Optimal. Leaf size=88 \[ -\frac {16 i a^3 \sqrt {\cot (c+d x)}}{3 d}-\frac {2 \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}-\frac {8 (-1)^{3/4} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
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Rubi [A] time = 0.17, antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3673, 3556, 3592, 3533, 208} \[ -\frac {16 i a^3 \sqrt {\cot (c+d x)}}{3 d}-\frac {2 \sqrt {\cot (c+d x)} \left (a^3 \cot (c+d x)+i a^3\right )}{3 d}-\frac {8 (-1)^{3/4} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3533
Rule 3556
Rule 3592
Rule 3673
Rubi steps
\begin {align*} \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x))^3 \, dx &=\int \frac {(i a+a \cot (c+d x))^3}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {2 \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}-\frac {1}{3} (2 i a) \int \frac {(-2 i a-4 a \cot (c+d x)) (i a+a \cot (c+d x))}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {16 i a^3 \sqrt {\cot (c+d x)}}{3 d}-\frac {2 \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}-\frac {1}{3} (2 i a) \int \frac {6 a^2-6 i a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {16 i a^3 \sqrt {\cot (c+d x)}}{3 d}-\frac {2 \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}-\frac {\left (48 i a^5\right ) \operatorname {Subst}\left (\int \frac {1}{-6 a^2-6 i a^2 x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=-\frac {8 (-1)^{3/4} a^3 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {16 i a^3 \sqrt {\cot (c+d x)}}{3 d}-\frac {2 \sqrt {\cot (c+d x)} \left (i a^3+a^3 \cot (c+d x)\right )}{3 d}\\ \end {align*}
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Mathematica [A] time = 2.74, size = 125, normalized size = 1.42 \[ -\frac {2 a^3 e^{-3 i c} \sqrt {\cot (c+d x)} (\cos (3 (c+d x))+i \sin (3 (c+d x))) \left (\cot (c+d x)-12 i \sqrt {i \tan (c+d x)} \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )+9 i\right )}{3 d (\cos (d x)+i \sin (d x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 297, normalized size = 3.38 \[ -\frac {3 \, \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - 3 \, \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \log \left (\frac {{\left (8 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {64 i \, a^{6}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, a^{3}}\right ) - {\left (-80 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )} + 64 i \, a^{3}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.48, size = 779, normalized size = 8.85 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.73, size = 145, normalized size = 1.65 \[ \frac {3 \, {\left (\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - \frac {18 i \, a^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {2 \, a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3 \,d x \]
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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