Optimal. Leaf size=49 \[ \frac{4 (-1)^{3/4} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}-\frac{2 a^2}{d \sqrt{\cot (c+d x)}} \]
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Rubi [A] time = 0.115364, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3673, 3542, 3533, 208} \[ \frac{4 (-1)^{3/4} a^2 \tanh ^{-1}\left ((-1)^{3/4} \sqrt{\cot (c+d x)}\right )}{d}-\frac{2 a^2}{d \sqrt{\cot (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3542
Rule 3533
Rule 208
Rubi steps
\begin{align*} \int \sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^2 \, dx &=\int \frac{(i a+a \cot (c+d x))^2}{\cot ^{\frac{3}{2}}(c+d x)} \, dx\\ &=-\frac{2 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}{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}{d \sqrt{\cot (c+d x)}}\\ \end{align*}
Mathematica [C] time = 1.69571, size = 83, normalized size = 1.69 \[ \frac{2 a^2 (i \tan (c+d x))^{3/2} \cot ^{\frac{3}{2}}(c+d x) \left (\sqrt{i \tan (c+d x)}-2 \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.269, size = 426, normalized size = 8.7 \begin{align*}{\frac{{a}^{2}\sqrt{2} \left ( \cos \left ( dx+c \right ) +1 \right ) ^{2} \left ( \cos \left ( dx+c \right ) -1 \right ) }{d\cos \left ( dx+c \right ) \left ( \sin \left ( dx+c \right ) \right ) ^{3}}\sqrt{{\frac{\cos \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}} \left ( 2\,i\sqrt{{\frac{\cos \left ( dx+c \right ) -1}{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) -1+\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{-{\frac{\cos \left ( dx+c \right ) -1-\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}}{\it EllipticPi} \left ( \sqrt{-{\frac{\cos \left ( dx+c \right ) -1-\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) \sin \left ( dx+c \right ) -2\,\sqrt{-{\frac{\cos \left ( dx+c \right ) -1-\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) -1+\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) -1}{\sin \left ( dx+c \right ) }}}{\it EllipticPi} \left ( \sqrt{-{\frac{\cos \left ( dx+c \right ) -1-\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}},1/2+i/2,1/2\,\sqrt{2} \right ) \sin \left ( dx+c \right ) +2\,\sqrt{-{\frac{\cos \left ( dx+c \right ) -1-\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) -1+\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}}\sqrt{{\frac{\cos \left ( dx+c \right ) -1}{\sin \left ( dx+c \right ) }}}{\it EllipticF} \left ( \sqrt{-{\frac{\cos \left ( dx+c \right ) -1-\sin \left ( dx+c \right ) }{\sin \left ( dx+c \right ) }}},1/2\,\sqrt{2} \right ) \sin \left ( dx+c \right ) -\sqrt{2}\cos \left ( dx+c \right ) +\sqrt{2} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.7014, size = 177, normalized size = 3.61 \begin{align*} -\frac{{\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} + 4 \, a^{2} \sqrt{\tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] time = 1.39623, size = 799, normalized size = 16.31 \begin{align*} \frac{\sqrt{-\frac{16 i \, a^{4}}{d^{2}}}{\left (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 ) - \sqrt{-\frac{16 i \, a^{4}}{d^{2}}}{\left (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 (8 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} - 8 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}}}{4 \,{\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} \left (\int - \tan ^{2}{\left (c + d x \right )} \sqrt{\cot{\left (c + d x \right )}}\, dx + \int 2 i \tan{\left (c + d x \right )} \sqrt{\cot{\left (c + d x \right )}}\, dx + \int \sqrt{\cot{\left (c + d x \right )}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \sqrt{\cot \left (d x + c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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