Optimal. Leaf size=114 \[ -\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}-\frac {3 a \sin (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.09, antiderivative size = 114, 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} \[ -\frac {a \sin ^6(c+d x)}{6 d}-\frac {a \sin ^5(c+d x)}{5 d}+\frac {3 a \sin ^4(c+d x)}{4 d}+\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}-\frac {3 a \sin (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^5(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x)) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)^3 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^7 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^3 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-3 a^5+\frac {a^7}{x^2}+\frac {a^6}{x}-3 a^4 x+3 a^3 x^2+3 a^2 x^3-a x^4-x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=-\frac {a \csc (c+d x)}{d}+\frac {a \log (\sin (c+d x))}{d}-\frac {3 a \sin (c+d x)}{d}-\frac {3 a \sin ^2(c+d x)}{2 d}+\frac {a \sin ^3(c+d x)}{d}+\frac {3 a \sin ^4(c+d x)}{4 d}-\frac {a \sin ^5(c+d x)}{5 d}-\frac {a \sin ^6(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 102, normalized size = 0.89 \[ -\frac {a \sin ^5(c+d x)}{5 d}+\frac {a \sin ^3(c+d x)}{d}-\frac {3 a \sin (c+d x)}{d}-\frac {a \csc (c+d x)}{d}+\frac {a \left (-2 \sin ^6(c+d x)+9 \sin ^4(c+d x)-18 \sin ^2(c+d x)+12 \log (\sin (c+d x))\right )}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.70, size = 113, normalized size = 0.99 \[ \frac {48 \, a \cos \left (d x + c\right )^{6} + 96 \, a \cos \left (d x + c\right )^{4} + 384 \, a \cos \left (d x + c\right )^{2} + 240 \, a \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 5 \, {\left (8 \, a \cos \left (d x + c\right )^{6} + 12 \, a \cos \left (d x + c\right )^{4} + 24 \, a \cos \left (d x + c\right )^{2} - 19 \, a\right )} \sin \left (d x + c\right ) - 768 \, a}{240 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 101, normalized size = 0.89 \[ -\frac {10 \, a \sin \left (d x + c\right )^{6} + 12 \, a \sin \left (d x + c\right )^{5} - 45 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} + 90 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 180 \, a \sin \left (d x + c\right ) + \frac {60 \, {\left (a \sin \left (d x + c\right ) + a\right )}}{\sin \left (d x + c\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.29, size = 150, normalized size = 1.32 \[ \frac {a \left (\cos ^{6}\left (d x +c \right )\right )}{6 d}+\frac {a \left (\cos ^{4}\left (d x +c \right )\right )}{4 d}+\frac {a \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a \left (\cos ^{8}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )}-\frac {16 a \sin \left (d x +c \right )}{5 d}-\frac {\left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{d}-\frac {6 a \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}-\frac {8 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a}{5 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.36, size = 91, normalized size = 0.80 \[ -\frac {10 \, a \sin \left (d x + c\right )^{6} + 12 \, a \sin \left (d x + c\right )^{5} - 45 \, a \sin \left (d x + c\right )^{4} - 60 \, a \sin \left (d x + c\right )^{3} + 90 \, a \sin \left (d x + c\right )^{2} - 60 \, a \log \left (\sin \left (d x + c\right )\right ) + 180 \, a \sin \left (d x + c\right ) + \frac {60 \, a}{\sin \left (d x + c\right )}}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.33, size = 340, normalized size = 2.98 \[ \frac {a\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {a\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}-\frac {6\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}+\frac {18\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {104\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3\,d}+\frac {44\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}-\frac {32\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{d}+\frac {32\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3\,d}-\frac {13\,a\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {14\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {112\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {136\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {96\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {32\,a\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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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