Optimal. Leaf size=30 \[ -\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d} \]
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Rubi [A]
time = 0.04, antiderivative size = 30, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2912, 12, 37}
\begin {gather*} -\frac {\csc ^4(c+d x) (a \sin (c+d x)+a)^4}{4 a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 37
Rule 2912
Rubi steps
\begin {align*} \int \cot (c+d x) \csc ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\frac {\text {Subst}\left (\int \frac {a^5 (a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{a d}\\ &=\frac {a^4 \text {Subst}\left (\int \frac {(a+x)^3}{x^5} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {\csc ^4(c+d x) (a+a \sin (c+d x))^4}{4 a d}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 20, normalized size = 0.67 \begin {gather*} -\frac {a^3 (1+\csc (c+d x))^4}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.13, size = 49, normalized size = 1.63
method | result | size |
derivativedivides | \(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )}-\frac {3}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )^{3}}\right )}{d}\) | \(49\) |
default | \(\frac {a^{3} \left (-\frac {1}{4 \sin \left (d x +c \right )^{4}}-\frac {1}{\sin \left (d x +c \right )}-\frac {3}{2 \sin \left (d x +c \right )^{2}}-\frac {1}{\sin \left (d x +c \right )^{3}}\right )}{d}\) | \(49\) |
risch | \(-\frac {2 i a^{3} \left (3 i {\mathrm e}^{6 i \left (d x +c \right )}+{\mathrm e}^{7 i \left (d x +c \right )}-8 i {\mathrm e}^{4 i \left (d x +c \right )}-7 \,{\mathrm e}^{5 i \left (d x +c \right )}+3 i {\mathrm e}^{2 i \left (d x +c \right )}+7 \,{\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )^{4}}\) | \(102\) |
norman | \(\frac {-\frac {a^{3}}{64 d}-\frac {a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {31 a^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {5 a^{3} \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {31 a^{3} \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {11 a^{3} \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {31 a^{3} \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {5 a^{3} \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d}-\frac {31 a^{3} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {a^{3} \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a^{3} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}+\frac {37 a^{3} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}+\frac {37 a^{3} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{16 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}\) | \(263\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 54, normalized size = 1.80 \begin {gather*} -\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 72 vs.
\(2 (28) = 56\).
time = 0.33, size = 72, normalized size = 2.40 \begin {gather*} \frac {6 \, a^{3} \cos \left (d x + c\right )^{2} - 7 \, a^{3} + 4 \, {\left (a^{3} \cos \left (d x + c\right )^{2} - 2 \, a^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (d \cos \left (d x + c\right )^{4} - 2 \, d \cos \left (d x + c\right )^{2} + d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 54, normalized size = 1.80 \begin {gather*} -\frac {4 \, a^{3} \sin \left (d x + c\right )^{3} + 6 \, a^{3} \sin \left (d x + c\right )^{2} + 4 \, a^{3} \sin \left (d x + c\right ) + a^{3}}{4 \, d \sin \left (d x + c\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.61, size = 54, normalized size = 1.80 \begin {gather*} -\frac {4\,a^3\,{\sin \left (c+d\,x\right )}^3+6\,a^3\,{\sin \left (c+d\,x\right )}^2+4\,a^3\,\sin \left (c+d\,x\right )+a^3}{4\,d\,{\sin \left (c+d\,x\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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