Integrand size = 26, antiderivative size = 141 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=-\frac {(-1)^{3/4} \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{8 a^3 d}+\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )} \]
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Time = 0.33 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {3754, 3639, 3677, 12, 3630, 3614, 214} \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=-\frac {(-1)^{3/4} \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{8 a^3 d}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (a \cot (c+d x)+i a)^2}+\frac {\sqrt {\cot (c+d x)}}{6 d (a \cot (c+d x)+i a)^3} \]
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Rule 12
Rule 214
Rule 3614
Rule 3630
Rule 3639
Rule 3677
Rule 3754
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cot ^{\frac {3}{2}}(c+d x)}{(i a+a \cot (c+d x))^3} \, dx \\ & = \frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {\int \frac {-\frac {i a}{2}+\frac {7}{2} a \cot (c+d x)}{\sqrt {\cot (c+d x)} (i a+a \cot (c+d x))^2} \, dx}{6 a^2} \\ & = \frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\int -\frac {6 i a^2 \sqrt {\cot (c+d x)}}{i a+a \cot (c+d x)} \, dx}{24 a^4} \\ & = \frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}-\frac {i \int \frac {\sqrt {\cot (c+d x)}}{i a+a \cot (c+d x)} \, dx}{4 a^2} \\ & = \frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {i \int \frac {-\frac {a}{2}+\frac {1}{2} i a \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{8 a^4} \\ & = \frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac {i \text {Subst}\left (\int \frac {1}{\frac {a}{2}+\frac {1}{2} i a x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d} \\ & = -\frac {(-1)^{3/4} \text {arctanh}\left ((-1)^{3/4} \sqrt {\cot (c+d x)}\right )}{8 a^3 d}+\frac {\sqrt {\cot (c+d x)}}{6 d (i a+a \cot (c+d x))^3}+\frac {i \sqrt {\cot (c+d x)}}{6 a d (i a+a \cot (c+d x))^2}+\frac {\sqrt {\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {\sqrt {\cot (c+d x)} \left (\frac {-3+10 i \cot (c+d x)+3 \cot ^2(c+d x)}{(i+\cot (c+d x))^3}+3 \sqrt [4]{-1} \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sqrt {\tan (c+d x)}\right )}{24 a^3 d} \]
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Time = 2.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {\frac {\arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \sqrt {2}-4 i \sqrt {2}}-\frac {-\left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )-\frac {10 i \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\sqrt {\cot }\left (d x +c \right )}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}}{a^{3} d}\) | \(89\) |
default | \(\frac {\frac {\arctan \left (\frac {2 \left (\sqrt {\cot }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4 \sqrt {2}-4 i \sqrt {2}}-\frac {-\left (\cot ^{\frac {5}{2}}\left (d x +c \right )\right )-\frac {10 i \left (\cot ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\sqrt {\cot }\left (d x +c \right )}{8 \left (i+\cot \left (d x +c \right )\right )^{3}}}{a^{3} d}\) | \(89\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 307 vs. \(2 (113) = 226\).
Time = 0.26 (sec) , antiderivative size = 307, normalized size of antiderivative = 2.18 \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {{\left (12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) - 12 \, a^{3} d \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-2 \, {\left (8 \, {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {-\frac {i}{64 \, a^{6} d^{2}}} - i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}\right ) + \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} {\left (-4 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 4 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{48 \, a^{3} d} \]
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\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\frac {i \int \frac {1}{\tan ^{3}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 3 i \tan ^{2}{\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} - 3 \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )} + i \cot ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx}{a^{3}} \]
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Exception generated. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} \cot \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\cot ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^3} \, dx=\int \frac {1}{{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
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