Optimal. Leaf size=131 \[ \frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))} \]
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Rubi [A] time = 0.20, antiderivative size = 131, 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} \[ \frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (2 a^2 b^2+a^4-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d} \]
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))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b^3 \left (b^2-x^2\right )^2}{x^3 (a+x)^2} \, 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)^2} \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^4}{a^2 x^3}-\frac {2 b^4}{a^3 x^2}+\frac {-2 a^2 b^2+3 b^4}{a^4 x}-\frac {\left (a^2-b^2\right )^2}{a^3 (a+x)^2}+\frac {a^4+2 a^2 b^2-3 b^4}{a^4 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^2 d}\\ &=\frac {2 b \csc (c+d x)}{a^3 d}-\frac {\csc ^2(c+d x)}{2 a^2 d}-\frac {\left (2 a^2-3 b^2\right ) \log (\sin (c+d x))}{a^4 d}+\frac {\left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{a^4 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a^3 b^2 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 116, normalized size = 0.89 \[ \frac {\frac {2 a \left (a^2-b^2\right )^2}{b^2 (a+b \sin (c+d x))}-2 \left (2 a^2-3 b^2\right ) \log (\sin (c+d x))-a^2 \csc ^2(c+d x)+\frac {2 \left (a^4+2 a^2 b^2-3 b^4\right ) \log (a+b \sin (c+d x))}{b^2}+4 a b \csc (c+d x)}{2 a^4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.16, size = 335, normalized size = 2.56 \[ -\frac {3 \, a^{2} b^{3} \sin \left (d x + c\right ) + 2 \, a^{5} - 5 \, a^{3} b^{2} + 6 \, a b^{4} - 2 \, {\left (a^{5} - 2 \, a^{3} b^{2} + 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4} - {\left (a^{5} + 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (a^{4} b + 2 \, a^{2} b^{3} - 3 \, b^{5} - {\left (a^{4} b + 2 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) - 2 \, {\left (2 \, a^{3} b^{2} - 3 \, a b^{4} - {\left (2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (2 \, a^{2} b^{3} - 3 \, b^{5} - {\left (2 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right )}{2 \, {\left (a^{5} b^{2} d \cos \left (d x + c\right )^{2} - a^{5} b^{2} d + {\left (a^{4} b^{3} d \cos \left (d x + c\right )^{2} - a^{4} b^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 190, normalized size = 1.45 \[ -\frac {\frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{4}} - \frac {2 \, {\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b^{2}} + \frac {2 \, {\left (a^{4} \sin \left (d x + c\right ) + 2 \, a^{2} b^{2} \sin \left (d x + c\right ) - 3 \, b^{4} \sin \left (d x + c\right ) + 4 \, a^{3} b - 4 \, a b^{3}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{4} b} - \frac {6 \, a^{2} \sin \left (d x + c\right )^{2} - 9 \, b^{2} \sin \left (d x + c\right )^{2} + 4 \, a b \sin \left (d x + c\right ) - a^{2}}{a^{4} \sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 189, normalized size = 1.44 \[ \frac {a}{d \,b^{2} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2}{a d \left (a +b \sin \left (d x +c \right )\right )}+\frac {b^{2}}{d \,a^{3} \left (a +b \sin \left (d x +c \right )\right )}+\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}}+\frac {2 \ln \left (a +b \sin \left (d x +c \right )\right )}{a^{2} d}-\frac {3 b^{2} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,a^{4}}-\frac {1}{2 d \,a^{2} \sin \left (d x +c \right )^{2}}-\frac {2 \ln \left (\sin \left (d x +c \right )\right )}{a^{2} d}+\frac {3 \ln \left (\sin \left (d x +c \right )\right ) b^{2}}{d \,a^{4}}+\frac {2 b}{d \,a^{3} \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 147, normalized size = 1.12 \[ \frac {\frac {3 \, a b^{3} \sin \left (d x + c\right ) - a^{2} b^{2} + 2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{2}}{a^{3} b^{3} \sin \left (d x + c\right )^{3} + a^{4} b^{2} \sin \left (d x + c\right )^{2}} - \frac {2 \, {\left (2 \, a^{2} - 3 \, b^{2}\right )} \log \left (\sin \left (d x + c\right )\right )}{a^{4}} + \frac {2 \, {\left (a^{4} + 2 \, a^{2} b^{2} - 3 \, b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.92, size = 280, normalized size = 2.14 \[ \frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^3\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^2}{2}-8\,b^2\right )+\frac {a^2}{2}-3\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (2\,a^4-5\,a^2\,b^2+2\,b^4\right )}{a\,b}}{d\,\left (4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,b\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{b^2\,d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^2\,d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a^2-3\,b^2\right )}{a^4\,d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4+2\,a^2\,b^2-3\,b^4\right )}{a^4\,b^2\,d} \]
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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