Optimal. Leaf size=120 \[ \frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sin (c+d x))}+\frac {\left (3 a^2+b^2\right ) \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a^2 b^4 d}+\frac {\log (\sin (c+d x))}{a^2 d}-\frac {2 a \sin (c+d x)}{b^3 d}+\frac {\sin ^2(c+d x)}{2 b^2 d} \]
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Rubi [A] time = 0.16, antiderivative size = 120, 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} \[ \frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sin (c+d x))}+\frac {\left (3 a^2+b^2\right ) \left (a^2-b^2\right ) \log (a+b \sin (c+d x))}{a^2 b^4 d}+\frac {\log (\sin (c+d x))}{a^2 d}-\frac {2 a \sin (c+d x)}{b^3 d}+\frac {\sin ^2(c+d x)}{2 b^2 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \cot (c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b \left (b^2-x^2\right )^2}{x (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 (a+x)^2} \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-2 a+\frac {b^4}{a^2 x}+x-\frac {\left (a^2-b^2\right )^2}{a (a+x)^2}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right )}{a^2 (a+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{b^4 d}\\ &=\frac {\log (\sin (c+d x))}{a^2 d}+\frac {\left (a^2-b^2\right ) \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{a^2 b^4 d}-\frac {2 a \sin (c+d x)}{b^3 d}+\frac {\sin ^2(c+d x)}{2 b^2 d}+\frac {\left (a^2-b^2\right )^2}{a b^4 d (a+b \sin (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.52, size = 111, normalized size = 0.92 \[ \frac {\frac {2 \left (a^2-b^2\right )^2}{a b^4 (a+b \sin (c+d x))}+\frac {2 (a-b) (a+b) \left (3 a^2+b^2\right ) \log (a+b \sin (c+d x))}{a^2 b^4}+\frac {2 \log (\sin (c+d x))}{a^2}-\frac {4 a \sin (c+d x)}{b^3}+\frac {\sin ^2(c+d x)}{b^2}}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.81, size = 189, normalized size = 1.58 \[ \frac {6 \, a^{3} b^{2} \cos \left (d x + c\right )^{2} + 4 \, a^{5} - 15 \, a^{3} b^{2} + 4 \, a b^{4} + 4 \, {\left (3 \, a^{5} - 2 \, a^{3} b^{2} - a b^{4} + {\left (3 \, a^{4} b - 2 \, a^{2} b^{3} - b^{5}\right )} \sin \left (d x + c\right )\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + 4 \, {\left (b^{5} \sin \left (d x + c\right ) + a b^{4}\right )} \log \left (-\frac {1}{2} \, \sin \left (d x + c\right )\right ) - {\left (2 \, a^{2} b^{3} \cos \left (d x + c\right )^{2} + 8 \, a^{4} b - a^{2} b^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{2} b^{5} d \sin \left (d x + c\right ) + a^{3} b^{4} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 154, normalized size = 1.28 \[ \frac {\frac {2 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a^{2}} + \frac {b^{2} \sin \left (d x + c\right )^{2} - 4 \, a b \sin \left (d x + c\right )}{b^{4}} + \frac {2 \, {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{4}} - \frac {2 \, {\left (3 \, a^{4} b \sin \left (d x + c\right ) - 2 \, a^{2} b^{3} \sin \left (d x + c\right ) - b^{5} \sin \left (d x + c\right ) + 2 \, a^{5} - 2 \, a b^{4}\right )}}{{\left (b \sin \left (d x + c\right ) + a\right )} a^{2} b^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.76, size = 169, normalized size = 1.41 \[ \frac {\sin ^{2}\left (d x +c \right )}{2 b^{2} d}-\frac {2 a \sin \left (d x +c \right )}{b^{3} d}+\frac {a^{3}}{d \,b^{4} \left (a +b \sin \left (d x +c \right )\right )}-\frac {2 a}{d \,b^{2} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{a d \left (a +b \sin \left (d x +c \right )\right )}+\frac {3 a^{2} \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{4}}-\frac {2 \ln \left (a +b \sin \left (d x +c \right )\right )}{d \,b^{2}}-\frac {\ln \left (a +b \sin \left (d x +c \right )\right )}{a^{2} d}+\frac {\ln \left (\sin \left (d x +c \right )\right )}{a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 118, normalized size = 0.98 \[ \frac {\frac {2 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}}{a b^{5} \sin \left (d x + c\right ) + a^{2} b^{4}} + \frac {2 \, \log \left (\sin \left (d x + c\right )\right )}{a^{2}} + \frac {b \sin \left (d x + c\right )^{2} - 4 \, a \sin \left (d x + c\right )}{b^{3}} + \frac {2 \, {\left (3 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{2} b^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.08, size = 338, normalized size = 2.82 \[ \frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^2\,d}-\frac {\frac {6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{b^2}+\frac {6\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{b^2}+\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^4-3\,a^2\,b^2+b^4\right )}{a^2\,b^3}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (3\,a^4-2\,a^2\,b^2+b^4\right )}{a^2\,b^3}+\frac {2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (3\,a^4-2\,a^2\,b^2+b^4\right )}{a^2\,b^3}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}-\frac {\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )\,\left (3\,a^2-2\,b^2\right )}{b^4\,d}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (-3\,a^4+2\,a^2\,b^2+b^4\right )}{a^2\,b^4\,d} \]
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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