Optimal. Leaf size=116 \[ \frac{a^2 b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left (a^2-b^2\right )}+\frac{a \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{a \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]
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Rubi [A] time = 0.230734, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {2837, 12, 1647, 801} \[ \frac{a^2 b \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac{\sec ^2(c+d x) (b-a \sin (c+d x))}{2 d \left (a^2-b^2\right )}+\frac{a \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{a \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 1647
Rule 801
Rubi steps
\begin{align*} \int \frac{\sec (c+d x) \tan ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{x^2}{b^2 (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{x^2}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{\sec ^2(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{2 d}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^2 b^2}{a^2-b^2}+\frac{a b^2 x}{a^2-b^2}}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 b d}\\ &=-\frac{\sec ^2(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{2 d}+\frac{\operatorname{Subst}\left (\int \left (-\frac{a b}{2 (a+b)^2 (b-x)}+\frac{2 a^2 b^2}{(a-b)^2 (a+b)^2 (a+x)}-\frac{a b}{2 (a-b)^2 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 b d}\\ &=\frac{a \log (1-\sin (c+d x))}{4 (a+b)^2 d}-\frac{a \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac{a^2 b \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2 d}-\frac{\sec ^2(c+d x) \left (\frac{b}{a^2-b^2}-\frac{a \sin (c+d x)}{a^2-b^2}\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.455436, size = 108, normalized size = 0.93 \[ -\frac{-\frac{4 a^2 b \log (a+b \sin (c+d x))}{(a-b)^2 (a+b)^2}+\frac{1}{(a+b) (\sin (c+d x)-1)}+\frac{1}{(a-b) (\sin (c+d x)+1)}-\frac{a \log (1-\sin (c+d x))}{(a+b)^2}+\frac{a \log (\sin (c+d x)+1)}{(a-b)^2}}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.075, size = 123, normalized size = 1.1 \begin{align*}{\frac{{a}^{2}b\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}}}-{\frac{1}{d \left ( 4\,a+4\,b \right ) \left ( \sin \left ( dx+c \right ) -1 \right ) }}+{\frac{\ln \left ( \sin \left ( dx+c \right ) -1 \right ) a}{4\,d \left ( a+b \right ) ^{2}}}-{\frac{1}{d \left ( 4\,a-4\,b \right ) \left ( 1+\sin \left ( dx+c \right ) \right ) }}-{\frac{a\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{4\, \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 = 0.980875, size = 178, normalized size = 1.53 \begin{align*} \frac{\frac{4 \, a^{2} b \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac{a \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{a \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \,{\left (a \sin \left (d x + c\right ) - b\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2} + b^{2}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.09375, size = 369, normalized size = 3.18 \begin{align*} \frac{4 \, a^{2} b \cos \left (d x + c\right )^{2} \log \left (b \sin \left (d x + c\right ) + a\right ) -{\left (a^{3} + 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, a^{2} b + 2 \, b^{3} + 2 \,{\left (a^{3} - a b^{2}\right )} \sin \left (d x + c\right )}{4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d \cos \left (d x + c\right )^{2}} \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{\sin ^{2}{\left (c + d x \right )} \sec ^{3}{\left (c + d x \right )}}{a + b \sin{\left (c + d x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23537, size = 227, normalized size = 1.96 \begin{align*} \frac{\frac{4 \, a^{2} 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{a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} + \frac{2 \,{\left (a^{2} b \sin \left (d x + c\right )^{2} - a^{3} \sin \left (d x + c\right ) + a b^{2} \sin \left (d x + c\right ) - b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (\sin \left (d x + c\right )^{2} - 1\right )}}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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