Optimal. Leaf size=112 \[ -\frac {(a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))}{4 d}+\frac {(a-b) (a A+b (A-2 B)) \log (\sin (c+d x)+1)}{4 d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) ((a A+b B) \sin (c+d x)+a B+A b)}{2 d} \]
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Rubi [A] time = 0.18, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, Rules used = {2837, 819, 633, 31} \[ -\frac {(a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))}{4 d}+\frac {(a-b) (a A+b (A-2 B)) \log (\sin (c+d x)+1)}{4 d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) ((a A+b B) \sin (c+d x)+a B+A b)}{2 d} \]
Antiderivative was successfully verified.
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Rule 31
Rule 633
Rule 819
Rule 2837
Rubi steps
\begin {align*} \int \sec ^3(c+d x) (a+b \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {(a+x)^2 \left (A+\frac {B x}{b}\right )}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (A b+a B+(a A+b B) \sin (c+d x))}{2 d}-\frac {b \operatorname {Subst}\left (\int \frac {-a^2 A+A b^2+2 a b B+2 b B x}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{2 d}\\ &=\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (A b+a B+(a A+b B) \sin (c+d x))}{2 d}-\frac {((a-b) (a A+b (A-2 B))) \operatorname {Subst}\left (\int \frac {1}{-b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}+\frac {((a+b) (a A-b (A+2 B))) \operatorname {Subst}\left (\int \frac {1}{b-x} \, dx,x,b \sin (c+d x)\right )}{4 d}\\ &=-\frac {(a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))}{4 d}+\frac {(a-b) (a A+b (A-2 B)) \log (1+\sin (c+d x))}{4 d}+\frac {\sec ^2(c+d x) (a+b \sin (c+d x)) (A b+a B+(a A+b B) \sin (c+d x))}{2 d}\\ \end {align*}
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Mathematica [A] time = 1.53, size = 174, normalized size = 1.55 \[ \frac {\left (-6 a^3 A b+4 a A b^3+2 b^4 B\right ) \tan ^2(c+d x)-2 a^3 (a B-A b) \sec ^2(c+d x)+\left (a^2-b^2\right ) ((a+b) (a A-b (A+2 B)) \log (1-\sin (c+d x))-(a-b) (a A+b (A-2 B)) \log (\sin (c+d x)+1))-2 \left (a^2-b^2\right ) \left (a^2 A+2 a b B+A b^2\right ) \tan (c+d x) \sec (c+d x)}{4 d \left (b^2-a^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 136, normalized size = 1.21 \[ \frac {{\left (A a^{2} - 2 \, B a b - {\left (A - 2 \, B\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A a^{2} - 2 \, B a b - {\left (A + 2 \, B\right )} b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, B a^{2} + 4 \, A a b + 2 \, B b^{2} + 2 \, {\left (A a^{2} + 2 \, B a b + A b^{2}\right )} \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 146, normalized size = 1.30 \[ \frac {{\left (A a^{2} - 2 \, B a b - A b^{2} + 2 \, B b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (A a^{2} - 2 \, B a b - A b^{2} - 2 \, B b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (B b^{2} \sin \left (d x + c\right )^{2} + A a^{2} \sin \left (d x + c\right ) + 2 \, B a b \sin \left (d x + c\right ) + A b^{2} \sin \left (d x + c\right ) + B a^{2} + 2 \, A a b\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.58, size = 231, normalized size = 2.06 \[ \frac {a^{2} A \sec \left (d x +c \right ) \tan \left (d x +c \right )}{2 d}+\frac {a^{2} A \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {B \,a^{2}}{2 d \cos \left (d x +c \right )^{2}}+\frac {A a b}{d \cos \left (d x +c \right )^{2}}+\frac {B a b \left (\sin ^{3}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{2}}+\frac {B a b \sin \left (d x +c \right )}{d}-\frac {B a b \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {A \,b^{2} \left (\sin ^{3}\left (d x +c \right )\right )}{2 d \cos \left (d x +c \right )^{2}}+\frac {A \,b^{2} \sin \left (d x +c \right )}{2 d}-\frac {A \,b^{2} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2 d}+\frac {B \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {B \,b^{2} \ln \left (\cos \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 122, normalized size = 1.09 \[ \frac {{\left (A a^{2} - 2 \, B a b - {\left (A - 2 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A a^{2} - 2 \, B a b - {\left (A + 2 \, B\right )} b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (B a^{2} + 2 \, A a b + B b^{2} + {\left (A a^{2} + 2 \, B a b + A b^{2}\right )} \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.36, size = 118, normalized size = 1.05 \[ \frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (a-b\right )\,\left (A\,a+A\,b-2\,B\,b\right )}{4\,d}-\frac {\sin \left (c+d\,x\right )\,\left (\frac {A\,a^2}{2}+B\,a\,b+\frac {A\,b^2}{2}\right )+\frac {B\,a^2}{2}+\frac {B\,b^2}{2}+A\,a\,b}{d\,\left ({\sin \left (c+d\,x\right )}^2-1\right )}+\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (a+b\right )\,\left (A\,b-A\,a+2\,B\,b\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (A + B \sin {\left (c + d x \right )}\right ) \left (a + b \sin {\left (c + d x \right )}\right )^{2} \sec ^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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