3.130 \(\int \frac {e^{c (a+b x)}}{\text {csch}^2(a c+b c x)^{3/2}} \, dx\)

Optimal. Leaf size=162 \[ \frac {e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {3 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {3 x \text {csch}(a c+b c x)}{8 \sqrt {\text {csch}^2(a c+b c x)}} \]

[Out]

1/16*csch(b*c*x+a*c)/b/c/exp(2*c*(b*x+a))/(csch(b*c*x+a*c)^2)^(1/2)-3/16*exp(2*c*(b*x+a))*csch(b*c*x+a*c)/b/c/
(csch(b*c*x+a*c)^2)^(1/2)+1/32*exp(4*c*(b*x+a))*csch(b*c*x+a*c)/b/c/(csch(b*c*x+a*c)^2)^(1/2)+3/8*x*csch(b*c*x
+a*c)/(csch(b*c*x+a*c)^2)^(1/2)

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Rubi [A]  time = 0.15, antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6720, 2282, 12, 266, 43} \[ \frac {e^{-2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}-\frac {3 e^{2 c (a+b x)} \text {csch}(a c+b c x)}{16 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {e^{4 c (a+b x)} \text {csch}(a c+b c x)}{32 b c \sqrt {\text {csch}^2(a c+b c x)}}+\frac {3 x \text {csch}(a c+b c x)}{8 \sqrt {\text {csch}^2(a c+b c x)}} \]

Antiderivative was successfully verified.

[In]

Int[E^(c*(a + b*x))/(Csch[a*c + b*c*x]^2)^(3/2),x]

[Out]

Csch[a*c + b*c*x]/(16*b*c*E^(2*c*(a + b*x))*Sqrt[Csch[a*c + b*c*x]^2]) - (3*E^(2*c*(a + b*x))*Csch[a*c + b*c*x
])/(16*b*c*Sqrt[Csch[a*c + b*c*x]^2]) + (E^(4*c*(a + b*x))*Csch[a*c + b*c*x])/(32*b*c*Sqrt[Csch[a*c + b*c*x]^2
]) + (3*x*Csch[a*c + b*c*x])/(8*Sqrt[Csch[a*c + b*c*x]^2])

Rule 12

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

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])

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 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 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])

Rubi steps

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

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Mathematica [A]  time = 0.07, size = 76, normalized size = 0.47 \[ \frac {\left (e^{-2 c (a+b x)}-3 e^{2 c (a+b x)}+\frac {1}{2} e^{4 c (a+b x)}+6 b c x\right ) \text {csch}^3(c (a+b x))}{16 b c \text {csch}^2(c (a+b x))^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[E^(c*(a + b*x))/(Csch[a*c + b*c*x]^2)^(3/2),x]

[Out]

((E^(-2*c*(a + b*x)) - 3*E^(2*c*(a + b*x)) + E^(4*c*(a + b*x))/2 + 6*b*c*x)*Csch[c*(a + b*x)]^3)/(16*b*c*(Csch
[c*(a + b*x)]^2)^(3/2))

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fricas [A]  time = 0.61, size = 126, normalized size = 0.78 \[ \frac {3 \, \cosh \left (b c x + a c\right )^{3} + 9 \, \cosh \left (b c x + a c\right ) \sinh \left (b c x + a c\right )^{2} - \sinh \left (b c x + a c\right )^{3} + 6 \, {\left (2 \, b c x - 1\right )} \cosh \left (b c x + a c\right ) - 3 \, {\left (4 \, b c x + \cosh \left (b c x + a c\right )^{2} + 2\right )} \sinh \left (b c x + a c\right )}{32 \, {\left (b c \cosh \left (b c x + a c\right ) - b c \sinh \left (b c x + a c\right )\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(csch(b*c*x+a*c)^2)^(3/2),x, algorithm="fricas")

[Out]

1/32*(3*cosh(b*c*x + a*c)^3 + 9*cosh(b*c*x + a*c)*sinh(b*c*x + a*c)^2 - sinh(b*c*x + a*c)^3 + 6*(2*b*c*x - 1)*
cosh(b*c*x + a*c) - 3*(4*b*c*x + cosh(b*c*x + a*c)^2 + 2)*sinh(b*c*x + a*c))/(b*c*cosh(b*c*x + a*c) - b*c*sinh
(b*c*x + a*c))

