3.152 \(\int \text {csch}(c+b x) \sinh (a+b x) \, dx\)

Optimal. Leaf size=26 \[ \frac {\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]

[Out]

x*cosh(a-c)+ln(sinh(b*x+c))*sinh(a-c)/b

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5626, 3475, 8} \[ \frac {\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]

Antiderivative was successfully verified.

[In]

Int[Csch[c + b*x]*Sinh[a + b*x],x]

[Out]

x*Cosh[a - c] + (Log[Sinh[c + b*x]]*Sinh[a - c])/b

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 3475

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

Rule 5626

Int[Csch[w_]^(n_.)*Sinh[v_], x_Symbol] :> Dist[Sinh[v - w], Int[Coth[w]*Csch[w]^(n - 1), x], x] + Dist[Cosh[v
- w], Int[Csch[w]^(n - 1), x], x] /; GtQ[n, 0] && NeQ[w, v] && FreeQ[v - w, x]

Rubi steps

\begin {align*} \int \text {csch}(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int 1 \, dx+\sinh (a-c) \int \coth (c+b x) \, dx\\ &=x \cosh (a-c)+\frac {\log (\sinh (c+b x)) \sinh (a-c)}{b}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 26, normalized size = 1.00 \[ \frac {\sinh (a-c) \log (\sinh (b x+c))}{b}+x \cosh (a-c) \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[c + b*x]*Sinh[a + b*x],x]

[Out]

x*Cosh[a - c] + (Log[Sinh[c + b*x]]*Sinh[a - c])/b

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fricas [B]  time = 0.43, size = 86, normalized size = 3.31 \[ \frac {2 \, b x + {\left (\cosh \left (-a + c\right )^{2} - 2 \, \cosh \left (-a + c\right ) \sinh \left (-a + c\right ) + \sinh \left (-a + c\right )^{2} - 1\right )} \log \left (\frac {2 \, \sinh \left (b x + c\right )}{\cosh \left (b x + c\right ) - \sinh \left (b x + c\right )}\right )}{2 \, {\left (b \cosh \left (-a + c\right ) - b \sinh \left (-a + c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)*sinh(b*x+a),x, algorithm="fricas")

[Out]

1/2*(2*b*x + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*log(2*sinh(b*x + c)/(cosh(b*x
+ c) - sinh(b*x + c))))/(b*cosh(-a + c) - b*sinh(-a + c))

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giac [A]  time = 0.11, size = 51, normalized size = 1.96 \[ \frac {2 \, b x e^{\left (-a + c\right )} + {\left (e^{\left (2 \, a + c\right )} - e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left ({\left | e^{\left (2 \, b x + 2 \, c\right )} - 1 \right |}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)*sinh(b*x+a),x, algorithm="giac")

[Out]

1/2*(2*b*x*e^(-a + c) + (e^(2*a + c) - e^(3*c))*e^(-a - 2*c)*log(abs(e^(2*b*x + 2*c) - 1)))/b

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maple [B]  time = 0.14, size = 150, normalized size = 5.77 \[ x \,{\mathrm e}^{a -c}+{\mathrm e}^{-a -c} {\mathrm e}^{2 c} x -{\mathrm e}^{-a -c} {\mathrm e}^{2 a} x +\frac {{\mathrm e}^{-a -c} {\mathrm e}^{2 c} a}{b}-\frac {{\mathrm e}^{-a -c} {\mathrm e}^{2 a} a}{b}+\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 a}}{2 b}-\frac {\ln \left ({\mathrm e}^{2 b x +2 a}-{\mathrm e}^{2 a -2 c}\right ) {\mathrm e}^{-a -c} {\mathrm e}^{2 c}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+c)*sinh(b*x+a),x)

[Out]

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

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maxima [B]  time = 0.32, size = 84, normalized size = 3.23 \[ \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} + e^{c}\right )}{2 \, b} + \frac {{\left (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )} \log \left (e^{\left (-b x\right )} - e^{c}\right )}{2 \, b} + \frac {{\left (b x + a\right )} e^{\left (a - c\right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)*sinh(b*x+a),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 1.57, size = 65, normalized size = 2.50 \[ x\,{\mathrm {e}}^{c-a}+\frac {{\mathrm {e}}^{2\,c-2\,a}\,\ln \left ({\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}-{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{-2\,c}\right )\,\left (2\,b\,{\mathrm {e}}^{3\,a-3\,c}-2\,b\,{\mathrm {e}}^{a-c}\right )}{4\,b^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)/sinh(c + b*x),x)

[Out]

x*exp(c - a) + (exp(2*c - 2*a)*log(exp(2*a)*exp(2*b*x) - exp(2*a)*exp(-2*c))*(2*b*exp(3*a - 3*c) - 2*b*exp(a -
c)))/(4*b^2)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sinh {\left (a + b x \right )} \operatorname {csch}{\left (b x + c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)*sinh(b*x+a),x)

[Out]

Integral(sinh(a + b*x)*csch(b*x + c), x)

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