Optimal. Leaf size=114 \[ \frac {a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^2(c+d x)}{d}-\frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.12, antiderivative size = 114, 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 = {2836, 12, 88} \[ \frac {a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^2(c+d x)}{d}-\frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^3(c+d x) \cot ^2(c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^2 (a-x)^2 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^4}{x^2} \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-a^4+\frac {a^6}{x^2}+\frac {2 a^5}{x}-4 a^3 x-a^2 x^2+2 a x^3+x^4\right ) \, dx,x,a \sin (c+d x)\right )}{a^3 d}\\ &=-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d}-\frac {a^2 \sin (c+d x)}{d}-\frac {2 a^2 \sin ^2(c+d x)}{d}-\frac {a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}+\frac {a^2 \sin ^5(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 114, normalized size = 1.00 \[ \frac {a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin ^4(c+d x)}{2 d}-\frac {a^2 \sin ^3(c+d x)}{3 d}-\frac {2 a^2 \sin ^2(c+d x)}{d}-\frac {a^2 \sin (c+d x)}{d}-\frac {a^2 \csc (c+d x)}{d}+\frac {2 a^2 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 118, normalized size = 1.04 \[ -\frac {48 \, a^{2} \cos \left (d x + c\right )^{6} - 64 \, a^{2} \cos \left (d x + c\right )^{4} - 256 \, a^{2} \cos \left (d x + c\right )^{2} - 480 \, a^{2} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 512 \, a^{2} - 15 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{4} + 16 \, a^{2} \cos \left (d x + c\right )^{2} - 11 \, a^{2}\right )} \sin \left (d x + c\right )}{240 \, d \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 107, normalized size = 0.94 \[ \frac {6 \, a^{2} \sin \left (d x + c\right )^{5} + 15 \, a^{2} \sin \left (d x + c\right )^{4} - 10 \, a^{2} \sin \left (d x + c\right )^{3} - 60 \, a^{2} \sin \left (d x + c\right )^{2} + 60 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - 30 \, a^{2} \sin \left (d x + c\right ) - \frac {30 \, {\left (2 \, a^{2} \sin \left (d x + c\right ) + a^{2}\right )}}{\sin \left (d x + c\right )}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 130, normalized size = 1.14 \[ -\frac {32 a^{2} \sin \left (d x +c \right )}{15 d}-\frac {4 \sin \left (d x +c \right ) a^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}-\frac {16 \sin \left (d x +c \right ) a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{15 d}+\frac {\left (\cos ^{4}\left (d x +c \right )\right ) a^{2}}{2 d}+\frac {a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{d}+\frac {2 a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 94, normalized size = 0.82 \[ \frac {6 \, a^{2} \sin \left (d x + c\right )^{5} + 15 \, a^{2} \sin \left (d x + c\right )^{4} - 10 \, a^{2} \sin \left (d x + c\right )^{3} - 60 \, a^{2} \sin \left (d x + c\right )^{2} + 60 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) - 30 \, a^{2} \sin \left (d x + c\right ) - \frac {30 \, a^{2}}{\sin \left (d x + c\right )}}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.95, size = 333, normalized size = 2.92 \[ \frac {16\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d}-\frac {8\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{d}-\frac {16\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{d}+\frac {8\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{d}-\frac {2\,a^2\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {2\,a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}-\frac {2\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{3\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {176\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {328\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}+\frac {96\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {32\,a^2\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{5\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {5\,a^2\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}-\frac {a^2\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,d\,\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]
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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