### 3.153 $$\int \text{csch}^2(c+b x) \sinh (a+b x) \, dx$$

Optimal. Leaf size=36 $-\frac{\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac{\sinh (a-c) \text{csch}(b x+c)}{b}$

[Out]

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

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Rubi [A]  time = 0.0317207, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.267, Rules used = {5626, 2606, 8, 3770} $-\frac{\cosh (a-c) \tanh ^{-1}(\cosh (b x+c))}{b}-\frac{\sinh (a-c) \text{csch}(b x+c)}{b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[c + b*x]^2*Sinh[a + b*x],x]

[Out]

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

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]

Rule 2606

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a/f, Subst[
Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n -
1)/2] &&  !(IntegerQ[m/2] && LtQ[0, m, n + 1])

Rule 8

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

Rule 3770

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

Rubi steps

\begin{align*} \int \text{csch}^2(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int \text{csch}(c+b x) \, dx+\sinh (a-c) \int \coth (c+b x) \text{csch}(c+b x) \, dx\\ &=-\frac{\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}-\frac{(i \sinh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \text{csch}(c+b x))}{b}\\ &=-\frac{\tanh ^{-1}(\cosh (c+b x)) \cosh (a-c)}{b}-\frac{\text{csch}(c+b x) \sinh (a-c)}{b}\\ \end{align*}

Mathematica [C]  time = 0.0799594, size = 90, normalized size = 2.5 $-\frac{\sinh (a-c) \text{csch}(b x+c)}{b}-\frac{2 i \cosh (a-c) \tan ^{-1}\left (\frac{(\cosh (c)-\sinh (c)) \left (\sinh (c) \sinh \left (\frac{b x}{2}\right )+\cosh (c) \cosh \left (\frac{b x}{2}\right )\right )}{i \cosh (c) \cosh \left (\frac{b x}{2}\right )-i \sinh (c) \cosh \left (\frac{b x}{2}\right )}\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[c + b*x]^2*Sinh[a + b*x],x]

[Out]

((-2*I)*ArcTan[((Cosh[c] - Sinh[c])*(Cosh[c]*Cosh[(b*x)/2] + Sinh[c]*Sinh[(b*x)/2]))/(I*Cosh[c]*Cosh[(b*x)/2]
- I*Cosh[(b*x)/2]*Sinh[c])]*Cosh[a - c])/b - (Csch[c + b*x]*Sinh[a - c])/b

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Maple [B]  time = 0.039, size = 172, normalized size = 4.8 \begin{align*}{\frac{{{\rm e}^{bx+a}} \left ({{\rm e}^{2\,a}}-{{\rm e}^{2\,c}} \right ) }{b \left ( -{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) }}-{\frac{\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}-{\frac{\ln \left ({{\rm e}^{bx+a}}+{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,a}}}{2\,b}}+{\frac{\ln \left ({{\rm e}^{bx+a}}-{{\rm e}^{a-c}} \right ){{\rm e}^{-a-c}}{{\rm e}^{2\,c}}}{2\,b}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.13385, size = 139, normalized size = 3.86 \begin{align*} -\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 (e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-b x - a\right )}}{b{\left (e^{\left (-2 \, b x\right )} - e^{\left (2 \, c\right )}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)^2*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 + (e^(2*a) - e^(2*c))*e^(-b*x - a)/(b*(e^(-2*b*x) - e^(2*c)))

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Fricas [B]  time = 2.19585, size = 1709, normalized size = 47.47 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(4*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - 2*cosh(b*x + c)*sinh(-a + c)^2 - 2*(cosh(-a + c)^2 - 1)*cosh(
b*x + c) - ((cosh(-a + c)^2 + 1)*cosh(b*x + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c
)^2 + 1)*sinh(b*x + c)^2 + (cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(2*cosh(b*x + c)*cosh(-a
+ c)*sinh(-a + c) - cosh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 + 1)*cosh(b*x + c))*sinh(b*x + c) - 2*(cosh
(b*x + c)^2*cosh(-a + c) - cosh(-a + c))*sinh(-a + c) - 1)*log(cosh(b*x + c) + sinh(b*x + c) + 1) + ((cosh(-a
+ c)^2 + 1)*cosh(b*x + c)^2 + (cosh(-a + c)^2 - 2*cosh(-a + c)*sinh(-a + c) + sinh(-a + c)^2 + 1)*sinh(b*x + c
)^2 + (cosh(b*x + c)^2 - 1)*sinh(-a + c)^2 - cosh(-a + c)^2 - 2*(2*cosh(b*x + c)*cosh(-a + c)*sinh(-a + c) - c
osh(b*x + c)*sinh(-a + c)^2 - (cosh(-a + c)^2 + 1)*cosh(b*x + c))*sinh(b*x + c) - 2*(cosh(b*x + c)^2*cosh(-a +
c) - cosh(-a + c))*sinh(-a + c) - 1)*log(cosh(b*x + c) + sinh(b*x + c) - 1) - 2*(cosh(-a + c)^2 - 2*cosh(-a +
c)*sinh(-a + c) + sinh(-a + c)^2 - 1)*sinh(b*x + c))/(b*cosh(b*x + c)^2*cosh(-a + c) + (b*cosh(-a + c) - b*si
nh(-a + c))*sinh(b*x + c)^2 - b*cosh(-a + c) + 2*(b*cosh(b*x + c)*cosh(-a + c) - b*cosh(b*x + c)*sinh(-a + c))
*sinh(b*x + c) - (b*cosh(b*x + c)^2 - b)*sinh(-a + c))

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.18783, size = 140, normalized size = 3.89 \begin{align*} -\frac{{\left (e^{\left (2 \, a + c\right )} + e^{\left (3 \, c\right )}\right )} e^{\left (-a - 2 \, c\right )} \log \left (e^{\left (b x + c\right )} + 1\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 (b x + c\right )} - 1 \right |}\right ) + \frac{2 \,{\left (e^{\left (b x + 2 \, a\right )} - e^{\left (b x + 2 \, c\right )}\right )} e^{\left (-a\right )}}{e^{\left (2 \, b x + 2 \, c\right )} - 1}}{2 \, b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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