Optimal. Leaf size=80 \[ -\frac {a^2 \log (a+b \sin (c+d x))}{b d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)} \]
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Rubi [A] time = 0.16, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2837, 12, 1629} \[ -\frac {a^2 \log (a+b \sin (c+d x))}{b d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}+\frac {\log (\sin (c+d x)+1)}{2 d (a-b)} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1629
Rule 2837
Rubi steps
\begin {align*} \int \frac {\sin (c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {x^2}{b^2 (a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b}{2 (a+b) (b-x)}-\frac {a^2}{(a-b) (a+b) (a+x)}+\frac {b}{2 (a-b) (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b d}\\ &=-\frac {\log (1-\sin (c+d x))}{2 (a+b) d}+\frac {\log (1+\sin (c+d x))}{2 (a-b) d}-\frac {a^2 \log (a+b \sin (c+d x))}{b \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 72, normalized size = 0.90 \[ \frac {-2 a^2 \log (a+b \sin (c+d x))-b (a-b) \log (1-\sin (c+d x))+b (a+b) \log (\sin (c+d x)+1)}{2 b d (a-b) (a+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.05, size = 74, normalized size = 0.92 \[ -\frac {2 \, a^{2} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (a b + b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left (a b - b^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right )}{2 \, {\left (a^{2} b - b^{3}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 71, normalized size = 0.89 \[ -\frac {\frac {2 \, a^{2} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b - b^{3}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} + \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a + b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 81, normalized size = 1.01 \[ -\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{d \left (2 a +2 b \right )}-\frac {a^{2} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \left (a +b \right ) \left (a -b \right ) b}+\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{d \left (2 a -2 b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.38, size = 68, normalized size = 0.85 \[ -\frac {\frac {2 \, a^{2} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} b - b^{3}} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} + \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a + b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.94, size = 117, normalized size = 1.46 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{d\,\left (a-b\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d\,\left (a+b\right )}+\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b\,d}-\frac {a^2\,\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{b\,d\,\left (a^2-b^2\right )} \]
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 \[ \int \frac {\sin ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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