Optimal. Leaf size=80 \[ \frac {a^4 \sin ^2(c+d x)}{2 d}+\frac {4 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc ^2(c+d x)}{2 d}-\frac {4 a^4 \csc (c+d x)}{d}+\frac {6 a^4 \log (\sin (c+d x))}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 80, 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} \[ \frac {a^4 \sin ^2(c+d x)}{2 d}+\frac {4 a^4 \sin (c+d x)}{d}-\frac {a^4 \csc ^2(c+d x)}{2 d}-\frac {4 a^4 \csc (c+d x)}{d}+\frac {6 a^4 \log (\sin (c+d x))}{d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 43
Rule 2833
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^2(c+d x) (a+a \sin (c+d x))^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a^3 (a+x)^4}{x^3} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \frac {(a+x)^4}{x^3} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=\frac {a^2 \operatorname {Subst}\left (\int \left (4 a+\frac {a^4}{x^3}+\frac {4 a^3}{x^2}+\frac {6 a^2}{x}+x\right ) \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {4 a^4 \csc (c+d x)}{d}-\frac {a^4 \csc ^2(c+d x)}{2 d}+\frac {6 a^4 \log (\sin (c+d x))}{d}+\frac {4 a^4 \sin (c+d x)}{d}+\frac {a^4 \sin ^2(c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 54, normalized size = 0.68 \[ -\frac {a^4 \left (-\sin ^2(c+d x)-8 \sin (c+d x)+\csc ^2(c+d x)+8 \csc (c+d x)-12 \log (\sin (c+d x))\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.51, size = 98, normalized size = 1.22 \[ -\frac {2 \, a^{4} \cos \left (d x + c\right )^{4} - 16 \, a^{4} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) - 3 \, a^{4} \cos \left (d x + c\right )^{2} - a^{4} - 24 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - a^{4}\right )} \log \left (\frac {1}{2} \, \sin \left (d x + c\right )\right )}{4 \, {\left (d \cos \left (d x + c\right )^{2} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 67, normalized size = 0.84 \[ \frac {a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left ({\left | \sin \left (d x + c\right ) \right |}\right ) + 8 \, a^{4} \sin \left (d x + c\right ) - \frac {8 \, a^{4} \sin \left (d x + c\right ) + a^{4}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 79, normalized size = 0.99 \[ \frac {a^{4} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}+\frac {4 a^{4} \sin \left (d x +c \right )}{d}-\frac {4 a^{4}}{d \sin \left (d x +c \right )}+\frac {6 a^{4} \ln \left (\sin \left (d x +c \right )\right )}{d}-\frac {a^{4}}{2 d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 66, normalized size = 0.82 \[ \frac {a^{4} \sin \left (d x + c\right )^{2} + 12 \, a^{4} \log \left (\sin \left (d x + c\right )\right ) + 8 \, a^{4} \sin \left (d x + c\right ) - \frac {8 \, a^{4} \sin \left (d x + c\right ) + a^{4}}{\sin \left (d x + c\right )^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.62, size = 207, normalized size = 2.59 \[ \frac {6\,a^4\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,d}-\frac {-24\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-\frac {15\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{2}-16\,a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+a^4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+8\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+\frac {a^4}{2}}{d\,\left (4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}-\frac {2\,a^4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d}-\frac {6\,a^4\,\ln \left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}{d} \]
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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