Optimal. Leaf size=99 \[ -\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d}+\frac {2 \sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{a d}-\frac {\sin (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
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Rubi [A] time = 0.10, antiderivative size = 99, 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 = {2836, 12, 88} \[ -\frac {\sin ^5(c+d x)}{5 a d}+\frac {\sin ^4(c+d x)}{4 a d}+\frac {2 \sin ^3(c+d x)}{3 a d}-\frac {\sin ^2(c+d x)}{a d}-\frac {\sin (c+d x)}{a d}+\frac {\log (\sin (c+d x))}{a d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x) \cot (c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^3 (a+x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^2}{x} \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^4+\frac {a^5}{x}-2 a^3 x+2 a^2 x^2+a x^3-x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^6 d}\\ &=\frac {\log (\sin (c+d x))}{a d}-\frac {\sin (c+d x)}{a d}-\frac {\sin ^2(c+d x)}{a d}+\frac {2 \sin ^3(c+d x)}{3 a d}+\frac {\sin ^4(c+d x)}{4 a d}-\frac {\sin ^5(c+d x)}{5 a d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.69 \[ \frac {-12 \sin ^5(c+d x)+15 \sin ^4(c+d x)+40 \sin ^3(c+d x)-60 \sin ^2(c+d x)-60 \sin (c+d x)+60 \log (\sin (c+d x))}{60 a d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 70, normalized size = 0.71 \[ \frac {15 \, \cos \left (d x + c\right )^{4} + 30 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (3 \, \cos \left (d x + c\right )^{4} + 4 \, \cos \left (d x + c\right )^{2} + 8\right )} \sin \left (d x + c\right ) + 60 \, \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{60 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 88, normalized size = 0.89 \[ \frac {\frac {60 \, \log \left ({\left | \sin \left (d x + c\right ) \right |}\right )}{a} - \frac {12 \, a^{4} \sin \left (d x + c\right )^{5} - 15 \, a^{4} \sin \left (d x + c\right )^{4} - 40 \, a^{4} \sin \left (d x + c\right )^{3} + 60 \, a^{4} \sin \left (d x + c\right )^{2} + 60 \, a^{4} \sin \left (d x + c\right )}{a^{5}}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 94, normalized size = 0.95 \[ \frac {\ln \left (\sin \left (d x +c \right )\right )}{a d}-\frac {\sin \left (d x +c \right )}{a d}-\frac {\sin ^{2}\left (d x +c \right )}{a d}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{3 d a}+\frac {\sin ^{4}\left (d x +c \right )}{4 d a}-\frac {\sin ^{5}\left (d x +c \right )}{5 d a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 71, normalized size = 0.72 \[ -\frac {\frac {12 \, \sin \left (d x + c\right )^{5} - 15 \, \sin \left (d x + c\right )^{4} - 40 \, \sin \left (d x + c\right )^{3} + 60 \, \sin \left (d x + c\right )^{2} + 60 \, \sin \left (d x + c\right )}{a} - \frac {60 \, \log \left (\sin \left (d x + c\right )\right )}{a}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.29, size = 140, normalized size = 1.41 \[ \frac {\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{a\,d}-\frac {\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{a\,d}-\frac {8\,\sin \left (c+d\,x\right )}{15\,a\,d}+\frac {{\cos \left (c+d\,x\right )}^2}{2\,a\,d}+\frac {{\cos \left (c+d\,x\right )}^4}{4\,a\,d}-\frac {4\,{\cos \left (c+d\,x\right )}^2\,\sin \left (c+d\,x\right )}{15\,a\,d}-\frac {{\cos \left (c+d\,x\right )}^4\,\sin \left (c+d\,x\right )}{5\,a\,d} \]
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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