### 3.964 $$\int e^{c+d x} \cosh (a+b x) \coth ^2(a+b x) \, dx$$

Optimal. Leaf size=160 $\frac{6 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{4 e^{-a-x (b-d)+c} \, _2F_1\left (2,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{5 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)}$

[Out]

(-5*E^(-a + c - (b - d)*x))/(2*(b - d)) + E^(a + c + (b + d)*x)/(2*(b + d)) + (6*E^(-a + c - (b - d)*x)*Hyperg
eometric2F1[1, -(b - d)/(2*b), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d) - (4*E^(-a + c - (b - d)*x)*Hypergeome
tric2F1[2, -(b - d)/(2*b), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d)

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Rubi [A]  time = 0.305758, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.182, Rules used = {5511, 2194, 2227, 2251} $\frac{6 e^{-a-x (b-d)+c} \, _2F_1\left (1,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{4 e^{-a-x (b-d)+c} \, _2F_1\left (2,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}-\frac{5 e^{-a-x (b-d)+c}}{2 (b-d)}+\frac{e^{a+x (b+d)+c}}{2 (b+d)}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(c + d*x)*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

(-5*E^(-a + c - (b - d)*x))/(2*(b - d)) + E^(a + c + (b + d)*x)/(2*(b + d)) + (6*E^(-a + c - (b - d)*x)*Hyperg
eometric2F1[1, -(b - d)/(2*b), (b + d)/(2*b), E^(2*(a + b*x))])/(b - d) - (4*E^(-a + c - (b - d)*x)*Hypergeome
tric2F1[2, -(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 ^2(a+b x) \, dx &=\int \left (\frac{5}{2} e^{-a+c-(b-d) x}+\frac{1}{2} e^{-a+c-(b-d) x+2 (a+b x)}+\frac{4 e^{-a+c-(b-d) x}}{\left (-1+e^{2 (a+b x)}\right )^2}+\frac{6 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{5}{2} \int e^{-a+c-(b-d) x} \, dx+4 \int \frac{e^{-a+c-(b-d) x}}{\left (-1+e^{2 (a+b x)}\right )^2} \, dx+6 \int \frac{e^{-a+c-(b-d) x}}{-1+e^{2 (a+b x)}} \, dx\\ &=-\frac{5 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac{6 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{4 e^{-a+c-(b-d) x} \, _2F_1\left (2,-\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{5 e^{-a+c-(b-d) x}}{2 (b-d)}+\frac{e^{a+c+(b+d) x}}{2 (b+d)}+\frac{6 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{4 e^{-a+c-(b-d) x} \, _2F_1\left (2,-\frac{b-d}{2 b};\frac{b+d}{2 b};e^{2 (a+b x)}\right )}{b-d}\\ \end{align*}

Mathematica [A]  time = 1.20762, size = 145, normalized size = 0.91 $\frac{e^{c-\frac{a d}{b}} \text{csch}(a+b x) \left (e^{d \left (\frac{a}{b}+x\right )} \left (b^2 \cosh (2 (a+b x))-b d \sinh (2 (a+b x))-3 b^2+2 d^2\right )-4 d (b-d) e^{\frac{(b+d) (a+b x)}{b}} \sinh (a+b x) \, _2F_1\left (1,\frac{b+d}{2 b};\frac{3 b+d}{2 b};e^{2 (a+b x)}\right )\right )}{2 b (b-d) (b+d)}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[E^(c + d*x)*Cosh[a + b*x]*Coth[a + b*x]^2,x]

[Out]

(E^(c - (a*d)/b)*Csch[a + b*x]*(-4*(b - d)*d*E^(((b + d)*(a + b*x))/b)*Hypergeometric2F1[1, (b + d)/(2*b), (3*
b + d)/(2*b), E^(2*(a + b*x))]*Sinh[a + b*x] + E^(d*(a/b + x))*(-3*b^2 + 2*d^2 + b^2*Cosh[2*(a + b*x)] - b*d*S
inh[2*(a + b*x)])))/(2*b*(b - d)*(b + d))

