Optimal. Leaf size=232 \[ \frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}-\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}-\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^2 d}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2} \]
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Rubi [A] time = 0.330622, antiderivative size = 232, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {3673, 3558, 3595, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (\cot (c+d x)+i)}-\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}-\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^2 d}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
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Rule 3673
Rule 3558
Rule 3595
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{\sqrt{\cot (c+d x)}}{(a+i a \tan (c+d x))^2} \, dx &=\int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(i a+a \cot (c+d x))^2} \, dx\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac{\int \frac{\sqrt{\cot (c+d x)} \left (-\frac{3 i a}{2}+\frac{7}{2} a \cot (c+d x)\right )}{i a+a \cot (c+d x)} \, dx}{4 a^2}\\ &=\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac{\int \frac{-\frac{5 i a^2}{2}+\frac{9}{2} a^2 \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{8 a^4}\\ &=\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac{\operatorname{Subst}\left (\int \frac{\frac{5 i a^2}{2}-\frac{9 a^2 x^2}{2}}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{4 a^4 d}\\ &=\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+-\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^2 d}+\frac{\left (\frac{9}{16}+\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^2 d}\\ &=\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+-\frac{\left (\frac{9}{32}-\frac{5 i}{32}\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^2 d}+-\frac{\left (\frac{9}{32}-\frac{5 i}{32}\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^2 d}+-\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}+2 x}{-1-\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}+-\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{2}-2 x}{-1+\sqrt{2} x-x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}\\ &=\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}-\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^2 d}+-\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}\\ &=\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}-\frac{\left (\frac{9}{16}-\frac{5 i}{16}\right ) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^2 d}+\frac{5 \sqrt{\cot (c+d x)}}{8 a^2 d (i+\cot (c+d x))}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}-\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^2 d}+\frac{\left (\frac{9}{32}+\frac{5 i}{32}\right ) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{\sqrt{2} a^2 d}\\ \end{align*}
Mathematica [A] time = 0.879167, size = 232, normalized size = 1. \[ \frac{\cot ^{\frac{3}{2}}(c+d x) \csc (c+d x) \sec ^2(c+d x) \left (7 \sin (c+d x)+7 \sin (3 (c+d x))+5 i \cos (c+d x)-5 i \cos (3 (c+d x))-(5+9 i) \sqrt{\sin (2 (c+d x))} \sin ^{-1}(\cos (c+d x)-\sin (c+d x)) (\sin (2 (c+d x))-i \cos (2 (c+d x)))+(-5+9 i) \sin ^{\frac{3}{2}}(2 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )+(9+5 i) \sqrt{\sin (2 (c+d x))} \cos (2 (c+d x)) \log \left (\sin (c+d x)+\sqrt{\sin (2 (c+d x))}+\cos (c+d x)\right )\right )}{32 a^2 d (\cot (c+d x)+i)^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.376, size = 778, normalized size = 3.4 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.43151, size = 1442, normalized size = 6.22 \begin{align*} \frac{{\left (4 \, a^{2} d \sqrt{-\frac{i}{16 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left ({\left ({\left (8 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i \, a^{2} 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}} \sqrt{-\frac{i}{16 \, a^{4} d^{2}}} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 4 \, a^{2} d \sqrt{-\frac{i}{16 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left ({\left ({\left (-8 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i \, a^{2} 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}} \sqrt{-\frac{i}{16 \, a^{4} d^{2}}} + 2 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 4 \, a^{2} d \sqrt{\frac{49 i}{64 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac{{\left (8 \,{\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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}} \sqrt{\frac{49 i}{64 \, a^{4} d^{2}}} + 7\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + 4 \, a^{2} d \sqrt{\frac{49 i}{64 \, a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac{{\left (8 \,{\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} 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}} \sqrt{\frac{49 i}{64 \, a^{4} d^{2}}} - 7\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} 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}}{\left (-6 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 5 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{2} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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 \frac{\sqrt{\cot \left (d x + c\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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