Optimal. Leaf size=110 \[ -\frac {b^3 \log (a+b \sin (c+d x))}{a^2 d \left (a^2-b^2\right )}-\frac {b \log (\sin (c+d x))}{a^2 d}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac {\csc (c+d x)}{a d} \]
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Rubi [A] time = 0.19, antiderivative size = 110, normalized size of antiderivative = 1.00, 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} \[ -\frac {b^3 \log (a+b \sin (c+d x))}{a^2 d \left (a^2-b^2\right )}-\frac {b \log (\sin (c+d x))}{a^2 d}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac {\csc (c+d x)}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\csc ^2(c+d x) \sec (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {b^2}{x^2 (a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{x^2 (a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {b^3 \operatorname {Subst}\left (\int \left (\frac {1}{2 b^3 (a+b) (b-x)}+\frac {1}{a b^2 x^2}-\frac {1}{a^2 b^2 x}-\frac {1}{a^2 (a-b) (a+b) (a+x)}-\frac {1}{2 b^3 (-a+b) (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\csc (c+d x)}{a d}-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}-\frac {b \log (\sin (c+d x))}{a^2 d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {b^3 \log (a+b \sin (c+d x))}{a^2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 97, normalized size = 0.88 \[ \frac {\frac {2 b^3 \log (a+b \sin (c+d x))}{a^2 \left (b^2-a^2\right )}-\frac {2 b \log (\sin (c+d x))}{a^2}-\frac {\log (1-\sin (c+d x))}{a+b}+\frac {\log (\sin (c+d x)+1)}{a-b}-\frac {2 \csc (c+d x)}{a}}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 143, normalized size = 1.30 \[ -\frac {2 \, b^{3} \log \left (b \sin \left (d x + c\right ) + a\right ) \sin \left (d x + c\right ) + 2 \, a^{3} - 2 \, a b^{2} + 2 \, {\left (a^{2} b - b^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - {\left (a^{3} + a^{2} b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (a^{3} - a^{2} b\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{2 \, {\left (a^{4} - a^{2} b^{2}\right )} d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 113, normalized size = 1.03 \[ -\frac {\frac {2 \, b^{4} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - a^{2} b^{3}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a + b} + \frac {2 \, b \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} - \frac {2 \, {\left (b \sin \left (d x + c\right ) - a\right )}}{a^{2} \sin \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.42, size = 113, normalized size = 1.03 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{d \left (2 a +2 b \right )}-\frac {b^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,a^{2} \left (a +b \right ) \left (a -b \right )}-\frac {1}{d a \sin \left (d x +c \right )}-\frac {b \ln \left (\sin \left (d x +c \right )\right )}{a^{2} d}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{d \left (2 a -2 b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 95, normalized size = 0.86 \[ -\frac {\frac {2 \, b^{3} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - a^{2} b^{2}} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} + \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a + b} + \frac {2 \, b \log \left (\sin \left (d x + c\right )\right )}{a^{2}} + \frac {2}{a \sin \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.92, size = 98, normalized size = 0.89 \[ -\frac {\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )}{2\,\left (a+b\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )}{2\,\left (a-b\right )}+\frac {1}{a\,\sin \left (c+d\,x\right )}+\frac {b\,\ln \left (\sin \left (c+d\,x\right )\right )}{a^2}+\frac {b^3\,\ln \left (a+b\,\sin \left (c+d\,x\right )\right )}{a^4-a^2\,b^2}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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