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Maple [F]  time = 0.163, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{3} \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 16 \, b d \int \frac{e^{\left (d x + c\right )}}{{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (7 \, b x + 7 \, a\right )} - 3 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (5 \, b x + 5 \, a\right )} + 3 \,{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (3 \, b x + 3 \, a\right )} -{\left (15 \, b^{2} - 8 \, b d + d^{2}\right )} e^{\left (b x + a\right )}}\,{d x} - \frac{{\left (15 \, b^{3} e^{c} + 39 \, b^{2} d e^{c} + 25 \, b d^{2} e^{c} + d^{3} e^{c} -{\left (15 \, b^{3} e^{c} - 23 \, b^{2} d e^{c} + 9 \, b d^{2} e^{c} - d^{3} e^{c}\right )} e^{\left (6 \, b x + 6 \, a\right )} +{\left (105 \, b^{3} e^{c} - 11 \, b^{2} d e^{c} - 17 \, b d^{2} e^{c} + 3 \, d^{3} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} -{\left (105 \, b^{3} e^{c} + 59 \, b^{2} d e^{c} - b d^{2} e^{c} - 3 \, d^{3} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{2 \,{\left ({\left (15 \, b^{4} - 8 \, b^{3} d - 14 \, b^{2} d^{2} + 8 \, b d^{3} - d^{4}\right )} e^{\left (5 \, b x + 5 \, a\right )} - 2 \,{\left (15 \, b^{4} - 8 \, b^{3} d - 14 \, b^{2} d^{2} + 8 \, b d^{3} - d^{4}\right )} e^{\left (3 \, b x + 3 \, a\right )} +{\left (15 \, b^{4} - 8 \, b^{3} d - 14 \, b^{2} d^{2} + 8 \, b d^{3} - d^{4}\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)^3*csch(b*x+a)^2,x, algorithm="maxima")

[Out]

16*b*d*integrate(e^(d*x + c)/((15*b^2 - 8*b*d + d^2)*e^(7*b*x + 7*a) - 3*(15*b^2 - 8*b*d + d^2)*e^(5*b*x + 5*a
) + 3*(15*b^2 - 8*b*d + d^2)*e^(3*b*x + 3*a) - (15*b^2 - 8*b*d + d^2)*e^(b*x + a)), x) - 1/2*(15*b^3*e^c + 39*
b^2*d*e^c + 25*b*d^2*e^c + d^3*e^c - (15*b^3*e^c - 23*b^2*d*e^c + 9*b*d^2*e^c - d^3*e^c)*e^(6*b*x + 6*a) + (10
5*b^3*e^c - 11*b^2*d*e^c - 17*b*d^2*e^c + 3*d^3*e^c)*e^(4*b*x + 4*a) - (105*b^3*e^c + 59*b^2*d*e^c - b*d^2*e^c
- 3*d^3*e^c)*e^(2*b*x + 2*a))*e^(d*x)/((15*b^4 - 8*b^3*d - 14*b^2*d^2 + 8*b*d^3 - d^4)*e^(5*b*x + 5*a) - 2*(1
5*b^4 - 8*b^3*d - 14*b^2*d^2 + 8*b*d^3 - d^4)*e^(3*b*x + 3*a) + (15*b^4 - 8*b^3*d - 14*b^2*d^2 + 8*b*d^3 - d^4
)*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 )^{3} \operatorname{csch}\left (b x + a\right )^{2} 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)^3*csch(b*x+a)^2,x, algorithm="fricas")

[Out]

integral(cosh(b*x + a)^3*csch(b*x + a)^2*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)**3*csch(b*x+a)**2,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 )^{3} \operatorname{csch}\left (b x + a\right )^{2} 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)^3*csch(b*x+a)^2,x, algorithm="giac")

[Out]

integrate(cosh(b*x + a)^3*csch(b*x + a)^2*e^(d*x + c), x)