Optimal. Leaf size=119 \[ \frac {a^2 \sin ^6(c+d x)}{6 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^4(c+d x)}{4 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}-\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 119, 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} \[ \frac {a^2 \sin ^6(c+d x)}{6 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d}-\frac {a^2 \sin ^4(c+d x)}{4 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}-\frac {a^2 \sin ^2(c+d x)}{2 d}+\frac {2 a^2 \sin (c+d x)}{d}+\frac {a^2 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 88
Rule 2836
Rubi steps
\begin {align*} \int \cos ^4(c+d x) \cot (c+d x) (a+a \sin (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a (a-x)^2 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^5 d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {(a-x)^2 (a+x)^4}{x} \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (2 a^5+\frac {a^6}{x}-a^4 x-4 a^3 x^2-a^2 x^3+2 a x^4+x^5\right ) \, dx,x,a \sin (c+d x)\right )}{a^4 d}\\ &=\frac {a^2 \log (\sin (c+d x))}{d}+\frac {2 a^2 \sin (c+d x)}{d}-\frac {a^2 \sin ^2(c+d x)}{2 d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}-\frac {a^2 \sin ^4(c+d x)}{4 d}+\frac {2 a^2 \sin ^5(c+d x)}{5 d}+\frac {a^2 \sin ^6(c+d x)}{6 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 78, normalized size = 0.66 \[ \frac {a^2 \left (10 \sin ^6(c+d x)+24 \sin ^5(c+d x)-15 \sin ^4(c+d x)-80 \sin ^3(c+d x)-30 \sin ^2(c+d x)+120 \sin (c+d x)+60 \log (\sin (c+d x))\right )}{60 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.76, size = 99, normalized size = 0.83 \[ -\frac {10 \, a^{2} \cos \left (d x + c\right )^{6} - 15 \, a^{2} \cos \left (d x + c\right )^{4} - 30 \, a^{2} \cos \left (d x + c\right )^{2} - 60 \, a^{2} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) - 8 \, {\left (3 \, a^{2} \cos \left (d x + c\right )^{4} + 4 \, a^{2} \cos \left (d x + c\right )^{2} + 8 \, a^{2}\right )} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 95, normalized size = 0.80 \[ \frac {10 \, a^{2} \sin \left (d x + c\right )^{6} + 24 \, a^{2} \sin \left (d x + c\right )^{5} - 15 \, a^{2} \sin \left (d x + c\right )^{4} - 80 \, a^{2} \sin \left (d x + c\right )^{3} - 30 \, a^{2} \sin \left (d x + c\right )^{2} + 60 \, a^{2} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 120 \, a^{2} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.47, size = 122, normalized size = 1.03 \[ -\frac {a^{2} \left (\cos ^{6}\left (d x +c \right )\right )}{6 d}+\frac {16 a^{2} \sin \left (d x +c \right )}{15 d}+\frac {2 \sin \left (d x +c \right ) a^{2} \left (\cos ^{4}\left (d x +c \right )\right )}{5 d}+\frac {8 \sin \left (d x +c \right ) a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{15 d}+\frac {\left (\cos ^{4}\left (d x +c \right )\right ) a^{2}}{4 d}+\frac {a^{2} \left (\cos ^{2}\left (d x +c \right )\right )}{2 d}+\frac {a^{2} \ln \left (\sin \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 94, normalized size = 0.79 \[ \frac {10 \, a^{2} \sin \left (d x + c\right )^{6} + 24 \, a^{2} \sin \left (d x + c\right )^{5} - 15 \, a^{2} \sin \left (d x + c\right )^{4} - 80 \, a^{2} \sin \left (d x + c\right )^{3} - 30 \, a^{2} \sin \left (d x + c\right )^{2} + 60 \, a^{2} \log \left (\sin \left (d x + c\right )\right ) + 120 \, a^{2} \sin \left (d x + c\right )}{60 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 9.07, size = 132, normalized size = 1.11 \[ \frac {5\,a^2\,\sin \left (c+d\,x\right )}{4\,d}-\frac {a^2\,\ln \left (\frac {1}{{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}\right )}{d}+\frac {a^2\,\ln \left (\frac {\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}\right )}{d}+\frac {19\,a^2\,\cos \left (2\,c+2\,d\,x\right )}{64\,d}-\frac {a^2\,\cos \left (6\,c+6\,d\,x\right )}{192\,d}+\frac {5\,a^2\,\sin \left (3\,c+3\,d\,x\right )}{24\,d}+\frac {a^2\,\sin \left (5\,c+5\,d\,x\right )}{40\,d} \]
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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