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giac [A]  time = 0.15, size = 204, normalized size = 1.26 \[ \frac {{\left (12 \, b c x e^{\left (-a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 2 \, {\left (3 \, e^{\left (2 \, b c x + 2 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-2 \, b c x - 3 \, a c\right )} + {\left (e^{\left (4 \, b c x + 9 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right ) - 6 \, e^{\left (2 \, b c x + 7 \, a c\right )} \mathrm {sgn}\left (e^{\left (b c x + a c\right )} - e^{\left (-b c x - a c\right )}\right )\right )} e^{\left (-6 \, a c\right )}\right )} e^{\left (a c\right )}}{32 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(csch(b*c*x+a*c)^2)^(3/2),x, algorithm="giac")

[Out]

1/32*(12*b*c*x*e^(-a*c)*sgn(e^(b*c*x + a*c) - e^(-b*c*x - a*c)) - 2*(3*e^(2*b*c*x + 2*a*c)*sgn(e^(b*c*x + a*c)
- e^(-b*c*x - a*c)) - sgn(e^(b*c*x + a*c) - e^(-b*c*x - a*c)))*e^(-2*b*c*x - 3*a*c) + (e^(4*b*c*x + 9*a*c)*sg
n(e^(b*c*x + a*c) - e^(-b*c*x - a*c)) - 6*e^(2*b*c*x + 7*a*c)*sgn(e^(b*c*x + a*c) - e^(-b*c*x - a*c)))*e^(-6*a
*c))*e^(a*c)/(b*c)

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maple [A]  time = 0.54, size = 216, normalized size = 1.33 \[ \frac {3 x \,{\mathrm e}^{c \left (b x +a \right )}}{8 \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {{\mathrm e}^{5 c \left (b x +a \right )}}{32 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}-\frac {3 \,{\mathrm e}^{3 c \left (b x +a \right )}}{16 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}}+\frac {{\mathrm e}^{-c \left (b x +a \right )}}{16 b c \left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right ) \sqrt {\frac {{\mathrm e}^{2 c \left (b x +a \right )}}{\left ({\mathrm e}^{2 c \left (b x +a \right )}-1\right )^{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))/(csch(b*c*x+a*c)^2)^(3/2),x)

[Out]

3/8*x/(exp(2*c*(b*x+a))-1)/(1/(exp(2*c*(b*x+a))-1)^2*exp(2*c*(b*x+a)))^(1/2)*exp(c*(b*x+a))+1/32/b/c/(exp(2*c*
(b*x+a))-1)/(1/(exp(2*c*(b*x+a))-1)^2*exp(2*c*(b*x+a)))^(1/2)*exp(5*c*(b*x+a))-3/16/b/c/(exp(2*c*(b*x+a))-1)/(
1/(exp(2*c*(b*x+a))-1)^2*exp(2*c*(b*x+a)))^(1/2)*exp(3*c*(b*x+a))+1/16/b/c/(exp(2*c*(b*x+a))-1)/(1/(exp(2*c*(b
*x+a))-1)^2*exp(2*c*(b*x+a)))^(1/2)*exp(-c*(b*x+a))

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maxima [A]  time = 0.42, size = 62, normalized size = 0.38 \[ \frac {{\left (e^{\left (6 \, b c x + 6 \, a c\right )} - 6 \, e^{\left (4 \, b c x + 4 \, a c\right )} + 2\right )} e^{\left (-2 \, b c x - 2 \, a c\right )}}{32 \, b c} + \frac {3 \, {\left (b c x + a c\right )}}{8 \, b c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(csch(b*c*x+a*c)^2)^(3/2),x, algorithm="maxima")

[Out]

1/32*(e^(6*b*c*x + 6*a*c) - 6*e^(4*b*c*x + 4*a*c) + 2)*e^(-2*b*c*x - 2*a*c)/(b*c) + 3/8*(b*c*x + a*c)/(b*c)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {e}}^{c\,\left (a+b\,x\right )}}{{\left (\frac {1}{{\mathrm {sinh}\left (a\,c+b\,c\,x\right )}^2}\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(a + b*x))/(1/sinh(a*c + b*c*x)^2)^(3/2),x)

[Out]

int(exp(c*(a + b*x))/(1/sinh(a*c + b*c*x)^2)^(3/2), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{a c} \int \frac {e^{b c x}}{\left (\operatorname {csch}^{2}{\left (a c + b c x \right )}\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))/(csch(b*c*x+a*c)**2)**(3/2),x)

[Out]

exp(a*c)*Integral(exp(b*c*x)/(csch(a*c + b*c*x)**2)**(3/2), x)

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