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] - Cot[x]^2/2

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Rubi [A]  time = 0.0327547, antiderivative size = 15, normalized size of antiderivative = 1., 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*}

Mathematica [A]  time = 0.0264791, size = 15, normalized size = 1. \[ -\frac{\csc ^2(x)}{2}+i \cot (x) \]

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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Maple [A]  time = 0.051, size = 15, normalized size = 1. \begin{align*} -{\frac{1}{2\, \left ( \tan \left ( x \right ) \right ) ^{2}}}+{\frac{i}{\tan \left ( x \right ) }} \end{align*}

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 = 1.23016, size = 16, normalized size = 1.07 \begin{align*} \frac{2 i \, \tan \left (x\right ) - 1}{2 \, \tan \left (x\right )^{2}} \end{align*}

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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Fricas [B]  time = 1.57706, size = 140, normalized size = 9.33 \begin{align*} \frac{2 \,{\left ({\left (3 \, e^{\left (2 i \, x\right )} - 1\right )} e^{\left (2 i \, x\right )} - 2 \, e^{\left (2 i \, x\right )}\right )} e^{\left (-2 i \, x\right )}}{3 \,{\left (e^{\left (6 i \, x\right )} - 3 \, e^{\left (4 i \, x\right )} + 3 \, e^{\left (2 i \, x\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*((3*e^(2*I*x) - 1)*e^(2*I*x) - 2*e^(2*I*x))*e^(-2*I*x)/(e^(6*I*x) - 3*e^(4*I*x) + 3*e^(2*I*x) - 1)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.25502, size = 16, normalized size = 1.07 \begin{align*} -\frac{-2 i \, \tan \left (x\right ) + 1}{2 \, \tan \left (x\right )^{2}} \end{align*}

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