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

Optimal. Leaf size=29 \[ \frac{d \log (\sinh (a+b x))}{b^2}-\frac{(c+d x) \coth (a+b x)}{b} \]

[Out]

-(((c + d*x)*Coth[a + b*x])/b) + (d*Log[Sinh[a + b*x]])/b^2

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Rubi [A]  time = 0.0310169, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {4184, 3475} \[ \frac{d \log (\sinh (a+b x))}{b^2}-\frac{(c+d x) \coth (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)*Csch[a + b*x]^2,x]

[Out]

-(((c + d*x)*Coth[a + b*x])/b) + (d*Log[Sinh[a + b*x]])/b^2

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0933455, size = 52, normalized size = 1.79 \[ \frac{d \log (\sinh (a+b x))}{b^2}-\frac{c \coth (a+b x)}{b}-\frac{d x \coth (a)}{b}+\frac{d x \text{csch}(a) \sinh (b x) \text{csch}(a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((d*x*Coth[a])/b) - (c*Coth[a + b*x])/b + (d*Log[Sinh[a + b*x]])/b^2 + (d*x*Csch[a]*Csch[a + b*x]*Sinh[b*x])/
b

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Maple [A]  time = 0.016, size = 56, normalized size = 1.9 \begin{align*} -2\,{\frac{dx}{b}}-2\,{\frac{da}{{b}^{2}}}-2\,{\frac{dx+c}{b \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }}+{\frac{d\ln \left ({{\rm e}^{2\,bx+2\,a}}-1 \right ) }{{b}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*d/b*x-2*d/b^2*a-2/b*(d*x+c)/(exp(2*b*x+2*a)-1)+d/b^2*ln(exp(2*b*x+2*a)-1)

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Maxima [B]  time = 1.12236, size = 123, normalized size = 4.24 \begin{align*} -d{\left (\frac{2 \, x e^{\left (2 \, b x + 2 \, a\right )}}{b e^{\left (2 \, b x + 2 \, a\right )} - b} - \frac{\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} - \frac{\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}}\right )} + \frac{2 \, c}{b{\left (e^{\left (-2 \, b x - 2 \, a\right )} - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-d*(2*x*e^(2*b*x + 2*a)/(b*e^(2*b*x + 2*a) - b) - log((e^(b*x + a) + 1)*e^(-a))/b^2 - log((e^(b*x + a) - 1)*e^
(-a))/b^2) + 2*c/(b*(e^(-2*b*x - 2*a) - 1))

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Fricas [B]  time = 1.60578, size = 431, normalized size = 14.86 \begin{align*} -\frac{2 \, b d x \cosh \left (b x + a\right )^{2} + 4 \, b d x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + 2 \, b d x \sinh \left (b x + a\right )^{2} + 2 \, b c -{\left (d \cosh \left (b x + a\right )^{2} + 2 \, d \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d \sinh \left (b x + a\right )^{2} - d\right )} \log \left (\frac{2 \, \sinh \left (b x + a\right )}{\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )}\right )}{b^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} \sinh \left (b x + a\right )^{2} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*d*x*cosh(b*x + a)^2 + 4*b*d*x*cosh(b*x + a)*sinh(b*x + a) + 2*b*d*x*sinh(b*x + a)^2 + 2*b*c - (d*cosh(b*
x + a)^2 + 2*d*cosh(b*x + a)*sinh(b*x + a) + d*sinh(b*x + a)^2 - d)*log(2*sinh(b*x + a)/(cosh(b*x + a) - sinh(
b*x + a))))/(b^2*cosh(b*x + a)^2 + 2*b^2*cosh(b*x + a)*sinh(b*x + a) + b^2*sinh(b*x + a)^2 - b^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1226, size = 108, normalized size = 3.72 \begin{align*} -\frac{2 \, b d x e^{\left (2 \, b x + 2 \, a\right )} - d e^{\left (2 \, b x + 2 \, a\right )} \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right ) + 2 \, b c + d \log \left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}{b^{2} e^{\left (2 \, b x + 2 \, a\right )} - b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*b*d*x*e^(2*b*x + 2*a) - d*e^(2*b*x + 2*a)*log(e^(2*b*x + 2*a) - 1) + 2*b*c + d*log(e^(2*b*x + 2*a) - 1))/(
b^2*e^(2*b*x + 2*a) - b^2)