3.6 \(\int \frac {\csc ^2(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=9 \[ \log (\sin (x))-i x \]

[Out]

-I*x+ln(sin(x))

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Rubi [A]  time = 0.03, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3487, 31} \[ \log (\sin (x))-i x \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^2/(I + Cot[x]),x]

[Out]

(-I)*x + Log[Sin[x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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 ^2(x)}{i+\cot (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{i+x} \, dx,x,\cot (x)\right )\\ &=-i x+\log (\sin (x))\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \[ \log (\sin (x))-i x \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^2/(I + Cot[x]),x]

[Out]

(-I)*x + Log[Sin[x]]

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fricas [A]  time = 0.43, size = 11, normalized size = 1.22 \[ -2 i \, x + \log \left (e^{\left (2 i \, x\right )} - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(I+cot(x)),x, algorithm="fricas")

[Out]

-2*I*x + log(e^(2*I*x) - 1)

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giac [B]  time = 0.49, size = 15, normalized size = 1.67 \[ -2 \, \log \left (\tan \left (\frac {1}{2} \, x\right ) - i\right ) + \log \left (\tan \left (\frac {1}{2} \, x\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(I+cot(x)),x, algorithm="giac")

[Out]

-2*log(tan(1/2*x) - I) + log(tan(1/2*x))

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maple [A]  time = 0.29, size = 9, normalized size = 1.00 \[ -\ln \left (i+\cot \relax (x )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^2/(I+cot(x)),x)

[Out]

-ln(I+cot(x))

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maxima [A]  time = 1.08, size = 7, normalized size = 0.78 \[ -\log \left (\cot \relax (x) + i\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^2/(I+cot(x)),x, algorithm="maxima")

[Out]

-log(cot(x) + I)

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mupad [B]  time = 0.25, size = 11, normalized size = 1.22 \[ -\mathrm {atan}\left (2\,\mathrm {tan}\relax (x)-\mathrm {i}\right )\,2{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^2*(cot(x) + 1i)),x)

[Out]

-atan(2*tan(x) - 1i)*2i

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sympy [A]  time = 4.82, size = 7, normalized size = 0.78 \[ - \log {\left (\cot {\relax (x )} + i \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**2/(I+cot(x)),x)

[Out]

-log(cot(x) + I)

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