Optimal. Leaf size=135 \[ \frac{A b-a B}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac{(A+B) \log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
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Rubi [A] time = 0.193716, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {2837, 801} \[ \frac{A b-a B}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\left (a^2 (-B)+2 a A b-b^2 B\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac{(A+B) \log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{(A-B) \log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) (A+B \sin (c+d x))}{(a+b \sin (c+d x))^2} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{A+\frac{B x}{b}}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \left (\frac{A+B}{2 b (a+b)^2 (b-x)}+\frac{-A b+a B}{(a-b) b (a+b) (a+x)^2}+\frac{-2 a A b+a^2 B+b^2 B}{(a-b)^2 b (a+b)^2 (a+x)}+\frac{A-B}{2 (a-b)^2 b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{(A+B) \log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac{(A-B) \log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac{\left (2 a A b-a^2 B-b^2 B\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2 d}+\frac{A b-a B}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 1.27208, size = 178, normalized size = 1.32 \[ \frac{b \left (A-\frac{a B}{b}\right ) \left (\frac{1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )-\frac{B ((b-a) \log (1-\sin (c+d x))+(a+b) \log (\sin (c+d x)+1)-2 b \log (a+b \sin (c+d x)))}{2 b (b-a) (a+b)}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.135, size = 240, normalized size = 1.8 \begin{align*}{\frac{Ab}{d \left ( a+b \right ) \left ( a-b \right ) \left ( a+b\sin \left ( dx+c \right ) \right ) }}-{\frac{aB}{d \left ( a+b \right ) \left ( a-b \right ) \left ( a+b\sin \left ( dx+c \right ) \right ) }}-2\,{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) Aab}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B{a}^{2}}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}+{\frac{\ln \left ( a+b\sin \left ( dx+c \right ) \right ) B{b}^{2}}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) A}{2\,d \left ( a+b \right ) ^{2}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) B}{2\,d \left ( a+b \right ) ^{2}}}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) A}{2\,d \left ( a-b \right ) ^{2}}}-{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) B}{2\,d \left ( a-b \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.99445, size = 198, normalized size = 1.47 \begin{align*} \frac{\frac{2 \,{\left (B a^{2} - 2 \, A a b + B b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} + \frac{{\left (A - B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} - \frac{{\left (A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \,{\left (B a - A b\right )}}{a^{3} - a b^{2} +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 3.37979, size = 671, normalized size = 4.97 \begin{align*} -\frac{2 \, B a^{3} - 2 \, A a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3} - 2 \,{\left (B a^{3} - 2 \, A a^{2} b + B a b^{2} +{\left (B a^{2} b - 2 \, A a b^{2} + B b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left ({\left (A - B\right )} a^{3} + 2 \,{\left (A - B\right )} a^{2} b +{\left (A - B\right )} a b^{2} +{\left ({\left (A - B\right )} a^{2} b + 2 \,{\left (A - B\right )} a b^{2} +{\left (A - B\right )} b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left ({\left (A + B\right )} a^{3} - 2 \,{\left (A + B\right )} a^{2} b +{\left (A + B\right )} a b^{2} +{\left ({\left (A + B\right )} a^{2} b - 2 \,{\left (A + B\right )} a b^{2} +{\left (A + B\right )} b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \,{\left ({\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} d \sin \left (d x + c\right ) +{\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (A + B \sin{\left (c + d x \right )}\right ) \sec{\left (c + d x \right )}}{\left (a + b \sin{\left (c + d x \right )}\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33868, size = 277, normalized size = 2.05 \begin{align*} \frac{\frac{2 \,{\left (B a^{2} b - 2 \, A a b^{2} + B b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac{{\left (A + B\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac{{\left (A - B\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} - \frac{2 \,{\left (B a^{2} b \sin \left (d x + c\right ) - 2 \, A a b^{2} \sin \left (d x + c\right ) + B b^{3} \sin \left (d x + c\right ) + 2 \, B a^{3} - 3 \, A a^{2} b + A b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (b \sin \left (d x + c\right ) + a\right )}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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