Optimal. Leaf size=65 \[ -\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 i a \sqrt {\cot (c+d x)}}{d}-\frac {2 (-1)^{3/4} a \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
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Rubi [A] time = 0.10, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3673, 3528, 3533, 208} \[ -\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 i a \sqrt {\cot (c+d x)}}{d}-\frac {2 (-1)^{3/4} a \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3528
Rule 3533
Rule 3673
Rubi steps
\begin {align*} \int \cot ^{\frac {5}{2}}(c+d x) (a+i a \tan (c+d x)) \, dx &=\int \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x)) \, dx\\ &=-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\int \sqrt {\cot (c+d x)} (-a+i a \cot (c+d x)) \, dx\\ &=-\frac {2 i a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\int \frac {-i a-a \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {2 i a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {\left (2 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{i a-a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=-\frac {2 (-1)^{3/4} a \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 i a \sqrt {\cot (c+d x)}}{d}-\frac {2 a \cot ^{\frac {3}{2}}(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 1.20, size = 116, normalized size = 1.78 \[ -\frac {2 a e^{-i c} \sin (c+d x) \sqrt {\cot (c+d x)} (\cot (c+d x)+i) (\cos (d x)-i \sin (d x)) \left (\cot (c+d x)-3 i \sqrt {i \tan (c+d x)} \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )+3 i\right )}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 287, normalized size = 4.42 \[ -\frac {3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \log \left (\frac {{\left ({\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} + 2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a}\right ) - 3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \log \left (\frac {{\left ({\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {-\frac {4 i \, a^{2}}{d^{2}}} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} + 2 i \, a e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{a}\right ) - {\left (-32 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 16 i \, a\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{12 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )}} \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )} \cot \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.18, size = 776, normalized size = 11.94 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.68, size = 139, normalized size = 2.14 \[ -\frac {3 \, {\left (-\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a + \frac {24 i \, a}{\sqrt {\tan \left (d x + c\right )}} + \frac {8 \, a}{\tan \left (d x + c\right )^{\frac {3}{2}}}}{12 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{5/2}\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right ) \,d x \]
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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