Optimal. Leaf size=105 \[ \frac {b \csc (c+d x)}{a^2 d}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^3 b^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\sin (c+d x)}{b d} \]
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Rubi [A] time = 0.18, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2837, 12, 894} \[ -\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^3 b^2 d}-\frac {\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}+\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}+\frac {\sin (c+d x)}{b d} \]
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 ^3(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^3 \left (b^2-x^2\right )^2}{x^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {\left (b^2-x^2\right )^2}{x^3 (a+x)} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (1+\frac {b^4}{a x^3}-\frac {b^4}{a^2 x^2}+\frac {-2 a^2 b^2+b^4}{a^3 x}-\frac {\left (a^2-b^2\right )^2}{a^3 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {b \csc (c+d x)}{a^2 d}-\frac {\csc ^2(c+d x)}{2 a d}-\frac {\left (2 a^2-b^2\right ) \log (\sin (c+d x))}{a^3 d}-\frac {\left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^3 b^2 d}+\frac {\sin (c+d x)}{b d}\\ \end {align*}
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Mathematica [A] time = 0.39, size = 97, normalized size = 0.92 \[ \frac {\frac {2 b \csc (c+d x)}{a^2}+\frac {\frac {2 b^2 \left (b^2-2 a^2\right ) \log (\sin (c+d x))-2 \left (a^2-b^2\right )^2 \log (a+b \sin (c+d x))}{a^3}+2 b \sin (c+d x)}{b^2}-\frac {\csc ^2(c+d x)}{a}}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 174, normalized size = 1.66 \[ \frac {a^{2} b^{2} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 2 \, {\left (2 \, a^{2} b^{2} - b^{4} - {\left (2 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) + 2 \, {\left (a^{3} b \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )}{2 \, {\left (a^{3} b^{2} d \cos \left (d x + c\right )^{2} - a^{3} b^{2} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 130, normalized size = 1.24 \[ \frac {\frac {2 \, \sin \left (d x + c\right )}{b} - \frac {2 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{3}} - \frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{3} b^{2}} + \frac {6 \, a^{2} \sin \left (d x + c\right )^{2} - 3 \, b^{2} \sin \left (d x + c\right )^{2} + 2 \, a b \sin \left (d x + c\right ) - a^{2}}{a^{3} \sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.48, size = 140, normalized size = 1.33 \[ \frac {\sin \left (d x +c \right )}{b d}-\frac {a \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}}+\frac {2 \ln \left (a +b \sin \left (d x +c \right )\right )}{d a}-\frac {b^{2} \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{3} d}-\frac {1}{2 d a \sin \left (d x +c \right )^{2}}-\frac {2 \ln \left (\sin \left (d x +c \right )\right )}{a d}+\frac {b^{2} \ln \left (\sin \left (d x +c \right )\right )}{a^{3} d}+\frac {b}{d \,a^{2} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 99, normalized size = 0.94 \[ \frac {\frac {2 \, \sin \left (d x + c\right )}{b} - \frac {2 \, {\left (2 \, a^{2} - b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{3}} - \frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{3} b^{2}} + \frac {2 \, b \sin \left (d x + c\right ) - a}{a^{2} \sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.98, size = 238, normalized size = 2.27 \[ \frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,a^2\,d}-\frac {\frac {a}{2}-2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{2}-\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (4\,a^2+b^2\right )}{b}}{d\,\left (4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a\,d}+\frac {a\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-b^2\right )}{a^3\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}{a^3\,b^2\,d} \]
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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