3.1.61 \(\int \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx\) [61]

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2},x\right ) \]

[Out]

Unintegrable(1/(d*x+c)^2/(a+b*coth(f*x+e))^2,x)

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Rubi [A]
time = 0.04, 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 \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[1/((c + d*x)^2*(a + b*Coth[e + f*x])^2),x]

[Out]

Defer[Int][1/((c + d*x)^2*(a + b*Coth[e + f*x])^2), x]

Rubi steps

\begin {align*} \int \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx &=\int \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx\\ \end {align*}

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Mathematica [A]
time = 80.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((c + d*x)^2*(a + b*Coth[e + f*x])^2),x]

[Out]

Integrate[1/((c + d*x)^2*(a + b*Coth[e + f*x])^2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (d x +c \right )^{2} \left (a +b \coth \left (f x +e \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d*x+c)^2/(a+b*coth(f*x+e))^2,x)

[Out]

int(1/(d*x+c)^2/(a+b*coth(f*x+e))^2,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(1/(d*x+c)^2/(a+b*coth(f*x+e))^2,x, algorithm="maxima")

[Out]

-(a^2*c*f - 2*a*b*c*f + (c*f - 2*d)*b^2 + (a^2*d*f - 2*a*b*d*f + b^2*d*f)*x - ((a^2*d*f - b^2*d*f)*x*e^(2*e) +
(a^2*c*f - b^2*c*f)*e^(2*e))*e^(2*f*x))/(a^4*c^2*d*f - 2*a^2*b^2*c^2*d*f + b^4*c^2*d*f + (a^4*d^3*f - 2*a^2*b
^2*d^3*f + b^4*d^3*f)*x^2 + 2*(a^4*c*d^2*f - 2*a^2*b^2*c*d^2*f + b^4*c*d^2*f)*x - ((a^4*d^3*f + 2*a^3*b*d^3*f
- 2*a*b^3*d^3*f - b^4*d^3*f)*x^2*e^(2*e) + 2*(a^4*c*d^2*f + 2*a^3*b*c*d^2*f - 2*a*b^3*c*d^2*f - b^4*c*d^2*f)*x
*e^(2*e) + (a^4*c^2*d*f + 2*a^3*b*c^2*d*f - 2*a*b^3*c^2*d*f - b^4*c^2*d*f)*e^(2*e))*e^(2*f*x)) - integrate(-4*
(a*b*d*f*x + a*b*c*f + b^2*d)/(a^4*c^3*f - 2*a^2*b^2*c^3*f + b^4*c^3*f + (a^4*d^3*f - 2*a^2*b^2*d^3*f + b^4*d^
3*f)*x^3 + 3*(a^4*c*d^2*f - 2*a^2*b^2*c*d^2*f + b^4*c*d^2*f)*x^2 + 3*(a^4*c^2*d*f - 2*a^2*b^2*c^2*d*f + b^4*c^
2*d*f)*x - ((a^4*d^3*f + 2*a^3*b*d^3*f - 2*a*b^3*d^3*f - b^4*d^3*f)*x^3*e^(2*e) + 3*(a^4*c*d^2*f + 2*a^3*b*c*d
^2*f - 2*a*b^3*c*d^2*f - b^4*c*d^2*f)*x^2*e^(2*e) + 3*(a^4*c^2*d*f + 2*a^3*b*c^2*d*f - 2*a*b^3*c^2*d*f - b^4*c
^2*d*f)*x*e^(2*e) + (a^4*c^3*f + 2*a^3*b*c^3*f - 2*a*b^3*c^3*f - b^4*c^3*f)*e^(2*e))*e^(2*f*x)), 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(1/(d*x+c)^2/(a+b*coth(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*d^2*x^2 + 2*a^2*c*d*x + a^2*c^2 + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*coth(f*x + e)^2 + 2*(a
*b*d^2*x^2 + 2*a*b*c*d*x + a*b*c^2)*coth(f*x + e)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \coth {\left (e + f x \right )}\right )^{2} \left (c + d x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d*x+c)**2/(a+b*coth(f*x+e))**2,x)

[Out]

Integral(1/((a + b*coth(e + f*x))**2*(c + d*x)**2), 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(1/(d*x+c)^2/(a+b*coth(f*x+e))^2,x, algorithm="giac")

[Out]

integrate(1/((d*x + c)^2*(b*coth(f*x + e) + a)^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {coth}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*coth(e + f*x))^2*(c + d*x)^2),x)

[Out]

int(1/((a + b*coth(e + f*x))^2*(c + d*x)^2), x)

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