3.126 \(\int e^{c (a+b x)} \text{csch}^2(a c+b c x)^{5/2} \, dx\)

Optimal. Leaf size=147 \[ -\frac{8 \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac{32 \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac{4 \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^4} \]

[Out]

(-4*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4) + (32*Sqrt[Csch[a*c + b*c*x]^
2]*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3) - (8*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c
*(1 - E^(2*c*(a + b*x)))^2)

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Rubi [A]  time = 0.171003, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6720, 2282, 12, 266, 43} \[ -\frac{8 \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^2}+\frac{32 \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac{4 \sinh (a c+b c x) \sqrt{\text{csch}^2(a c+b c x)}}{b c \left (1-e^{2 c (a+b x)}\right )^4} \]

Antiderivative was successfully verified.

[In]

Int[E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(5/2),x]

[Out]

(-4*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c*(1 - E^(2*c*(a + b*x)))^4) + (32*Sqrt[Csch[a*c + b*c*x]^
2]*Sinh[a*c + b*c*x])/(3*b*c*(1 - E^(2*c*(a + b*x)))^3) - (8*Sqrt[Csch[a*c + b*c*x]^2]*Sinh[a*c + b*c*x])/(b*c
*(1 - E^(2*c*(a + b*x)))^2)

Rule 6720

Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a*v^m)^FracPart[p])/v^(m*FracPart[p]), Int
[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] &&  !IntegerQ[p] &&  !FreeQ[v, x] &&  !(EqQ[a, 1] && EqQ[m, 1]) &&
!(EqQ[v, x] && EqQ[m, 1])

Rule 2282

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int e^{c (a+b x)} \text{csch}^2(a c+b c x)^{5/2} \, dx &=\left (\sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \int e^{c (a+b x)} \text{csch}^5(a c+b c x) \, dx\\ &=\frac{\left (\sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{32 x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (32 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{x^5}{\left (-1+x^2\right )^5} \, dx,x,e^{c (a+b x)}\right )}{b c}\\ &=\frac{\left (16 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{(-1+x)^5} \, dx,x,e^{2 c (a+b x)}\right )}{b c}\\ &=\frac{\left (16 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)\right ) \operatorname{Subst}\left (\int \left (\frac{1}{(-1+x)^5}+\frac{2}{(-1+x)^4}+\frac{1}{(-1+x)^3}\right ) \, dx,x,e^{2 c (a+b x)}\right )}{b c}\\ &=-\frac{4 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^4}+\frac{32 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)}{3 b c \left (1-e^{2 c (a+b x)}\right )^3}-\frac{8 \sqrt{\text{csch}^2(a c+b c x)} \sinh (a c+b c x)}{b c \left (1-e^{2 c (a+b x)}\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.064242, size = 72, normalized size = 0.49 \[ -\frac{4 \left (-4 e^{2 c (a+b x)}+6 e^{4 c (a+b x)}+1\right ) \sinh (c (a+b x)) \sqrt{\text{csch}^2(c (a+b x))}}{3 b c \left (e^{2 c (a+b x)}-1\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c*(a + b*x))*(Csch[a*c + b*c*x]^2)^(5/2),x]

[Out]

(-4*(1 - 4*E^(2*c*(a + b*x)) + 6*E^(4*c*(a + b*x)))*Sqrt[Csch[c*(a + b*x)]^2]*Sinh[c*(a + b*x)])/(3*b*c*(-1 +
E^(2*c*(a + b*x)))^4)

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Maple [A]  time = 0.164, size = 80, normalized size = 0.5 \begin{align*} -{\frac{ \left ( 24\,{{\rm e}^{4\,c \left ( bx+a \right ) }}-16\,{{\rm e}^{2\,c \left ( bx+a \right ) }}+4 \right ){{\rm e}^{-c \left ( bx+a \right ) }}}{3\, \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{3}cb}\sqrt{{\frac{{{\rm e}^{2\,c \left ( bx+a \right ) }}}{ \left ({{\rm e}^{2\,c \left ( bx+a \right ) }}-1 \right ) ^{2}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(5/2),x)

[Out]

-4/3/(exp(2*c*(b*x+a))-1)^3*(1/(exp(2*c*(b*x+a))-1)^2*exp(2*c*(b*x+a)))^(1/2)*(6*exp(4*c*(b*x+a))-4*exp(2*c*(b
*x+a))+1)/c/b*exp(-c*(b*x+a))

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Maxima [A]  time = 1.55461, size = 282, normalized size = 1.92 \begin{align*} -\frac{8 \, e^{\left (4 \, b c x + 4 \, a c\right )}}{b c{\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} + \frac{16 \, e^{\left (2 \, b c x + 2 \, a c\right )}}{3 \, b c{\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} - \frac{4}{3 \, b c{\left (e^{\left (8 \, b c x + 8 \, a c\right )} - 4 \, e^{\left (6 \, b c x + 6 \, a c\right )} + 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(5/2),x, algorithm="maxima")

