Optimal. Leaf size=91 \[ \frac{A+B}{8 d (a-a \sin (c+d x))}-\frac{a (A-B)}{8 d (a \sin (c+d x)+a)^2}+\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{A}{4 d (a \sin (c+d x)+a)} \]
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Rubi [A] time = 0.138204, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {2836, 77, 206} \[ \frac{A+B}{8 d (a-a \sin (c+d x))}-\frac{a (A-B)}{8 d (a \sin (c+d x)+a)^2}+\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a d}-\frac{A}{4 d (a \sin (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2836
Rule 77
Rule 206
Rubi steps
\begin{align*} \int \frac{\sec ^3(c+d x) (A+B \sin (c+d x))}{a+a \sin (c+d x)} \, dx &=\frac{a^3 \operatorname{Subst}\left (\int \frac{A+\frac{B x}{a}}{(a-x)^2 (a+x)^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{a^3 \operatorname{Subst}\left (\int \left (\frac{A+B}{8 a^3 (a-x)^2}+\frac{A-B}{4 a^2 (a+x)^3}+\frac{A}{4 a^3 (a+x)^2}+\frac{3 A+B}{8 a^3 \left (a^2-x^2\right )}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac{A+B}{8 d (a-a \sin (c+d x))}-\frac{a (A-B)}{8 d (a+a \sin (c+d x))^2}-\frac{A}{4 d (a+a \sin (c+d x))}+\frac{(3 A+B) \operatorname{Subst}\left (\int \frac{1}{a^2-x^2} \, dx,x,a \sin (c+d x)\right )}{8 d}\\ &=\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{8 a d}+\frac{A+B}{8 d (a-a \sin (c+d x))}-\frac{a (A-B)}{8 d (a+a \sin (c+d x))^2}-\frac{A}{4 d (a+a \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.25448, size = 75, normalized size = 0.82 \[ \frac{\frac{A+B}{a-a \sin (c+d x)}+\frac{B-A}{a (\sin (c+d x)+1)^2}+\frac{(3 A+B) \tanh ^{-1}(\sin (c+d x))}{a}-\frac{2 A}{a \sin (c+d x)+a}}{8 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.134, size = 169, normalized size = 1.9 \begin{align*} -{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) A}{16\,da}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) B}{16\,da}}-{\frac{A}{8\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{B}{8\,da \left ( \sin \left ( dx+c \right ) -1 \right ) }}-{\frac{A}{4\,da \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{A}{8\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{B}{8\,da \left ( 1+\sin \left ( dx+c \right ) \right ) ^{2}}}+{\frac{3\,\ln \left ( 1+\sin \left ( dx+c \right ) \right ) A}{16\,da}}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) B}{16\,da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01793, size = 153, normalized size = 1.68 \begin{align*} \frac{\frac{{\left (3 \, A + B\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a} - \frac{{\left (3 \, A + B\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a} - \frac{2 \,{\left ({\left (3 \, A + B\right )} \sin \left (d x + c\right )^{2} +{\left (3 \, A + B\right )} \sin \left (d x + c\right ) - 2 \, A + 2 \, B\right )}}{a \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - a \sin \left (d x + c\right ) - a}}{16 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50304, size = 423, normalized size = 4.65 \begin{align*} -\frac{2 \,{\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2} -{\left ({\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) +{\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left ({\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) +{\left (3 \, A + B\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \,{\left (3 \, A + B\right )} \sin \left (d x + c\right ) - 2 \, A - 6 \, B}{16 \,{\left (a d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + a d \cos \left (d x + c\right )^{2}\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*} \frac{\int \frac{A \sec ^{3}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx + \int \frac{B \sin{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.33967, size = 198, normalized size = 2.18 \begin{align*} \frac{\frac{2 \,{\left (3 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a} - \frac{2 \,{\left (3 \, A + B\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a} + \frac{2 \,{\left (3 \, A \sin \left (d x + c\right ) + B \sin \left (d x + c\right ) - 5 \, A - 3 \, B\right )}}{a{\left (\sin \left (d x + c\right ) - 1\right )}} - \frac{9 \, A \sin \left (d x + c\right )^{2} + 3 \, B \sin \left (d x + c\right )^{2} + 26 \, A \sin \left (d x + c\right ) + 6 \, B \sin \left (d x + c\right ) + 21 \, A - B}{a{\left (\sin \left (d x + c\right ) + 1\right )}^{2}}}{32 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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