3.10.56 \(\int e^{c+d x} \cosh (a+b x) \coth (a+b x) \, dx\) [956]

Optimal. Leaf size=103 \[ -\frac {3 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac {e^{a+c+(b+d) x}}{2 (b+d)}+\frac {2 e^{-a+c-(b-d) x} \, _2F_1\left (1,-\frac {b-d}{2 b};\frac {b+d}{2 b};e^{2 (a+b x)}\right )}{b-d} \]

[Out]

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

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Rubi [A]
time = 0.14, antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5622, 2225, 2259, 2283} \begin {gather*} \frac {2 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac {b-d}{2 b};\frac {b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac {3 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac {e^{a+x (b+d)+c}}{2 (b+d)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

(-3*E^(-a + c - (b - d)*x))/(2*(b - d)) + E^(a + c + (b + d)*x)/(2*(b + d)) + (2*E^(-a + c - (b - d)*x)*Hyperg
eometric2F1[1, -1/2*(b - d)/b, (b + d)/(2*b), E^(2*(a + b*x))])/(b - d)

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2259

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rule 2283

Int[((a_) + (b_.)*(F_)^((e_.)*((c_.) + (d_.)*(x_))))^(p_)*(G_)^((h_.)*((f_.) + (g_.)*(x_))), x_Symbol] :> Simp
[a^p*(G^(h*(f + g*x))/(g*h*Log[G]))*Hypergeometric2F1[-p, g*h*(Log[G]/(d*e*Log[F])), g*h*(Log[G]/(d*e*Log[F]))
+ 1, Simplify[(-b/a)*F^(e*(c + d*x))]], x] /; FreeQ[{F, G, a, b, c, d, e, f, g, h, p}, x] && (ILtQ[p, 0] || G
tQ[a, 0])

Rule 5622

Int[(F_)^((c_.)*((a_.) + (b_.)*(x_)))*(G_)[(d_.) + (e_.)*(x_)]^(m_.)*(H_)[(d_.) + (e_.)*(x_)]^(n_.), x_Symbol]
:> Int[ExpandTrigToExp[F^(c*(a + b*x)), G[d + e*x]^m*H[d + e*x]^n, x], x] /; FreeQ[{F, a, b, c, d, e}, x] &&
IGtQ[m, 0] && IGtQ[n, 0] && HyperbolicQ[G] && HyperbolicQ[H]

Rubi steps

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

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Mathematica [A]
time = 0.39, size = 93, normalized size = 0.90 \begin {gather*} \frac {e^c \left (-2 e^{(b+d) x} \, _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 ) (\cosh (a)+\sinh (a))+\frac {e^{d x} (b \cosh (a+b x)-d \sinh (a+b x))}{b-d}\right )}{b+d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(c + d*x)*Cosh[a + b*x]*Coth[a + b*x],x]

[Out]

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

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Maple [F]
time = 2.73, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{d x +c} \left (\cosh ^{2}\left (b x +a \right )\right ) \mathrm {csch}\left (b x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x+c)*cosh(b*x+a)^2*csch(b*x+a),x)

[Out]

int(exp(d*x+c)*cosh(b*x+a)^2*csch(b*x+a),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)*cosh(b*x+a)^2*csch(b*x+a),x, algorithm="maxima")

[Out]

-4*b*integrate(e^(d*x + c)/((3*b - d)*e^(5*b*x + 5*a) - 2*(3*b - d)*e^(3*b*x + 3*a) + (3*b - d)*e^(b*x + a)),
x) + 1/2*(5*b^2*e^c + 6*b*d*e^c + d^2*e^c + (3*b^2*e^c - 4*b*d*e^c + d^2*e^c)*e^(4*b*x + 4*a) - 2*(6*b^2*e^c +
b*d*e^c - d^2*e^c)*e^(2*b*x + 2*a))*e^(d*x)/((3*b^3 - b^2*d - 3*b*d^2 + d^3)*e^(3*b*x + 3*a) - (3*b^3 - b^2*d
- 3*b*d^2 + d^3)*e^(b*x + 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)*cosh(b*x+a)^2*csch(b*x+a),x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^2*csch(b*x + a)*e^(d*x + c), x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x+c)*cosh(b*x+a)**2*csch(b*x+a),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3433 deep

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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)*cosh(b*x+a)^2*csch(b*x+a),x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^2*csch(b*x + a)*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 {cosh}\left (a+b\,x\right )}^2\,{\mathrm {e}}^{c+d\,x}}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((cosh(a + b*x)^2*exp(c + d*x))/sinh(a + b*x),x)

[Out]

int((cosh(a + b*x)^2*exp(c + d*x))/sinh(a + b*x), x)

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