3.9.83 \(\int e^{c+d x} \text {csch}^3(a+b x) \, dx\) [883]

Optimal. Leaf size=100 \[ -\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {(b-d) e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {1}{2} \left (3+\frac {d}{b}\right );e^{2 (a+b x)}\right )}{b^2} \]

[Out]

-1/2*d*exp(d*x+c)*csch(b*x+a)/b^2-1/2*exp(d*x+c)*coth(b*x+a)*csch(b*x+a)/b+(b-d)*exp(b*x+d*x+a+c)*hypergeom([1
, 1/2*(b+d)/b],[3/2+1/2*d/b],exp(2*b*x+2*a))/b^2

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Rubi [A]
time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5599, 5601} \begin {gather*} \frac {(b-d) e^{a+b x+c+d x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {1}{2} \left (\frac {d}{b}+3\right );e^{2 (a+b x)}\right )}{b^2}-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Csch[a + b*x]^3,x]

[Out]

-1/2*(d*E^(c + d*x)*Csch[a + b*x])/b^2 - (E^(c + d*x)*Coth[a + b*x]*Csch[a + b*x])/(2*b) + ((b - d)*E^(a + c +
b*x + d*x)*Hypergeometric2F1[1, (b + d)/(2*b), (3 + d/b)/2, E^(2*(a + b*x))])/b^2

Rule 5599

Int[Csch[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-b)*c*Log[F]*F^(c*(a +
b*x))*(Csch[d + e*x]^(n - 2)/(e^2*(n - 1)*(n - 2))), x] + (-Dist[(e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n -
1)*(n - 2)), Int[F^(c*(a + b*x))*Csch[d + e*x]^(n - 2), x], x] - Simp[F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*(
Cosh[d + e*x]/(e*(n - 1))), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 5601

Int[Csch[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[(-2)^n*E^(n*(d + e*x))
*(F^(c*(a + b*x))/(e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 + b*c*(Log[F]/(2*e)), 1 + n/2 + b*c*(Log[F]/(2*
e)), E^(2*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps

\begin {align*} \int e^{c+d x} \text {csch}^3(a+b x) \, dx &=-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}-\frac {1}{2} \left (1-\frac {d^2}{b^2}\right ) \int e^{c+d x} \text {csch}(a+b x) \, dx\\ &=-\frac {d e^{c+d x} \text {csch}(a+b x)}{2 b^2}-\frac {e^{c+d x} \coth (a+b x) \text {csch}(a+b x)}{2 b}+\frac {(b-d) e^{a+c+b x+d x} \, _2F_1\left (1,\frac {b+d}{2 b};\frac {1}{2} \left (3+\frac {d}{b}\right );e^{2 (a+b x)}\right )}{b^2}\\ \end {align*}

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Mathematica [A]
time = 1.70, size = 94, normalized size = 0.94 \begin {gather*} \frac {e^c \left (-e^{d x} (d+b \coth (a+b x)) \text {csch}(a+b x)+\frac {2 (b-d) e^{(b+d) x} \text {csch}(a) \, _2F_1\left (1,\frac {b+d}{2 b};\frac {3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )}{-1+\coth (a)}\right )}{2 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(E^c*(-(E^(d*x)*(d + b*Coth[a + b*x])*Csch[a + b*x]) + (2*(b - d)*E^((b + d)*x)*Csch[a]*Hypergeometric2F1[1, (
b + d)/(2*b), (3*b + d)/(2*b), E^(2*b*x)*(Cosh[a] + Sinh[a])^2])/(-1 + Coth[a])))/(2*b^2)

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Maple [F]
time = 1.43, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d x +c} \mathrm {csch}\left (b x +a \right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

48*(b^2*e^c + b*d*e^c)*integrate(e^(b*x + d*x + a)/(15*b^2 - 8*b*d + d^2 + (15*b^2 - 8*b*d + d^2)*e^(8*b*x + 8
*a) - 4*(15*b^2 - 8*b*d + d^2)*e^(6*b*x + 6*a) + 6*(15*b^2 - 8*b*d + d^2)*e^(4*b*x + 4*a) - 4*(15*b^2 - 8*b*d
+ d^2)*e^(2*b*x + 2*a)), x) + 8*((5*b*e^c - d*e^c)*e^(3*b*x + 3*a) - 6*b*e^(b*x + a + c))*e^(d*x)/(15*b^2 - 8*
b*d + d^2 - (15*b^2 - 8*b*d + d^2)*e^(6*b*x + 6*a) + 3*(15*b^2 - 8*b*d + d^2)*e^(4*b*x + 4*a) - 3*(15*b^2 - 8*
b*d + d^2)*e^(2*b*x + 2*a))

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csch(b*x + a)^3*e^(d*x + c), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int e^{d x} \operatorname {csch}^{3}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(csch(b*x + a)^3*e^(d*x + c), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c + d*x)/sinh(a + b*x)^3,x)

[Out]

int(exp(c + d*x)/sinh(a + b*x)^3, x)

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