3.1 \(\int \frac{\cos ^4(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=78 \[ -\frac{i x}{16}+\frac{i}{8 (-\cot (x)+i)}+\frac{3 i}{16 (\cot (x)+i)}+\frac{1}{32 (-\cot (x)+i)^2}+\frac{5}{32 (\cot (x)+i)^2}-\frac{i}{24 (\cot (x)+i)^3} \]

[Out]

(-I/16)*x + 1/(32*(I - Cot[x])^2) + (I/8)/(I - Cot[x]) - (I/24)/(I + Cot[x])^3 + 5/(32*(I + Cot[x])^2) + ((3*I
)/16)/(I + Cot[x])

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Rubi [A]  time = 0.0694971, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3516, 848, 88, 203} \[ -\frac{i x}{16}+\frac{i}{8 (-\cot (x)+i)}+\frac{3 i}{16 (\cot (x)+i)}+\frac{1}{32 (-\cot (x)+i)^2}+\frac{5}{32 (\cot (x)+i)^2}-\frac{i}{24 (\cot (x)+i)^3} \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]^4/(I + Cot[x]),x]

[Out]

(-I/16)*x + 1/(32*(I - Cot[x])^2) + (I/8)/(I - Cot[x]) - (I/24)/(I + Cot[x])^3 + 5/(32*(I + Cot[x])^2) + ((3*I
)/16)/(I + Cot[x])

Rule 3516

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[b/f, Subst[Int
[(x^m*(a + x)^n)/(b^2 + x^2)^(m/2 + 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && IntegerQ[m/
2]

Rule 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\cos ^4(x)}{i+\cot (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{x^4}{(i+x) \left (1+x^2\right )^3} \, dx,x,\cot (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{x^4}{(-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{5}{16 (i+x)^3}+\frac{3 i}{16 (i+x)^2}-\frac{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{5}{32 (i+\cot (x))^2}+\frac{3 i}{16 (i+\cot (x))}+\frac{1}{16} i \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\cot (x)\right )\\ &=-\frac{i x}{16}+\frac{1}{32 (i-\cot (x))^2}+\frac{i}{8 (i-\cot (x))}-\frac{i}{24 (i+\cot (x))^3}+\frac{5}{32 (i+\cot (x))^2}+\frac{3 i}{16 (i+\cot (x))}\\ \end{align*}

Mathematica [A]  time = 0.0873442, size = 60, normalized size = 0.77 \[ -\frac{1}{192} i \left (12 x+3 \sin (2 x)-3 \sin (4 x)-\sin (6 x)-18 i \cos ^2(x)-6 i \cos (2 x)-6 i \cos (4 x)-i \cos (6 x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]^4/(I + Cot[x]),x]

[Out]

(-I/192)*(12*x - (18*I)*Cos[x]^2 - (6*I)*Cos[2*x] - (6*I)*Cos[4*x] - I*Cos[6*x] + 3*Sin[2*x] - 3*Sin[4*x] - Si
n[6*x])

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Maple [A]  time = 0.184, size = 56, normalized size = 0.7 \begin{align*}{\frac{-{\frac{i}{16}}}{\tan \left ( x \right ) +i}}+{\frac{1}{32\, \left ( \tan \left ( x \right ) +i \right ) ^{2}}}+{\frac{\ln \left ( \tan \left ( x \right ) +i \right ) }{32}}-{\frac{{\frac{i}{24}}}{ \left ( \tan \left ( x \right ) -i \right ) ^{3}}}+{\frac{1}{32\, \left ( \tan \left ( x \right ) -i \right ) ^{2}}}-{\frac{\ln \left ( \tan \left ( x \right ) -i \right ) }{32}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)^4/(I+cot(x)),x)

[Out]

-1/16*I/(tan(x)+I)+1/32/(tan(x)+I)^2+1/32*ln(tan(x)+I)-1/24*I/(tan(x)-I)^3+1/32/(tan(x)-I)^2-1/32*ln(tan(x)-I)

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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.

[In]

integrate(cos(x)^4/(I+cot(x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 1.58688, size = 223, normalized size = 2.86 \begin{align*} \frac{1}{384} \,{\left (48 i \, x e^{\left (6 i \, x\right )} +{\left (-72 i \, x e^{\left (4 i \, x\right )} - 3 \, e^{\left (8 i \, x\right )} - 24 \, e^{\left (6 i \, x\right )} + 24 \, e^{\left (2 i \, x\right )} + 3\right )} e^{\left (2 i \, x\right )} + 6 \, e^{\left (8 i \, x\right )} - 36 \, e^{\left (4 i \, x\right )} - 12 \, e^{\left (2 i \, x\right )} - 2\right )} e^{\left (-6 i \, x\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^4/(I+cot(x)),x, algorithm="fricas")

[Out]

1/384*(48*I*x*e^(6*I*x) + (-72*I*x*e^(4*I*x) - 3*e^(8*I*x) - 24*e^(6*I*x) + 24*e^(2*I*x) + 3)*e^(2*I*x) + 6*e^
(8*I*x) - 36*e^(4*I*x) - 12*e^(2*I*x) - 2)*e^(-6*I*x)

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Sympy [A]  time = 1.02413, size = 54, normalized size = 0.69 \begin{align*} - \frac{i x}{16} - \frac{e^{4 i x}}{128} - \frac{3 e^{2 i x}}{64} - \frac{e^{- 2 i x}}{32} - \frac{3 e^{- 4 i x}}{128} - \frac{e^{- 6 i x}}{192} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)**4/(I+cot(x)),x)

[Out]

-I*x/16 - exp(4*I*x)/128 - 3*exp(2*I*x)/64 - exp(-2*I*x)/32 - 3*exp(-4*I*x)/128 - exp(-6*I*x)/192

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Giac [A]  time = 1.25301, size = 85, normalized size = 1.09 \begin{align*} \frac{3 \, \tan \left (x\right )^{2} + 10 i \, \tan \left (x\right ) - 9}{64 \,{\left (-i \, \tan \left (x\right ) + 1\right )}^{2}} + \frac{11 \, \tan \left (x\right )^{3} - 33 i \, \tan \left (x\right )^{2} - 27 \, \tan \left (x\right ) - 3 i}{192 \,{\left (\tan \left (x\right ) - i\right )}^{3}} + \frac{1}{32} \, \log \left (\tan \left (x\right ) + i\right ) - \frac{1}{32} \, \log \left (\tan \left (x\right ) - i\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)^4/(I+cot(x)),x, algorithm="giac")

[Out]

1/64*(3*tan(x)^2 + 10*I*tan(x) - 9)/(-I*tan(x) + 1)^2 + 1/192*(11*tan(x)^3 - 33*I*tan(x)^2 - 27*tan(x) - 3*I)/
(tan(x) - I)^3 + 1/32*log(tan(x) + I) - 1/32*log(tan(x) - I)