Integrand size = 13, antiderivative size = 78 \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=-\frac {5 i x}{16}+\frac {1}{32 (i-\cot (x))^2}-\frac {i}{8 (i-\cot (x))}-\frac {i}{24 (i+\cot (x))^3}-\frac {3}{32 (i+\cot (x))^2}+\frac {3 i}{16 (i+\cot (x))} \]
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Time = 0.09 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3568, 46, 209} \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=-\frac {5 i x}{16}-\frac {i}{8 (-\cot (x)+i)}+\frac {3 i}{16 (\cot (x)+i)}+\frac {1}{32 (-\cot (x)+i)^2}-\frac {3}{32 (\cot (x)+i)^2}-\frac {i}{24 (\cot (x)+i)^3} \]
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Rule 46
Rule 209
Rule 3568
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{(i-x)^3 (i+x)^4} \, dx,x,\cot (x)\right ) \\ & = \text {Subst}\left (\int \left (-\frac {1}{16 (-i+x)^3}-\frac {i}{8 (-i+x)^2}+\frac {i}{8 (i+x)^4}+\frac {3}{16 (i+x)^3}-\frac {3 i}{16 (i+x)^2}+\frac {5 i}{16 \left (1+x^2\right )}\right ) \, dx,x,\cot (x)\right ) \\ & = \frac {1}{32 (i-\cot (x))^2}-\frac {i}{8 (i-\cot (x))}-\frac {i}{24 (i+\cot (x))^3}-\frac {3}{32 (i+\cot (x))^2}+\frac {3 i}{16 (i+\cot (x))}+\frac {5}{16} i \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\cot (x)\right ) \\ & = -\frac {5 i x}{16}+\frac {1}{32 (i-\cot (x))^2}-\frac {i}{8 (i-\cot (x))}-\frac {i}{24 (i+\cot (x))^3}-\frac {3}{32 (i+\cot (x))^2}+\frac {3 i}{16 (i+\cot (x))} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.65 \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=\frac {1}{192} (-15 \cos (2 x)+6 \cos (4 x)-i (60 x-i \cos (6 x)-45 \sin (2 x)+9 \sin (4 x)-\sin (6 x))) \]
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Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {5 i x}{16}-\frac {{\mathrm e}^{-6 i x}}{192}+\frac {\cos \left (4 x \right )}{32}-\frac {3 i \sin \left (4 x \right )}{64}-\frac {5 \cos \left (2 x \right )}{64}+\frac {15 i \sin \left (2 x \right )}{64}\) | \(39\) |
default | \(\frac {i}{2 \tan \left (x \right )-2 i}-\frac {i}{24 \left (\tan \left (x \right )-i\right )^{3}}-\frac {7}{32 \left (\tan \left (x \right )-i\right )^{2}}-\frac {5 \ln \left (\tan \left (x \right )-i\right )}{32}+\frac {3 i}{16 \left (\tan \left (x \right )+i\right )}+\frac {1}{32 \left (\tan \left (x \right )+i\right )^{2}}+\frac {5 \ln \left (\tan \left (x \right )+i\right )}{32}\) | \(66\) |
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Time = 0.26 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.50 \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=\frac {1}{384} \, {\left (-120 i \, x e^{\left (6 i \, x\right )} - 3 \, e^{\left (10 i \, x\right )} + 30 \, e^{\left (8 i \, x\right )} - 60 \, e^{\left (4 i \, x\right )} + 15 \, e^{\left (2 i \, x\right )} - 2\right )} e^{\left (-6 i \, x\right )} \]
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Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.69 \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=- \frac {5 i x}{16} - \frac {e^{4 i x}}{128} + \frac {5 e^{2 i x}}{64} - \frac {5 e^{- 2 i x}}{32} + \frac {5 e^{- 4 i x}}{128} - \frac {e^{- 6 i x}}{192} \]
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Exception generated. \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=\text {Exception raised: RuntimeError} \]
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none
Time = 0.25 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81 \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=\frac {15 \, \tan \left (x\right )^{2} + 18 i \, \tan \left (x\right ) - 5}{64 \, {\left (-i \, \tan \left (x\right ) + 1\right )}^{2}} + \frac {55 \, \tan \left (x\right )^{3} - 69 i \, \tan \left (x\right )^{2} - 15 \, \tan \left (x\right ) - 7 i}{192 \, {\left (\tan \left (x\right ) - i\right )}^{3}} + \frac {5}{32} \, \log \left (\tan \left (x\right ) + i\right ) - \frac {5}{32} \, \log \left (\tan \left (x\right ) - i\right ) \]
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Time = 12.56 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.62 \[ \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx=-\frac {x\,5{}\mathrm {i}}{16}+\frac {\frac {11\,{\mathrm {tan}\left (x\right )}^4}{16}-\frac {{\mathrm {tan}\left (x\right )}^3\,3{}\mathrm {i}}{16}+\frac {31\,{\mathrm {tan}\left (x\right )}^2}{48}-\frac {\mathrm {tan}\left (x\right )\,7{}\mathrm {i}}{48}+\frac {1}{6}}{{\left (\mathrm {tan}\left (x\right )+1{}\mathrm {i}\right )}^2\,{\left (1+\mathrm {tan}\left (x\right )\,1{}\mathrm {i}\right )}^3} \]
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