Optimal. Leaf size=78 \[ -\frac {5 i x}{16}-\frac {i}{8 (-\cot (x)+i)}+\frac {3 i}{16 (\cot (x)+i)}+\frac {1}{32 (-\cot (x)+i)^2}-\frac {3}{32 (\cot (x)+i)^2}-\frac {i}{24 (\cot (x)+i)^3} \]
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Rubi [A] time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3487, 44, 203} \[ -\frac {5 i x}{16}-\frac {i}{8 (-\cot (x)+i)}+\frac {3 i}{16 (\cot (x)+i)}+\frac {1}{32 (-\cot (x)+i)^2}-\frac {3}{32 (\cot (x)+i)^2}-\frac {i}{24 (\cot (x)+i)^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 3487
Rubi steps
\begin {align*} \int \frac {\sin ^4(x)}{i+\cot (x)} \, dx &=\operatorname {Subst}\left (\int \frac {1}{(i-x)^3 (i+x)^4} \, dx,x,\cot (x)\right )\\ &=\operatorname {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 \operatorname {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*}
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Mathematica [A] time = 0.12, size = 51, normalized size = 0.65 \[ \frac {1}{192} (-15 \cos (2 x)+6 \cos (4 x)-i (60 x-45 \sin (2 x)+9 \sin (4 x)-\sin (6 x)-i \cos (6 x))) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 39, normalized size = 0.50 \[ \frac {1}{384} \, {\left (-120 i \, x e^{\left (6 i \, x\right )} - 3 \, e^{\left (10 i \, x\right )} + 30 \, e^{\left (8 i \, x\right )} - 60 \, e^{\left (4 i \, x\right )} + 15 \, e^{\left (2 i \, x\right )} - 2\right )} e^{\left (-6 i \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 63, normalized size = 0.81 \[ \frac {15 \, \tan \relax (x)^{2} + 18 i \, \tan \relax (x) - 5}{64 \, {\left (-i \, \tan \relax (x) + 1\right )}^{2}} + \frac {55 \, \tan \relax (x)^{3} - 69 i \, \tan \relax (x)^{2} - 15 \, \tan \relax (x) - 7 i}{192 \, {\left (\tan \relax (x) - i\right )}^{3}} + \frac {5}{32} \, \log \left (\tan \relax (x) + i\right ) - \frac {5}{32} \, \log \left (\tan \relax (x) - i\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 66, normalized size = 0.85 \[ \frac {3 i}{16 \left (i+\tan \relax (x )\right )}+\frac {1}{32 \left (i+\tan \relax (x )\right )^{2}}+\frac {5 \ln \left (i+\tan \relax (x )\right )}{32}+\frac {i}{2 \tan \relax (x )-2 i}-\frac {i}{24 \left (\tan \relax (x )-i\right )^{3}}-\frac {7}{32 \left (\tan \relax (x )-i\right )^{2}}-\frac {5 \ln \left (\tan \relax (x )-i\right )}{32} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.34, size = 48, normalized size = 0.62 \[ -\frac {x\,5{}\mathrm {i}}{16}+\frac {\frac {11\,{\mathrm {tan}\relax (x)}^4}{16}-\frac {{\mathrm {tan}\relax (x)}^3\,3{}\mathrm {i}}{16}+\frac {31\,{\mathrm {tan}\relax (x)}^2}{48}-\frac {\mathrm {tan}\relax (x)\,7{}\mathrm {i}}{48}+\frac {1}{6}}{{\left (\mathrm {tan}\relax (x)+1{}\mathrm {i}\right )}^2\,{\left (1+\mathrm {tan}\relax (x)\,1{}\mathrm {i}\right )}^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 54, normalized size = 0.69 \[ - \frac {5 i x}{16} - \frac {e^{4 i x}}{128} + \frac {5 e^{2 i x}}{64} - \frac {5 e^{- 2 i x}}{32} + \frac {5 e^{- 4 i x}}{128} - \frac {e^{- 6 i x}}{192} \]
Verification of antiderivative is not currently implemented for this CAS.
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