3.1.46 \(\int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx\) [46]

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

[Out]

Unintegrable((a+b*coth(f*x+e))^2/(d*x+c)^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 {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 19.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \coth (e+f x))^2}{(c+d x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

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

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

[Out]

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

[Out]

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

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

[Out]

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

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