[Out]

-8*e^(4*b*c*x + 4*a*c)/(b*c*(e^(8*b*c*x + 8*a*c) - 4*e^(6*b*c*x + 6*a*c) + 6*e^(4*b*c*x + 4*a*c) - 4*e^(2*b*c*
x + 2*a*c) + 1)) + 16/3*e^(2*b*c*x + 2*a*c)/(b*c*(e^(8*b*c*x + 8*a*c) - 4*e^(6*b*c*x + 6*a*c) + 6*e^(4*b*c*x +
4*a*c) - 4*e^(2*b*c*x + 2*a*c) + 1)) - 4/3/(b*c*(e^(8*b*c*x + 8*a*c) - 4*e^(6*b*c*x + 6*a*c) + 6*e^(4*b*c*x +
4*a*c) - 4*e^(2*b*c*x + 2*a*c) + 1))

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Fricas [B]  time = 1.56371, size = 797, normalized size = 5.42 \begin{align*} -\frac{4 \,{\left (7 \, \cosh \left (b c x + a c\right )^{2} + 10 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right ) + 7 \, \sinh \left (b c x + a c\right )^{2} - 4\right )}}{3 \,{\left (b c \cosh \left (b c x + a c\right )^{6} + 6 \, b c \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{5} + b c \sinh \left (b c x + a c\right )^{6} - 4 \, b c \cosh \left (b c x + a c\right )^{4} +{\left (15 \, b c \cosh \left (b c x + a c\right )^{2} - 4 \, b c\right )} \sinh \left (b c x + a c\right )^{4} + 7 \, b c \cosh \left (b c x + a c\right )^{2} + 4 \,{\left (5 \, b c \cosh \left (b c x + a c\right )^{3} - 4 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )^{3} +{\left (15 \, b c \cosh \left (b c x + a c\right )^{4} - 24 \, b c \cosh \left (b c x + a c\right )^{2} + 7 \, b c\right )} \sinh \left (b c x + a c\right )^{2} - 4 \, b c + 2 \,{\left (3 \, b c \cosh \left (b c x + a c\right )^{5} - 8 \, b c \cosh \left (b c x + a c\right )^{3} + 5 \, b c \cosh \left (b c x + a c\right )\right )} \sinh \left (b c x + a c\right )\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(5/2),x, algorithm="fricas")

[Out]

-4/3*(7*cosh(b*c*x + a*c)^2 + 10*cosh(b*c*x + a*c)*sinh(b*c*x + a*c) + 7*sinh(b*c*x + a*c)^2 - 4)/(b*c*cosh(b*
c*x + a*c)^6 + 6*b*c*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^5 + b*c*sinh(b*c*x + a*c)^6 - 4*b*c*cosh(b*c*x + a*c)
^4 + (15*b*c*cosh(b*c*x + a*c)^2 - 4*b*c)*sinh(b*c*x + a*c)^4 + 7*b*c*cosh(b*c*x + a*c)^2 + 4*(5*b*c*cosh(b*c*
x + a*c)^3 - 4*b*c*cosh(b*c*x + a*c))*sinh(b*c*x + a*c)^3 + (15*b*c*cosh(b*c*x + a*c)^4 - 24*b*c*cosh(b*c*x +
a*c)^2 + 7*b*c)*sinh(b*c*x + a*c)^2 - 4*b*c + 2*(3*b*c*cosh(b*c*x + a*c)^5 - 8*b*c*cosh(b*c*x + a*c)^3 + 5*b*c
*cosh(b*c*x + a*c))*sinh(b*c*x + a*c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} e^{a c} \int \left (\operatorname{csch}^{2}{\left (a c + b c x \right )}\right )^{\frac{5}{2}} e^{b c x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)**2)**(5/2),x)

[Out]

exp(a*c)*Integral((csch(a*c + b*c*x)**2)**(5/2)*exp(b*c*x), x)

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Giac [A]  time = 1.15435, size = 104, normalized size = 0.71 \begin{align*} -\frac{4 \,{\left (6 \, e^{\left (4 \, b c x + 4 \, a c\right )} - 4 \, e^{\left (2 \, b c x + 2 \, a c\right )} + 1\right )}}{3 \, b c{\left (e^{\left (2 \, b c x + 2 \, a c\right )} - 1\right )}^{4} \mathrm{sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*(csch(b*c*x+a*c)^2)^(5/2),x, algorithm="giac")

[Out]

-4/3*(6*e^(4*b*c*x + 4*a*c) - 4*e^(2*b*c*x + 2*a*c) + 1)/(b*c*(e^(2*b*c*x + 2*a*c) - 1)^4*sgn(e^(b*c*x + a*c)
- e^(-b*c*x - a*c)))