3.8 \(\int \frac {\csc ^4(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=15 \[ -\frac {\cot ^2(x)}{2}+i \cot (x) \]

[Out]

I*cot(x)-1/2*cot(x)^2

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Rubi [A]  time = 0.03, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3487} \[ -\frac {\cot ^2(x)}{2}+i \cot (x) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^4/(I + Cot[x]),x]

[Out]

I*Cot[x] - Cot[x]^2/2

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rubi steps

\begin {align*} \int \frac {\csc ^4(x)}{i+\cot (x)} \, dx &=\operatorname {Subst}(\int (i-x) \, dx,x,\cot (x))\\ &=i \cot (x)-\frac {\cot ^2(x)}{2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 15, normalized size = 1.00 \[ -\frac {\csc ^2(x)}{2}+i \cot (x) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^4/(I + Cot[x]),x]

[Out]

I*Cot[x] - Csc[x]^2/2

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fricas [A]  time = 0.52, size = 16, normalized size = 1.07 \[ \frac {2}{e^{\left (4 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/(e^(4*I*x) - 2*e^(2*I*x) + 1)

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giac [A]  time = 1.97, size = 12, normalized size = 0.80 \[ -\frac {-2 i \, \tan \relax (x) + 1}{2 \, \tan \relax (x)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-2*I*tan(x) + 1)/tan(x)^2

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maple [A]  time = 0.30, size = 15, normalized size = 1.00 \[ -\frac {1}{2 \tan \relax (x )^{2}}+\frac {i}{\tan \relax (x )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2/tan(x)^2+I/tan(x)

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maxima [A]  time = 1.02, size = 12, normalized size = 0.80 \[ \frac {2 i \, \tan \relax (x) - 1}{2 \, \tan \relax (x)^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*I*tan(x) - 1)/tan(x)^2

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mupad [B]  time = 0.16, size = 9, normalized size = 0.60 \[ -\frac {\mathrm {cot}\relax (x)\,\left (\mathrm {cot}\relax (x)-2{}\mathrm {i}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^4*(cot(x) + 1i)),x)

[Out]

-(cot(x)*(cot(x) - 2i))/2

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{4}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(csc(x)**4/(cot(x) + I), x)

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