3.7 \(\int \frac {\csc ^3(x)}{i+\cot (x)} \, dx\)

Optimal. Leaf size=12 \[ -\csc (x)+i \tanh ^{-1}(\cos (x)) \]

[Out]

I*arctanh(cos(x))-csc(x)

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Rubi [A]  time = 0.03, antiderivative size = 12, 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 = {3501, 3770} \[ -\csc (x)+i \tanh ^{-1}(\cos (x)) \]

Antiderivative was successfully verified.

[In]

Int[Csc[x]^3/(I + Cot[x]),x]

[Out]

I*ArcTanh[Cos[x]] - Csc[x]

Rule 3501

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d^2*
(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1))/(b*f*(m + n - 1)), x] + Dist[(d^2*(m - 2))/(a*(m + n -
1)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2
+ b^2, 0] && LtQ[n, 0] && GtQ[m, 1] &&  !ILtQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \frac {\csc ^3(x)}{i+\cot (x)} \, dx &=-\csc (x)-i \int \csc (x) \, dx\\ &=i \tanh ^{-1}(\cos (x))-\csc (x)\\ \end {align*}

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Mathematica [B]  time = 0.05, size = 26, normalized size = 2.17 \[ -\csc (x)+i \left (\log \left (\cos \left (\frac {x}{2}\right )\right )-\log \left (\sin \left (\frac {x}{2}\right )\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[x]^3/(I + Cot[x]),x]

[Out]

-Csc[x] + I*(Log[Cos[x/2]] - Log[Sin[x/2]])

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fricas [B]  time = 0.70, size = 48, normalized size = 4.00 \[ \frac {{\left (i \, e^{\left (2 i \, x\right )} - i\right )} \log \left (e^{\left (i \, x\right )} + 1\right ) + {\left (-i \, e^{\left (2 i \, x\right )} + i\right )} \log \left (e^{\left (i \, x\right )} - 1\right ) - 2 i \, e^{\left (i \, x\right )}}{e^{\left (2 i \, x\right )} - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(I+cot(x)),x, algorithm="fricas")

[Out]

((I*e^(2*I*x) - I)*log(e^(I*x) + 1) + (-I*e^(2*I*x) + I)*log(e^(I*x) - 1) - 2*I*e^(I*x))/(e^(2*I*x) - 1)

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giac [B]  time = 2.45, size = 30, normalized size = 2.50 \[ -\frac {-2 i \, \tan \left (\frac {1}{2} \, x\right ) + 1}{2 \, \tan \left (\frac {1}{2} \, x\right )} - i \, \log \left (\tan \left (\frac {1}{2} \, x\right )\right ) - \frac {1}{2} \, \tan \left (\frac {1}{2} \, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(I+cot(x)),x, algorithm="giac")

[Out]

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

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maple [B]  time = 0.25, size = 24, normalized size = 2.00 \[ -\frac {\tan \left (\frac {x}{2}\right )}{2}-i \ln \left (\tan \left (\frac {x}{2}\right )\right )-\frac {1}{2 \tan \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(x)^3/(I+cot(x)),x)

[Out]

-1/2*tan(1/2*x)-I*ln(tan(1/2*x))-1/2/tan(1/2*x)

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maxima [B]  time = 0.54, size = 33, normalized size = 2.75 \[ -\frac {\cos \relax (x) + 1}{2 \, \sin \relax (x)} - \frac {\sin \relax (x)}{2 \, {\left (\cos \relax (x) + 1\right )}} - i \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)^3/(I+cot(x)),x, algorithm="maxima")

[Out]

-1/2*(cos(x) + 1)/sin(x) - 1/2*sin(x)/(cos(x) + 1) - I*log(sin(x)/(cos(x) + 1))

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mupad [B]  time = 0.23, size = 23, normalized size = 1.92 \[ -\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{2}-\frac {1}{2\,\mathrm {tan}\left (\frac {x}{2}\right )}-\ln \left (\mathrm {tan}\left (\frac {x}{2}\right )\right )\,1{}\mathrm {i} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(x)^3*(cot(x) + 1i)),x)

[Out]

- tan(x/2)/2 - log(tan(x/2))*1i - 1/(2*tan(x/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{3}{\relax (x )}}{\cot {\relax (x )} + i}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(x)**3/(I+cot(x)),x)

[Out]

Integral(csc(x)**3/(cot(x) + I), x)

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