### 3.154 $$\int \text{csch}^3(c+b x) \sinh (a+b x) \, dx$$

Optimal. Leaf size=39 $-\frac{\cosh (a-c) \coth (b x+c)}{b}-\frac{\sinh (a-c) \text{csch}^2(b x+c)}{2 b}$

[Out]

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

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Rubi [A]  time = 0.0443071, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.333, Rules used = {5626, 2606, 30, 3767, 8} $-\frac{\cosh (a-c) \coth (b x+c)}{b}-\frac{\sinh (a-c) \text{csch}^2(b x+c)}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Int[Csch[c + b*x]^3*Sinh[a + b*x],x]

[Out]

-((Cosh[a - c]*Coth[c + b*x])/b) - (Csch[c + b*x]^2*Sinh[a - c])/(2*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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

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

Rubi steps

\begin{align*} \int \text{csch}^3(c+b x) \sinh (a+b x) \, dx &=\cosh (a-c) \int \text{csch}^2(c+b x) \, dx+\sinh (a-c) \int \coth (c+b x) \text{csch}^2(c+b x) \, dx\\ &=-\frac{(i \cosh (a-c)) \operatorname{Subst}(\int 1 \, dx,x,-i \coth (c+b x))}{b}+\frac{\sinh (a-c) \operatorname{Subst}(\int x \, dx,x,-i \text{csch}(c+b x))}{b}\\ &=-\frac{\cosh (a-c) \coth (c+b x)}{b}-\frac{\text{csch}^2(c+b x) \sinh (a-c)}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.172487, size = 35, normalized size = 0.9 $-\frac{\text{csch}(c) \text{csch}^2(b x+c) (\cosh (a)-\cosh (a-c) \cosh (2 b x+c))}{2 b}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[Csch[c + b*x]^3*Sinh[a + b*x],x]

[Out]

-((Cosh[a] - Cosh[a - c]*Cosh[c + 2*b*x])*Csch[c]*Csch[c + b*x]^2)/(2*b)

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Maple [A]  time = 0.028, size = 57, normalized size = 1.5 \begin{align*}{\frac{ \left ( -2\,{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}}+{{\rm e}^{2\,c}} \right ){{\rm e}^{3\,a-c}}}{b \left ( -{{\rm e}^{2\,bx+2\,a+2\,c}}+{{\rm e}^{2\,a}} \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(csch(b*x+c)^3*sinh(b*x+a),x)

[Out]

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

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Maxima [B]  time = 1.08096, size = 177, normalized size = 4.54 \begin{align*} -\frac{2 \, e^{\left (-2 \, b x + 3 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} - e^{\left (-4 \, b x + a\right )} - e^{\left (a + 4 \, c\right )}\right )}} + \frac{e^{\left (2 \, a + 3 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} - e^{\left (-4 \, b x + a\right )} - e^{\left (a + 4 \, c\right )}\right )}} + \frac{e^{\left (5 \, c\right )}}{b{\left (2 \, e^{\left (-2 \, b x + a + 2 \, c\right )} - e^{\left (-4 \, b x + a\right )} - e^{\left (a + 4 \, c\right )}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)^3*sinh(b*x+a),x, algorithm="maxima")

[Out]

-2*e^(-2*b*x + 3*c)/(b*(2*e^(-2*b*x + a + 2*c) - e^(-4*b*x + a) - e^(a + 4*c))) + e^(2*a + 3*c)/(b*(2*e^(-2*b*
x + a + 2*c) - e^(-4*b*x + a) - e^(a + 4*c))) + e^(5*c)/(b*(2*e^(-2*b*x + a + 2*c) - e^(-4*b*x + a) - e^(a + 4
*c)))

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Fricas [B]  time = 2.11493, size = 620, normalized size = 15.9 \begin{align*} -\frac{2 \,{\left ({\left (2 \, \cosh \left (-a + c\right ) - \sinh \left (-a + c\right )\right )} \sinh \left (b x + c\right ) - \cosh \left (b x + c\right ) \sinh \left (-a + c\right )\right )}}{b \cosh \left (b x + c\right )^{3} \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} +{\left (b \cosh \left (-a + c\right )^{2} - b \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{3} + 3 \,{\left (b \cosh \left (b x + c\right ) \cosh \left (-a + c\right )^{2} - b \cosh \left (b x + c\right ) \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )^{2} -{\left (b \cosh \left (b x + c\right )^{3} - b \cosh \left (b x + c\right )\right )} \sinh \left (-a + c\right )^{2} + 3 \,{\left (b \cosh \left (b x + c\right )^{2} \cosh \left (-a + c\right )^{2} - b \cosh \left (-a + c\right )^{2} -{\left (b \cosh \left (b x + c\right )^{2} - b\right )} \sinh \left (-a + c\right )^{2}\right )} \sinh \left (b x + c\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)^3*sinh(b*x+a),x, algorithm="fricas")

[Out]

-2*((2*cosh(-a + c) - sinh(-a + c))*sinh(b*x + c) - cosh(b*x + c)*sinh(-a + c))/(b*cosh(b*x + c)^3*cosh(-a + c
)^2 - b*cosh(b*x + c)*cosh(-a + c)^2 + (b*cosh(-a + c)^2 - b*sinh(-a + c)^2)*sinh(b*x + c)^3 + 3*(b*cosh(b*x +
c)*cosh(-a + c)^2 - b*cosh(b*x + c)*sinh(-a + c)^2)*sinh(b*x + c)^2 - (b*cosh(b*x + c)^3 - b*cosh(b*x + c))*s
inh(-a + c)^2 + 3*(b*cosh(b*x + c)^2*cosh(-a + c)^2 - b*cosh(-a + c)^2 - (b*cosh(b*x + c)^2 - b)*sinh(-a + c)^
2)*sinh(b*x + 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}^{3}{\left (b x + c \right )}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)**3*sinh(b*x+a),x)

[Out]

Integral(sinh(a + b*x)*csch(b*x + c)**3, x)

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Giac [A]  time = 1.17676, size = 72, normalized size = 1.85 \begin{align*} -\frac{{\left (2 \, e^{\left (2 \, b x + 2 \, a + 2 \, c\right )} - e^{\left (2 \, a\right )} - e^{\left (2 \, c\right )}\right )} e^{\left (-a - c\right )}}{b{\left (e^{\left (2 \, b x + 2 \, c\right )} - 1\right )}^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(b*x+c)^3*sinh(b*x+a),x, algorithm="giac")

[Out]

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