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

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 = 0.06, 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 = {} \[ \int \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx \]

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 = 125.76, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c+d x)^2 (a+b \coth (e+f x))^2} \, dx \]

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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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d^{2} x^{2} + 2 \, a^{2} c d x + a^{2} c^{2} + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \coth \left (f x + e\right )^{2} + 2 \, {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2}\right )} \coth \left (f x + e\right )}, x\right ) \]

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

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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maple [A]  time = 1.76, 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 \[ -\frac {a^{2} c f - 2 \, a b c f + {\left (c f - 2 \, d\right )} b^{2} + {\left (a^{2} d f - 2 \, a b d f + b^{2} d f\right )} x - {\left (a^{2} c f e^{\left (2 \, e\right )} - b^{2} c f e^{\left (2 \, e\right )} + {\left (a^{2} d f e^{\left (2 \, e\right )} - b^{2} d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}{a^{4} c^{2} d f - 2 \, a^{2} b^{2} c^{2} d f + b^{4} c^{2} d f + {\left (a^{4} d^{3} f - 2 \, a^{2} b^{2} d^{3} f + b^{4} d^{3} f\right )} x^{2} + 2 \, {\left (a^{4} c d^{2} f - 2 \, a^{2} b^{2} c d^{2} f + b^{4} c d^{2} f\right )} x - {\left (a^{4} c^{2} d f e^{\left (2 \, e\right )} + 2 \, a^{3} b c^{2} d f e^{\left (2 \, e\right )} - 2 \, a b^{3} c^{2} d f e^{\left (2 \, e\right )} - b^{4} c^{2} d f e^{\left (2 \, e\right )} + {\left (a^{4} d^{3} f e^{\left (2 \, e\right )} + 2 \, a^{3} b d^{3} f e^{\left (2 \, e\right )} - 2 \, a b^{3} d^{3} f e^{\left (2 \, e\right )} - b^{4} d^{3} f e^{\left (2 \, e\right )}\right )} x^{2} + 2 \, {\left (a^{4} c d^{2} f e^{\left (2 \, e\right )} + 2 \, a^{3} b c d^{2} f e^{\left (2 \, e\right )} - 2 \, a b^{3} c d^{2} f e^{\left (2 \, e\right )} - b^{4} c d^{2} f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}} - \int -\frac {4 \, {\left (a b d f x + a b c f + b^{2} d\right )}}{a^{4} c^{3} f - 2 \, a^{2} b^{2} c^{3} f + b^{4} c^{3} f + {\left (a^{4} d^{3} f - 2 \, a^{2} b^{2} d^{3} f + b^{4} d^{3} f\right )} x^{3} + 3 \, {\left (a^{4} c d^{2} f - 2 \, a^{2} b^{2} c d^{2} f + b^{4} c d^{2} f\right )} x^{2} + 3 \, {\left (a^{4} c^{2} d f - 2 \, a^{2} b^{2} c^{2} d f + b^{4} c^{2} d f\right )} x - {\left (a^{4} c^{3} f e^{\left (2 \, e\right )} + 2 \, a^{3} b c^{3} f e^{\left (2 \, e\right )} - 2 \, a b^{3} c^{3} f e^{\left (2 \, e\right )} - b^{4} c^{3} f e^{\left (2 \, e\right )} + {\left (a^{4} d^{3} f e^{\left (2 \, e\right )} + 2 \, a^{3} b d^{3} f e^{\left (2 \, e\right )} - 2 \, a b^{3} d^{3} f e^{\left (2 \, e\right )} - b^{4} d^{3} f e^{\left (2 \, e\right )}\right )} x^{3} + 3 \, {\left (a^{4} c d^{2} f e^{\left (2 \, e\right )} + 2 \, a^{3} b c d^{2} f e^{\left (2 \, e\right )} - 2 \, a b^{3} c d^{2} f e^{\left (2 \, e\right )} - b^{4} c d^{2} f e^{\left (2 \, e\right )}\right )} x^{2} + 3 \, {\left (a^{4} c^{2} d f e^{\left (2 \, e\right )} + 2 \, a^{3} b c^{2} d f e^{\left (2 \, e\right )} - 2 \, a b^{3} c^{2} d f e^{\left (2 \, e\right )} - b^{4} c^{2} d f e^{\left (2 \, e\right )}\right )} x\right )} e^{\left (2 \, f x\right )}}\,{d x} \]

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*c*f*e^(2*e) - b^2*c*f*e^(2*
e) + (a^2*d*f*e^(2*e) - b^2*d*f*e^(2*e))*x)*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*c^2*d*f*e
^(2*e) + 2*a^3*b*c^2*d*f*e^(2*e) - 2*a*b^3*c^2*d*f*e^(2*e) - b^4*c^2*d*f*e^(2*e) + (a^4*d^3*f*e^(2*e) + 2*a^3*
b*d^3*f*e^(2*e) - 2*a*b^3*d^3*f*e^(2*e) - b^4*d^3*f*e^(2*e))*x^2 + 2*(a^4*c*d^2*f*e^(2*e) + 2*a^3*b*c*d^2*f*e^
(2*e) - 2*a*b^3*c*d^2*f*e^(2*e) - b^4*c*d^2*f*e^(2*e))*x)*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*c^3*f*e^(2
*e) + 2*a^3*b*c^3*f*e^(2*e) - 2*a*b^3*c^3*f*e^(2*e) - b^4*c^3*f*e^(2*e) + (a^4*d^3*f*e^(2*e) + 2*a^3*b*d^3*f*e
^(2*e) - 2*a*b^3*d^3*f*e^(2*e) - b^4*d^3*f*e^(2*e))*x^3 + 3*(a^4*c*d^2*f*e^(2*e) + 2*a^3*b*c*d^2*f*e^(2*e) - 2
*a*b^3*c*d^2*f*e^(2*e) - b^4*c*d^2*f*e^(2*e))*x^2 + 3*(a^4*c^2*d*f*e^(2*e) + 2*a^3*b*c^2*d*f*e^(2*e) - 2*a*b^3
*c^2*d*f*e^(2*e) - b^4*c^2*d*f*e^(2*e))*x)*e^(2*f*x)), x)

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

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

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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