Optimal. Leaf size=91 \[ \frac {4 i a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2}{d \sqrt {\cot (c+d x)}}-\frac {4 (-1)^{3/4} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
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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, 3542, 3529, 3533, 208} \[ \frac {4 i a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4 a^2}{d \sqrt {\cot (c+d x)}}-\frac {4 (-1)^{3/4} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 208
Rule 3529
Rule 3533
Rule 3542
Rule 3673
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (c+d x))^2}{\cot ^{\frac {3}{2}}(c+d x)} \, dx &=\int \frac {(i a+a \cot (c+d x))^2}{\cot ^{\frac {7}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\int \frac {2 i a^2+2 a^2 \cot (c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4 i a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\int \frac {2 a^2-2 i a^2 \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4 i a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2}{d \sqrt {\cot (c+d x)}}+\int \frac {-2 i a^2-2 a^2 \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4 i a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2}{d \sqrt {\cot (c+d x)}}-\frac {\left (8 a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2 i a^2-2 a^2 x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=-\frac {4 (-1)^{3/4} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{d}-\frac {2 a^2}{5 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {4 i a^2}{3 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2}{d \sqrt {\cot (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 4.01, size = 152, normalized size = 1.67 \[ \frac {a^2 e^{-2 i c} (\sin (2 (c+d x))-i \cos (2 (c+d x))) \left (2 i \sec ^2(c+d x) (10 i \sin (2 (c+d x))+33 \cos (2 (c+d x))+27)-\frac {120 i \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )}{\sqrt {i \tan (c+d x)}}\right )}{30 d \sqrt {\cot (c+d x)} (\cos (d x)+i \sin (d x))^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 2.14, size = 391, normalized size = 4.30 \[ -\frac {15 \, \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - 15 \, \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (\frac {{\left (4 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {16 i \, a^{4}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{2 \, a^{2}}\right ) - {\left (-344 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 88 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 248 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 184 i \, a^{2}\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}}{60 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]
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 \[ \int \frac {{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}{\cot \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.28, size = 511, normalized size = 5.62 \[ \frac {a^{2} \left (-1+\cos \left (d x +c \right )\right ) \left (-30 i \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )+30 \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \EllipticPi \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {1}{2}+\frac {i}{2}, \frac {\sqrt {2}}{2}\right )-30 \sin \left (d x +c \right ) \sqrt {\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \EllipticF \left (\sqrt {\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+10 i \sqrt {2}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-10 i \sqrt {2}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+33 \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )-33 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {2}-3 \cos \left (d x +c \right ) \sqrt {2}+3 \sqrt {2}\right ) \left (1+\cos \left (d x +c \right )\right )^{2} \sqrt {2}}{15 d \left (\frac {\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 161, normalized size = 1.77 \[ -\frac {4 \, {\left (3 \, a^{2} - \frac {10 i \, a^{2}}{\tan \left (d x + c\right )} - \frac {30 \, a^{2}}{\tan \left (d x + c\right )^{2}}\right )} \tan \left (d x + c\right )^{\frac {5}{2}} - 15 \, {\left (\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + \left (i - 1\right ) \, \sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \left (i - 1\right ) \, \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{2}}{30 \, d} \]
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 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - a^{2} \left (\int \frac {\tan ^{2}{\left (c + d x \right )}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx + \int \left (- \frac {2 i \tan {\left (c + d x \right )}}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\right )\, dx + \int \left (- \frac {1}{\cot ^{\frac {3}{2}}{\left (c + d x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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