c\right )^{2} + 2 \, A a + 2 \, B b\right )} \sin \left (d x + c\right )}{16 \, d \cos \left (d x + c\right )^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 114, normalized size = 1.30 \[ \frac {{\left (3 \, A a - B b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (3 \, A a - B b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (3 \, A a \sin \left (d x + c\right )^{3} - B b \sin \left (d x + c\right )^{3} - 5 \, A a \sin \left (d x + c\right ) - B b \sin \left (d x + c\right ) - 2 \, B a - 2 \, A b\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 173, normalized size = 1.97 \[ \frac {a A \tan \left (d x +c \right ) \left (\sec ^{3}\left (d x +c \right )\right )}{4 d}+\frac {3 a A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{8 d}+\frac {3 a A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d}+\frac {a B}{4 d \cos \left (d x +c \right )^{4}}+\frac {A b}{4 d \cos \left (d x +c \right )^{4}}+\frac {B b \left (\sin ^{3}\left (d x +c \right )\right )}{4 d \cos \left (d x +c \right )^{4}}+\frac {B b \left (\sin ^{3}\left (d x +c \right )\right )}{8 d \cos \left (d x +c \right )^{2}}+\frac {b B \sin \left (d x +c \right )}{8 d}-\frac {B b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 112, normalized size = 1.27 \[ \frac {{\left (3 \, A a - B b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (3 \, A a - B b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left ({\left (3 \, A a - B b\right )} \sin \left (d x + c\right )^{3} - 2 \, B a - 2 \, A b - {\left (5 \, A a + B b\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{16 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 91, normalized size = 1.03 \[ \frac {\left (\frac {B\,b}{8}-\frac {3\,A\,a}{8}\right )\,{\sin \left (c+d\,x\right )}^3+\left (\frac {5\,A\,a}{8}+\frac {B\,b}{8}\right )\,\sin \left (c+d\,x\right )+\frac {A\,b}{4}+\frac {B\,a}{4}}{d\,\left ({\sin \left (c+d\,x\right )}^4-2\,{\sin \left (c+d\,x\right )}^2+1\right )}+\frac {\mathrm {atanh}\left (\sin \left (c+d\,x\right )\right )\,\left (\frac {3\,A\,a}{8}-\frac {B\,b}{8}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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