Optimal. Leaf size=96 \[ -\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}+\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (\csc (c+d x)+1)} \]
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Rubi [A] time = 0.15, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2709, 3770, 3767, 8, 3768, 3777} \[ -\frac {\cot ^3(c+d x)}{3 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}+\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (\csc (c+d x)+1)} \]
Antiderivative was successfully verified.
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Rule 8
Rule 2709
Rule 3767
Rule 3768
Rule 3770
Rule 3777
Rubi steps
\begin {align*} \int \frac {\cot ^4(c+d x)}{(a+a \sin (c+d x))^3} \, dx &=\frac {\int \left (4 a-4 a \csc (c+d x)+4 a \csc ^2(c+d x)-3 a \csc ^3(c+d x)+a \csc ^4(c+d x)-\frac {4 a}{1+\csc (c+d x)}\right ) \, dx}{a^4}\\ &=\frac {4 x}{a^3}+\frac {\int \csc ^4(c+d x) \, dx}{a^3}-\frac {3 \int \csc ^3(c+d x) \, dx}{a^3}-\frac {4 \int \csc (c+d x) \, dx}{a^3}+\frac {4 \int \csc ^2(c+d x) \, dx}{a^3}-\frac {4 \int \frac {1}{1+\csc (c+d x)} \, dx}{a^3}\\ &=\frac {4 x}{a^3}+\frac {4 \tanh ^{-1}(\cos (c+d x))}{a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (1+\csc (c+d x))}-\frac {3 \int \csc (c+d x) \, dx}{2 a^3}+\frac {4 \int -1 \, dx}{a^3}-\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,\cot (c+d x)\right )}{a^3 d}-\frac {4 \operatorname {Subst}(\int 1 \, dx,x,\cot (c+d x))}{a^3 d}\\ &=\frac {11 \tanh ^{-1}(\cos (c+d x))}{2 a^3 d}-\frac {5 \cot (c+d x)}{a^3 d}-\frac {\cot ^3(c+d x)}{3 a^3 d}+\frac {3 \cot (c+d x) \csc (c+d x)}{2 a^3 d}-\frac {4 \cot (c+d x)}{a^3 d (1+\csc (c+d x))}\\ \end {align*}
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Mathematica [B] time = 5.03, size = 251, normalized size = 2.61 \[ -\frac {\sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (\cot \left (\frac {1}{2} (c+d x)\right )+1\right )^5 \csc ^3(c+d x) \left (-4 \sin ^8\left (\frac {1}{2} (c+d x)\right )-8 \sin (c+d x) (7 \sin (c+d x)-2) \sin ^6\left (\frac {1}{2} (c+d x)\right )+\frac {1}{4} \sin ^4(c+d x) \left (28 \sin (c+d x)+\cot \left (\frac {1}{2} (c+d x)\right )-8\right )+\sin ^2(c+d x) \sin ^4\left (\frac {1}{2} (c+d x)\right ) \left (9-2 \sin (c+d x) \left (-33 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+33 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )+62\right )\right )-\frac {1}{2} \sin ^3(c+d x) \sin ^2\left (\frac {1}{2} (c+d x)\right ) \left (\sin (c+d x) \left (-66 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+66 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-28\right )+9\right )\right )}{12 a^3 d (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 302, normalized size = 3.15 \[ \frac {104 \, \cos \left (d x + c\right )^{4} + 38 \, \cos \left (d x + c\right )^{3} - 156 \, \cos \left (d x + c\right )^{2} + 33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) - 33 \, {\left (\cos \left (d x + c\right )^{4} - 2 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} - \cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 1\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 2 \, {\left (52 \, \cos \left (d x + c\right )^{3} + 33 \, \cos \left (d x + c\right )^{2} - 45 \, \cos \left (d x + c\right ) - 24\right )} \sin \left (d x + c\right ) - 42 \, \cos \left (d x + c\right ) + 48}{12 \, {\left (a^{3} d \cos \left (d x + c\right )^{4} - 2 \, a^{3} d \cos \left (d x + c\right )^{2} + a^{3} d - {\left (a^{3} d \cos \left (d x + c\right )^{3} + a^{3} d \cos \left (d x + c\right )^{2} - a^{3} d \cos \left (d x + c\right ) - a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 146, normalized size = 1.52 \[ -\frac {\frac {132 \, \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) \right |}\right )}{a^{3}} + \frac {192}{a^{3} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}} - \frac {242 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 57 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 9 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}} - \frac {a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 9 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 57 \, a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{9}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.67, size = 153, normalized size = 1.59 \[ \frac {\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d \,a^{3}}-\frac {3 \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d \,a^{3}}+\frac {19 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \,a^{3}}-\frac {1}{24 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}-\frac {19}{8 d \,a^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {11 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \,a^{3}}-\frac {8}{d \,a^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 199, normalized size = 2.07 \[ \frac {\frac {\frac {8 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {48 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {249 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1}{\frac {a^{3} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {\frac {57 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {9 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {\sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}}}{a^{3}} - \frac {132 \, \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{a^{3}}}{24 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 8.69, size = 153, normalized size = 1.59 \[ \frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24\,a^3\,d}-\frac {3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{8\,a^3\,d}-\frac {11\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,a^3\,d}-\frac {83\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+16\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{3}+\frac {1}{3}}{d\,\left (8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+8\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\right )}+\frac {19\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8\,a^3\,d} \]
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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