Optimal. Leaf size=104 \[ \frac{b}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{2 a b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
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Rubi [A] time = 0.114654, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2668, 710, 801} \[ \frac{b}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{2 a b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac{\log (1-\sin (c+d x))}{2 d (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 2668
Rule 710
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{b \operatorname{Subst}\left (\int \frac{1}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{b \operatorname{Subst}\left (\int \frac{a-x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=\frac{b}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}+\frac{b \operatorname{Subst}\left (\int \left (\frac{a-b}{2 b (a+b) (b-x)}-\frac{2 a}{(a-b) (a+b) (a+x)}+\frac{a+b}{2 (a-b) b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{\left (a^2-b^2\right ) d}\\ &=-\frac{\log (1-\sin (c+d x))}{2 (a+b)^2 d}+\frac{\log (1+\sin (c+d x))}{2 (a-b)^2 d}-\frac{2 a b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2 d}+\frac{b}{\left (a^2-b^2\right ) d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.205833, size = 102, normalized size = 0.98 \[ \frac{b \left (\frac{1}{\left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac{\log (1-\sin (c+d x))}{2 b (a+b)^2}+\frac{\log (\sin (c+d x)+1)}{2 b (a-b)^2}-\frac{2 a \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.091, size = 101, normalized size = 1. \begin{align*}{\frac{b}{d \left ( a+b \right ) \left ( a-b \right ) \left ( a+b\sin \left ( dx+c \right ) \right ) }}-2\,{\frac{ab\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}-{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) }{2\,d \left ( a+b \right ) ^{2}}}+{\frac{\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{2\, \left ( a-b \right ) ^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.05678, size = 159, normalized size = 1.53 \begin{align*} -\frac{\frac{4 \, a b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{2 \, b}{a^{3} - a b^{2} +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )} - \frac{\log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{\log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.20017, size = 443, normalized size = 4.26 \begin{align*} \frac{2 \, a^{2} b - 2 \, b^{3} - 4 \,{\left (a b^{2} \sin \left (d x + c\right ) + a^{2} b\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) +{\left (a^{3} + 2 \, a^{2} b + a b^{2} +{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \sin \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) -{\left (a^{3} - 2 \, a^{2} b + a b^{2} +{\left (a^{2} b - 2 \, a b^{2} + 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{\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 = 1.14376, size = 198, normalized size = 1.9 \begin{align*} -\frac{\frac{4 \, a b^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} - \frac{\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \,{\left (2 \, a b^{2} \sin \left (d x + c\right ) + 3 \, a^{2} b - 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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