Optimal. Leaf size=59 \[ \frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a b^2 d}+\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{b d} \]
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Rubi [A] time = 0.11, antiderivative size = 59, 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} \[ \frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a b^2 d}+\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b \left (b^2-x^2\right )}{x (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2-x^2}{x (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-1+\frac {b^2}{a x}+\frac {a^2-b^2}{a (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\log (\sin (c+d x))}{a d}+\frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a b^2 d}-\frac {\sin (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 53, normalized size = 0.90 \[ \frac {\left (a^2-b^2\right ) \log (a+b \sin (c+d x))-a b \sin (c+d x)+b^2 \log (\sin (c+d x))}{a b^2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 55, normalized size = 0.93 \[ \frac {b^{2} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - a b \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.72, size = 56, normalized size = 0.95 \[ \frac {\frac {\log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {\sin \left (d x + c\right )}{b} + \frac {{\left (a^{2} - b^{2}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 68, normalized size = 1.15 \[ -\frac {\sin \left (d x +c \right )}{b d}+\frac {a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{d a}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.59, size = 54, normalized size = 0.92 \[ \frac {\frac {\log \left (\sin \left (d x + c\right )\right )}{a} - \frac {\sin \left (d x + c\right )}{b} + \frac {{\left (a^{2} - b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a b^{2}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.69, size = 98, normalized size = 1.66 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a\,d}-\frac {\sin \left (c+d\,x\right )}{b\,d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (\frac {a}{b^2}-\frac {1}{a}\right )}{d}-\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b^2\,d} \]
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 \[ \int \frac {\cos ^{2}{\left (c + d x \right )} \cot {\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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