3.3.20 \(\int \csc (\coth (a+b x)) \, dx\) [220]

Optimal. Leaf size=67 \[ \frac {1}{2} \text {Int}\left (\frac {\csc (\coth (a+b x)) \text {csch}^2(a+b x)}{-1+\coth (a+b x)},x\right )-\frac {1}{2} \text {Int}\left (\frac {\csc (\coth (a+b x)) \text {csch}^2(a+b x)}{1+\coth (a+b x)},x\right ) \]

[Out]

1/2*Unintegrable(csc(coth(b*x+a))*csch(b*x+a)^2/(-1+coth(b*x+a)),x)-1/2*Unintegrable(csc(coth(b*x+a))*csch(b*x
+a)^2/(1+coth(b*x+a)),x)

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Rubi [A]
time = 0.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \csc (\coth (a+b x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Csc[Coth[a + b*x]],x]

[Out]

-1/2*Defer[Subst][Defer[Int][Csc[x]/(-1 + x), x], x, Coth[a + b*x]]/b + Defer[Subst][Defer[Int][Csc[x]/(1 + x)
, x], x, Coth[a + b*x]]/(2*b)

Rubi steps

\begin {align*} \int \csc (\coth (a+b x)) \, dx &=\frac {\text {Subst}\left (\int \frac {\csc (x)}{1-x^2} \, dx,x,\coth (a+b x)\right )}{b}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {\csc (x)}{2 (-1+x)}+\frac {\csc (x)}{2 (1+x)}\right ) \, dx,x,\coth (a+b x)\right )}{b}\\ &=-\frac {\text {Subst}\left (\int \frac {\csc (x)}{-1+x} \, dx,x,\coth (a+b x)\right )}{2 b}+\frac {\text {Subst}\left (\int \frac {\csc (x)}{1+x} \, dx,x,\coth (a+b x)\right )}{2 b}\\ \end {align*}

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Mathematica [A]
time = 1.94, size = 0, normalized size = 0.00 \begin {gather*} \int \csc (\coth (a+b x)) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Csc[Coth[a + b*x]],x]

[Out]

Integrate[Csc[Coth[a + b*x]], x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \csc \left (\coth \left (b x +a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(coth(b*x+a)),x)

[Out]

int(csc(coth(b*x+a)),x)

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(coth(b*x+a)),x, algorithm="maxima")

[Out]

integrate(csc(coth(b*x + a)), x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(coth(b*x+a)),x, algorithm="fricas")

[Out]

integral(csc(coth(b*x + a)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \csc {\left (\coth {\left (a + b x \right )} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(coth(b*x+a)),x)

[Out]

Integral(csc(coth(a + b*x)), x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(coth(b*x+a)),x, algorithm="giac")

[Out]

integrate(csc(coth(b*x + a)), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sin \left (\mathrm {coth}\left (a+b\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/sin(coth(a + b*x)),x)

[Out]

int(1/sin(coth(a + b*x)), x)

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