3.949 \(\int e^{c+d x} \coth (a+b x) \, dx\)

Optimal. Leaf size=53 \[ \frac {e^{c+d x}}{d}-\frac {2 e^{c+d x} \, _2F_1\left (1,\frac {d}{2 b};\frac {d}{2 b}+1;e^{2 (a+b x)}\right )}{d} \]

[Out]

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

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Rubi [A]  time = 0.06, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5485, 2194, 2251} \[ \frac {e^{c+d x}}{d}-\frac {2 e^{c+d x} \, _2F_1\left (1,\frac {d}{2 b};\frac {d}{2 b}+1;e^{2 (a+b x)}\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[E^(c + d*x)*Coth[a + b*x],x]

[Out]

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

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 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])

Rule 5485

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

Rubi steps

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

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Mathematica [B]  time = 1.94, size = 120, normalized size = 2.26 \[ \frac {\coth (a) e^{c+d x}}{d}-\frac {2 e^{2 a+c} \left (\frac {e^{d x} \, _2F_1\left (1,\frac {d}{2 b};\frac {d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac {e^{x (2 b+d)} \, _2F_1\left (1,\frac {d}{2 b}+1;\frac {d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{e^{2 a}-1} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\cosh \left (b x + a\right ) \operatorname {csch}\left (b x + a\right ) e^{\left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.30, size = 0, normalized size = 0.00 \[ \int {\mathrm e}^{d x +c} \cosh \left (b x +a \right ) \mathrm {csch}\left (b x +a \right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -4 \, b \int \frac {e^{\left (d x + c\right )}}{{\left (2 \, b - d\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \, {\left (2 \, b - d\right )} e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b - d}\,{d x} - \frac {{\left ({\left (2 \, b e^{c} - d e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b e^{c} - d e^{c}\right )} e^{\left (d x\right )}}{2 \, b d - d^{2} - {\left (2 \, b d - d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*b*integrate(e^(d*x + c)/((2*b - d)*e^(4*b*x + 4*a) - 2*(2*b - d)*e^(2*b*x + 2*a) + 2*b - d), x) - ((2*b*e^c
- d*e^c)*e^(2*b*x + 2*a) - 2*b*e^c - d*e^c)*e^(d*x)/(2*b*d - d^2 - (2*b*d - d^2)*e^(2*b*x + 2*a))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\mathrm {e}}^{c+d\,x}}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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