Optimal. Leaf size=93 \[ \frac {a^3 \log (a+b \sin (c+d x))}{b^2 d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}-\frac {\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac {\sin (c+d x)}{b d} \]
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Rubi [A] time = 0.18, antiderivative size = 93, 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, 1629} \[ \frac {a^3 \log (a+b \sin (c+d x))}{b^2 d \left (a^2-b^2\right )}-\frac {\log (1-\sin (c+d x))}{2 d (a+b)}-\frac {\log (\sin (c+d x)+1)}{2 d (a-b)}-\frac {\sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 1629
Rule 2837
Rubi steps
\begin {align*} \int \frac {\sin ^2(c+d x) \tan (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {b \operatorname {Subst}\left (\int \frac {x^3}{b^3 (a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^3}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {b^2}{2 (a+b) (b-x)}+\frac {a^3}{(a-b) (a+b) (a+x)}-\frac {b^2}{2 (a-b) (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 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^3 \log (a+b \sin (c+d x))}{b^2 \left (a^2-b^2\right ) d}-\frac {\sin (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 83, normalized size = 0.89 \[ -\frac {-\frac {2 a^3 \log (a+b \sin (c+d x))}{b^2 \left (a^2-b^2\right )}+\frac {\log (1-\sin (c+d x))}{a+b}+\frac {\log (\sin (c+d x)+1)}{a-b}+\frac {2 \sin (c+d x)}{b}}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.60, size = 100, normalized size = 1.08 \[ \frac {2 \, a^{3} \log \left (b \sin \left (d x + c\right ) + a\right ) - {\left (a b^{2} + b^{3}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (a b^{2} - b^{3}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{2} b^{2} - b^{4}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 85, normalized size = 0.91 \[ \frac {\frac {2 \, a^{3} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{2} - b^{4}} - \frac {\log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a - b} - \frac {\log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a + b} - \frac {2 \, \sin \left (d x + c\right )}{b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 95, normalized size = 1.02 \[ -\frac {\sin \left (d x +c \right )}{b d}-\frac {\ln \left (\sin \left (d x +c \right )-1\right )}{d \left (2 a +2 b \right )}+\frac {a^{3} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2} \left (a +b \right ) \left (a -b \right )}-\frac {\ln \left (1+\sin \left (d x +c \right )\right )}{d \left (2 a -2 b \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 82, normalized size = 0.88 \[ \frac {\frac {2 \, a^{3} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} b^{2} - b^{4}} - \frac {\log \left (\sin \left (d x + c\right ) + 1\right )}{a - b} - \frac {\log \left (\sin \left (d x + c\right ) - 1\right )}{a + b} - \frac {2 \, \sin \left (d x + c\right )}{b}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.27, size = 134, normalized size = 1.44 \[ -\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d\,\left (a+b\right )}-\frac {\sin \left (c+d\,x\right )}{b\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{d\,\left (a-b\right )}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b^2\,d}-\frac {a^3\,\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}{d\,\left (b^4-a^2\,b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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