Optimal. Leaf size=65 \[ -\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 65, 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 = {2833, 12, 43} \[ -\frac {a^3 \csc ^3(c+d x)}{3 d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {3 a^3 \csc (c+d x)}{d}+\frac {a^3 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^4 (a+x)^3}{x^4} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \frac {(a+x)^3}{x^4} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^3 \operatorname {Subst}\left (\int \left (\frac {a^3}{x^4}+\frac {3 a^2}{x^3}+\frac {3 a}{x^2}+\frac {1}{x}\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {3 a^3 \csc (c+d x)}{d}-\frac {3 a^3 \csc ^2(c+d x)}{2 d}-\frac {a^3 \csc ^3(c+d x)}{3 d}+\frac {a^3 \log (\sin (c+d x))}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 57, normalized size = 0.88 \[ a^3 \left (-\frac {\csc ^3(c+d x)}{3 d}-\frac {3 \csc ^2(c+d x)}{2 d}-\frac {3 \csc (c+d x)}{d}+\frac {\log (\sin (c+d x))}{d}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 91, normalized size = 1.40 \[ -\frac {18 \, a^{3} \cos \left (d x + c\right )^{2} - 9 \, a^{3} \sin \left (d x + c\right ) - 20 \, a^{3} - 6 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - a^{3}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right )}{6 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )} \sin \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 59, normalized size = 0.91 \[ \frac {6 \, a^{3} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) - \frac {18 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 64, normalized size = 0.98 \[ -\frac {3 a^{3}}{d \sin \left (d x +c \right )}+\frac {a^{3} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {3 a^{3}}{2 d \sin \left (d x +c \right )^{2}}-\frac {a^{3}}{3 d \sin \left (d x +c \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 58, normalized size = 0.89 \[ \frac {6 \, a^{3} \log \left (\sin \left (d x + c\right )\right ) - \frac {18 \, a^{3} \sin \left (d x + c\right )^{2} + 9 \, a^{3} \sin \left (d x + c\right ) + 2 \, a^{3}}{\sin \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.62, size = 147, normalized size = 2.26 \[ \frac {a^3\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {3\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,d}-\frac {{\mathrm {cot}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+3\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^3}{3}\right )}{8\,d}-\frac {13\,a^3\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,d}-\frac {a^3\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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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