Optimal. Leaf size=99 \[ -\frac {a^3 \cos ^3(c+d x)}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {13 a^3 x}{8} \]
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Rubi [A] time = 0.17, antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {2873, 2635, 8, 2592, 321, 206, 2565, 30, 2568} \[ -\frac {a^3 \cos ^3(c+d x)}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {13 a^3 \sin (c+d x) \cos (c+d x)}{8 d}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {13 a^3 x}{8} \]
Antiderivative was successfully verified.
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Rule 8
Rule 30
Rule 206
Rule 321
Rule 2565
Rule 2568
Rule 2592
Rule 2635
Rule 2873
Rubi steps
\begin {align*} \int \cos (c+d x) \cot (c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (3 a^3 \cos ^2(c+d x)+a^3 \cos (c+d x) \cot (c+d x)+3 a^3 \cos ^2(c+d x) \sin (c+d x)+a^3 \cos ^2(c+d x) \sin ^2(c+d x)\right ) \, dx\\ &=a^3 \int \cos (c+d x) \cot (c+d x) \, dx+a^3 \int \cos ^2(c+d x) \sin ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^2(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^2(c+d x) \sin (c+d x) \, dx\\ &=\frac {3 a^3 \cos (c+d x) \sin (c+d x)}{2 d}-\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{4} a^3 \int \cos ^2(c+d x) \, dx+\frac {1}{2} \left (3 a^3\right ) \int 1 \, dx-\frac {a^3 \operatorname {Subst}\left (\int \frac {x^2}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \operatorname {Subst}\left (\int x^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {3 a^3 x}{2}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}+\frac {1}{8} a^3 \int 1 \, dx-\frac {a^3 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cos (c+d x)\right )}{d}\\ &=\frac {13 a^3 x}{8}-\frac {a^3 \tanh ^{-1}(\cos (c+d x))}{d}+\frac {a^3 \cos (c+d x)}{d}-\frac {a^3 \cos ^3(c+d x)}{d}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{8 d}-\frac {a^3 \cos ^3(c+d x) \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 82, normalized size = 0.83 \[ \frac {a^3 \left (24 \sin (2 (c+d x))-\sin (4 (c+d x))+8 \cos (c+d x)-8 \cos (3 (c+d x))+32 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )-32 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+52 c+52 d x\right )}{32 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 101, normalized size = 1.02 \[ -\frac {8 \, a^{3} \cos \left (d x + c\right )^{3} - 13 \, a^{3} d x - 8 \, a^{3} \cos \left (d x + c\right ) + 4 \, a^{3} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 4 \, a^{3} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (2 \, a^{3} \cos \left (d x + c\right )^{3} - 13 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 144, normalized size = 1.45 \[ \frac {13 \, {\left (d x + c\right )} a^{3} + 8 \, a^{3} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right ) - \frac {2 \, {\left (11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 19 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 19 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 11 \, a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )}^{4}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.36, size = 111, normalized size = 1.12 \[ -\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{4 d}+\frac {13 a^{3} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{8 d}+\frac {13 a^{3} x}{8}+\frac {13 a^{3} c}{8 d}-\frac {a^{3} \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {a^{3} \cos \left (d x +c \right )}{d}+\frac {a^{3} \ln \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 99, normalized size = 1.00 \[ -\frac {32 \, a^{3} \cos \left (d x + c\right )^{3} - {\left (4 \, d x + 4 \, c - \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 24 \, {\left (2 \, d x + 2 \, c + \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 16 \, a^{3} {\left (2 \, \cos \left (d x + c\right ) - \log \left (\cos \left (d x + c\right ) + 1\right ) + \log \left (\cos \left (d x + c\right ) - 1\right )\right )}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.36, size = 244, normalized size = 2.46 \[ \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {13\,a^3\,\mathrm {atan}\left (\frac {169\,a^6}{16\,\left (\frac {13\,a^6}{2}-\frac {169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}+\frac {13\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2\,\left (\frac {13\,a^6}{2}-\frac {169\,a^6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{16}\right )}\right )}{4\,d}+\frac {-\frac {11\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}-4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}+\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+4\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{4}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ a^{3} \left (\int \cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 3 \sin {\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int 3 \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx + \int \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )} \csc {\left (c + d x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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