Optimal. Leaf size=91 \[ -\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {4 \sqrt [4]{-1} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
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Rubi [A] time = 0.17, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3673, 3543, 3528, 3533, 208} \[ -\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}+\frac {4 \sqrt [4]{-1} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3528
Rule 3533
Rule 3543
Rule 3673
Rubi steps
\begin {align*} \int \cot ^{\frac {7}{2}}(c+d x) (a+i a \tan (c+d x))^2 \, dx &=\int \cot ^{\frac {3}{2}}(c+d x) (i a+a \cot (c+d x))^2 \, dx\\ &=-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \cot ^{\frac {3}{2}}(c+d x) \left (-2 a^2+2 i a^2 \cot (c+d x)\right ) \, dx\\ &=-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \sqrt {\cot (c+d x)} \left (-2 i a^2-2 a^2 \cot (c+d x)\right ) \, dx\\ &=\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\int \frac {2 a^2-2 i a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}+\frac {\left (8 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{-2 a^2-2 i a^2 x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {4 \sqrt [4]{-1} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}+\frac {4 a^2 \sqrt {\cot (c+d x)}}{d}-\frac {4 i a^2 \cot ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 \cot ^{\frac {5}{2}}(c+d x)}{5 d}\\ \end {align*}
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Mathematica [A] time = 3.52, size = 104, normalized size = 1.14 \[ -\frac {a^2 \sqrt {\cot (c+d x)} \left (60 \sqrt {i \tan (c+d x)} \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )+\csc ^2(c+d x) (10 i \sin (2 (c+d x))+33 \cos (2 (c+d x))-27)\right )}{15 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.01, size = 340, normalized size = 3.74 \[ -\frac {15 \, \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 15 \, \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt {\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} - d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 8 \, {\left (43 \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 54 \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 23 \, a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{60 \, {\left (d e^{\left (4 i \, d x + 4 i \, c\right )} - 2 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \cot \left (d x + c\right )^{\frac {7}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.39, size = 1458, normalized size = 16.02 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.72, size = 158, normalized size = 1.74 \[ -\frac {15 \, {\left (-\left (2 i - 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - \left (2 i - 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i + 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i + 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2} - \frac {120 \, a^{2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {40 i \, a^{2}}{\tan \left (d x + c\right )^{\frac {3}{2}}} + \frac {12 \, a^{2}}{\tan \left (d x + c\right )^{\frac {5}{2}}}}{30 \, d} \]
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 \[ \int {\mathrm {cot}\left (c+d\,x\right )}^{7/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \]
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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