### 3.956 $$\int e^{c+d x} \cosh (a+b x) \coth (a+b x) \, dx$$

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

[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, -(b - d)/(2*b), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d)

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Rubi [A]  time = 0.207046, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.2, Rules used = {5511, 2194, 2227, 2251} $\frac{2 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{3 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)}$

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, -(b - d)/(2*b), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d)

Rule 5511

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]

Rule 2194

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 2227

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 2251

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

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*}

Mathematica [A]  time = 0.554539, size = 93, normalized size = 0.9 $\frac{e^c \left (\frac{e^{d x} (b \cosh (a+b x)-d \sinh (a+b x))}{b-d}-2 (\sinh (a)+\cosh (a)) e^{x (b+d)} \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 b x} (\cosh (a)+\sinh (a))^2\right )\right )}{b+d}$

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 = 0.119, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{2}{\rm csch} \left (bx+a\right )\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} -4 \, b \int \frac{e^{\left (d x + c\right )}}{{\left (3 \, b - d\right )} e^{\left (5 \, b x + 5 \, a\right )} - 2 \,{\left (3 \, b - d\right )} e^{\left (3 \, b x + 3 \, a\right )} +{\left (3 \, b - d\right )} e^{\left (b x + a\right )}}\,{d x} + \frac{{\left (5 \, b^{2} e^{c} + 6 \, b d e^{c} + d^{2} e^{c} +{\left (3 \, b^{2} e^{c} - 4 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (6 \, b^{2} e^{c} + b d e^{c} - d^{2} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{2 \,{\left ({\left (3 \, b^{3} - b^{2} d - 3 \, b d^{2} + d^{3}\right )} e^{\left (3 \, b x + 3 \, a\right )} -{\left (3 \, b^{3} - b^{2} d - 3 \, b d^{2} + d^{3}\right )} e^{\left (b x + a\right )}\right )}} \end{align*}

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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right ) e^{\left (d x + c\right )}, x\right ) \end{align*}

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(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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]

Timed out

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

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)