Optimal. Leaf size=156 \[ -\frac{b^4 \log (a+b \sin (c+d x))}{a d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}+\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}+\frac{\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.241525, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {2837, 12, 894} \[ -\frac{b^4 \log (a+b \sin (c+d x))}{a d \left (a^2-b^2\right )^2}+\frac{1}{4 d (a+b) (1-\sin (c+d x))}+\frac{1}{4 d (a-b) (\sin (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sin (c+d x))}{4 d (a+b)^2}-\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{4 d (a-b)^2}+\frac{\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 2837
Rule 12
Rule 894
Rubi steps
\begin{align*} \int \frac{\csc (c+d x) \sec ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{b}{x (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^4 \operatorname{Subst}\left (\int \frac{1}{x (a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac{b^4 \operatorname{Subst}\left (\int \left (\frac{1}{4 b^3 (a+b) (b-x)^2}+\frac{2 a+3 b}{4 b^4 (a+b)^2 (b-x)}+\frac{1}{a b^4 x}-\frac{1}{a (a-b)^2 (a+b)^2 (a+x)}-\frac{1}{4 (a-b) b^3 (b+x)^2}+\frac{-2 a+3 b}{4 (a-b)^2 b^4 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac{(2 a+3 b) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac{\log (\sin (c+d x))}{a d}-\frac{(2 a-3 b) \log (1+\sin (c+d x))}{4 (a-b)^2 d}-\frac{b^4 \log (a+b \sin (c+d x))}{a \left (a^2-b^2\right )^2 d}+\frac{1}{4 (a+b) d (1-\sin (c+d x))}+\frac{1}{4 (a-b) d (1+\sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.681881, size = 151, normalized size = 0.97 \[ \frac{b^4 \left (-\frac{1}{b^4 (a+b) (\sin (c+d x)-1)}+\frac{1}{b^4 (a-b) (\sin (c+d x)+1)}-\frac{(2 a+3 b) \log (1-\sin (c+d x))}{b^4 (a+b)^2}+\frac{4 \log (\sin (c+d x))}{a b^4}-\frac{(2 a-3 b) \log (\sin (c+d x)+1)}{b^4 (a-b)^2}-\frac{4 \log (a+b \sin (c+d x))}{a (a-b)^2 (a+b)^2}\right )}{4 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.089, size = 181, normalized size = 1.2 \begin{align*} -{\frac{{b}^{4}\ln \left ( a+b\sin \left ( dx+c \right ) \right ) }{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2}a}}-{\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}{2\,d \left ( a+b \right ) ^{2}}}-{\frac{3\,\ln \left ( \sin \left ( dx+c \right ) -1 \right ) b}{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 ) }{2\, \left ( a-b \right ) ^{2}d}}+{\frac{3\,b\ln \left ( 1+\sin \left ( dx+c \right ) \right ) }{4\, \left ( a-b \right ) ^{2}d}}+{\frac{\ln \left ( \sin \left ( dx+c \right ) \right ) }{da}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.999931, size = 211, normalized size = 1.35 \begin{align*} -\frac{\frac{4 \, b^{4} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4}} + \frac{{\left (2 \, a - 3 \, b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{{\left (2 \, a + 3 \, b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{2 \,{\left (b \sin \left (d x + c\right ) - a\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2} + b^{2}} - \frac{4 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.55277, size = 502, normalized size = 3.22 \begin{align*} -\frac{4 \, b^{4} \cos \left (d x + c\right )^{2} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, a^{4} + 2 \, a^{2} b^{2} - 4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} \log \left (-\frac{1}{2} \, \sin \left (d x + c\right )\right ) +{\left (2 \, a^{4} + a^{3} b - 4 \, a^{2} b^{2} - 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) +{\left (2 \, a^{4} - a^{3} b - 4 \, a^{2} b^{2} + 3 \, a b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \,{\left (a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )}{4 \,{\left (a^{5} - 2 \, a^{3} b^{2} + a 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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.24238, size = 284, normalized size = 1.82 \begin{align*} -\frac{\frac{4 \, b^{5} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{5} b - 2 \, a^{3} b^{3} + a b^{5}} + \frac{{\left (2 \, a - 3 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac{{\left (2 \, a + 3 \, b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac{4 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac{2 \,{\left (a^{3} \sin \left (d x + c\right )^{2} - 2 \, a b^{2} \sin \left (d x + c\right )^{2} + a^{2} b \sin \left (d x + c\right ) - b^{3} \sin \left (d x + c\right ) - 2 \, a^{3} + 3 \, a b^{2}\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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