3.32 \(\int \frac{\text{csch}^2(a+b x)}{(c+d x)^2} \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left (\frac{\text{csch}^2(a+b x)}{(c+d x)^2},x\right ) \]

[Out]

Unintegrable[Csch[a + b*x]^2/(c + d*x)^2, x]

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Rubi [A]  time = 0.0378597, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\text{csch}^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Csch[a + b*x]^2/(c + d*x)^2,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 18.6732, size = 0, normalized size = 0. \[ \int \frac{\text{csch}^2(a+b x)}{(c+d x)^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}}{ \left ( dx+c \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+a)^2/(d*x+c)^2,x)

[Out]

int(csch(b*x+a)^2/(d*x+c)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} 4 \, d \int \frac{1}{2 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} +{\left (b d^{3} x^{3} e^{a} + 3 \, b c d^{2} x^{2} e^{a} + 3 \, b c^{2} d x e^{a} + b c^{3} e^{a}\right )} e^{\left (b x\right )}\right )}}\,{d x} - 4 \, d \int -\frac{1}{2 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} -{\left (b d^{3} x^{3} e^{a} + 3 \, b c d^{2} x^{2} e^{a} + 3 \, b c^{2} d x e^{a} + b c^{3} e^{a}\right )} e^{\left (b x\right )}\right )}}\,{d x} + \frac{2}{b d^{2} x^{2} + 2 \, b c d x + b c^{2} -{\left (b d^{2} x^{2} e^{\left (2 \, a\right )} + 2 \, b c d x e^{\left (2 \, a\right )} + b c^{2} e^{\left (2 \, a\right )}\right )} e^{\left (2 \, b x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^2/(d*x+c)^2,x, algorithm="maxima")

[Out]

4*d*integrate(1/2/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + (b*d^3*x^3*e^a + 3*b*c*d^2*x^2*e^a + 3*b*
c^2*d*x*e^a + b*c^3*e^a)*e^(b*x)), x) - 4*d*integrate(-1/2/(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 -
(b*d^3*x^3*e^a + 3*b*c*d^2*x^2*e^a + 3*b*c^2*d*x*e^a + b*c^3*e^a)*e^(b*x)), x) + 2/(b*d^2*x^2 + 2*b*c*d*x + b*
c^2 - (b*d^2*x^2*e^(2*a) + 2*b*c*d*x*e^(2*a) + b*c^2*e^(2*a))*e^(2*b*x))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{csch}\left (b x + a\right )^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^2/(d*x+c)^2,x, algorithm="fricas")

[Out]

integral(csch(b*x + a)^2/(d^2*x^2 + 2*c*d*x + c^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)**2/(d*x+c)**2,x)

[Out]

Integral(csch(a + b*x)**2/(c + d*x)**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{csch}\left (b x + a\right )^{2}}{{\left (d x + c\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+a)^2/(d*x+c)^2,x, algorithm="giac")

[Out]

integrate(csch(b*x + a)^2/(d*x + c)^2, x)