3.10.49 \(\int e^{c+d x} \coth (a+b x) \, dx\) [949]

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

[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 = 0.05, 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 = {5593, 2225, 2283} \begin {gather*} \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} \end {gather*}

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 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 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 5593

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] Leaf count is larger than twice the leaf count of optimal. \(120\) vs. \(2(53)=106\).
time = 1.32, size = 120, normalized size = 2.26 \begin {gather*} \frac {e^{c+d x} \coth (a)}{d}-\frac {2 e^{2 a+c} \left (\frac {e^{d x} \, _2F_1\left (1,\frac {d}{2 b};1+\frac {d}{2 b};e^{2 (a+b x)}\right )}{d}-\frac {e^{(2 b+d) x} \, _2F_1\left (1,1+\frac {d}{2 b};2+\frac {d}{2 b};e^{2 (a+b x)}\right )}{2 b+d}\right )}{-1+e^{2 a}} \end {gather*}

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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Maple [F]
time = 1.52, 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 \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)*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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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)*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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{c} \int e^{d x} \cosh {\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \end {gather*}

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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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)*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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\mathrm {cosh}\left (a+b\,x\right )\,{\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)*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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