Optimal. Leaf size=258 \[ -\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.807919, antiderivative size = 258, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4241, 3559, 3596, 3598, 12, 3544, 205} \[ -\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \sqrt{\tan (c+d x)} \sqrt{\cot (c+d x)} \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{a^{5/2} d}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 4241
Rule 3559
Rule 3596
Rule 3598
Rule 12
Rule 3544
Rule 205
Rubi steps
\begin{align*} \int \frac{\cot ^{\frac{5}{2}}(c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{1}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{5/2}} \, dx\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{13 a}{2}-4 i a \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) (a+i a \tan (c+d x))^{3/2}} \, dx}{5 a^2}\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\frac{141 a^2}{4}-\frac{63}{2} i a^2 \tan (c+d x)}{\tan ^{\frac{5}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}} \, dx}{15 a^4}\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (\frac{1083 a^3}{8}-\frac{267}{2} i a^3 \tan (c+d x)\right )}{\tan ^{\frac{5}{2}}(c+d x)} \, dx}{15 a^6}\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{\left (2 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)} \left (-\frac{2121 i a^4}{16}-\frac{1083}{8} a^4 \tan (c+d x)\right )}{\tan ^{\frac{3}{2}}(c+d x)} \, dx}{45 a^7}\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{\left (4 \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int -\frac{45 a^5 \sqrt{a+i a \tan (c+d x)}}{32 \sqrt{\tan (c+d x)}} \, dx}{45 a^8}\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{\left (\sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \int \frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{\tan (c+d x)}} \, dx}{8 a^3}\\ &=\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}+\frac{\left (i \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{-i a-2 a^2 x^2} \, dx,x,\frac{\sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right )}{4 a d}\\ &=-\frac{\left (\frac{1}{8}-\frac{i}{8}\right ) \tanh ^{-1}\left (\frac{(1+i) \sqrt{a} \sqrt{\tan (c+d x)}}{\sqrt{a+i a \tan (c+d x)}}\right ) \sqrt{\cot (c+d x)} \sqrt{\tan (c+d x)}}{a^{5/2} d}+\frac{\cot ^{\frac{3}{2}}(c+d x)}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{7 \cot ^{\frac{3}{2}}(c+d x)}{10 a d (a+i a \tan (c+d x))^{3/2}}+\frac{89 \cot ^{\frac{3}{2}}(c+d x)}{20 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{707 i \sqrt{\cot (c+d x)} \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}-\frac{361 \cot ^{\frac{3}{2}}(c+d x) \sqrt{a+i a \tan (c+d x)}}{60 a^3 d}\\ \end{align*}
Mathematica [A] time = 1.9777, size = 199, normalized size = 0.77 \[ \frac{i e^{-6 i (c+d x)} \sqrt{\frac{a e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}} \sqrt{\cot (c+d x)} \left (33 e^{2 i (c+d x)}+348 e^{4 i (c+d x)}-1527 e^{6 i (c+d x)}+983 e^{8 i (c+d x)}+15 e^{5 i (c+d x)} \left (-1+e^{2 i (c+d x)}\right )^{3/2} \tanh ^{-1}\left (\frac{e^{i (c+d x)}}{\sqrt{-1+e^{2 i (c+d x)}}}\right )+3\right )}{60 \sqrt{2} a^3 d \left (-1+e^{2 i (c+d x)}\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.428, size = 499, normalized size = 1.9 \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 = 1.69396, size = 1247, normalized size = 4.83 \begin{align*} \frac{\sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 (983 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1527 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 348 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 33 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i\right )} e^{\left (i \, d x + i \, c\right )} - 30 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \sqrt{-\frac{i}{8 \, a^{5} d^{2}}} \log \left (\frac{1}{4} \,{\left (4 i \, a^{3} d \sqrt{-\frac{i}{8 \, a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 30 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )} \sqrt{-\frac{i}{8 \, a^{5} d^{2}}} \log \left (\frac{1}{4} \,{\left (-4 i \, a^{3} d \sqrt{-\frac{i}{8 \, a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \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 (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right )}{120 \,{\left (a^{3} d e^{\left (8 i \, d x + 8 i \, c\right )} - a^{3} d e^{\left (6 i \, d x + 6 i \, c\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\cot \left (d x + c\right )^{\frac{5}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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