3.3.32 \(\int e^{c (a+b x)} \coth ^2(d+e x) \, dx\) [232]

Optimal. Leaf size=113 \[ \frac {e^{c (a+b x)}}{b c}-\frac {4 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c} \]

[Out]

exp(c*(b*x+a))/b/c-4*exp(c*(b*x+a))*hypergeom([1, 1/2*b*c/e],[1+1/2*b*c/e],exp(2*e*x+2*d))/b/c+4*exp(c*(b*x+a)
)*hypergeom([2, 1/2*b*c/e],[1+1/2*b*c/e],exp(2*e*x+2*d))/b/c

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Rubi [A]
time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5593, 2225, 2283} \begin {gather*} -\frac {4 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};\frac {b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};\frac {b c}{2 e}+1;e^{2 (d+e x)}\right )}{b c}+\frac {e^{c (a+b x)}}{b c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

E^(c*(a + b*x))/(b*c) - (4*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))]
)/(b*c) + (4*E^(c*(a + b*x))*Hypergeometric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c)

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 (a+b x)} \coth ^2(d+e x) \, dx &=\int \left (e^{c (a+b x)}+\frac {4 e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^2}+\frac {4 e^{c (a+b x)}}{-1+e^{2 (d+e x)}}\right ) \, dx\\ &=4 \int \frac {e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^2} \, dx+4 \int \frac {e^{c (a+b x)}}{-1+e^{2 (d+e x)}} \, dx+\int e^{c (a+b x)} \, dx\\ &=\frac {e^{c (a+b x)}}{b c}-\frac {4 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c}\\ \end {align*}

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Mathematica [A]
time = 2.16, size = 145, normalized size = 1.28 \begin {gather*} e^{c (a+b x)} \left (\frac {1}{b c}+\frac {2 e^{2 d} \left (b c e^{2 e x} \, _2F_1\left (1,1+\frac {b c}{2 e};2+\frac {b c}{2 e};e^{2 (d+e x)}\right )-(b c+2 e) \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )\right )}{e (b c+2 e) \left (-1+e^{2 d}\right )}+\frac {\text {csch}(d) \text {csch}(d+e x) \sinh (e x)}{e}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 1.18, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{c \left (b x +a \right )} \left (\coth ^{2}\left (e x +d \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))*coth(e*x+d)^2,x)

[Out]

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

[Out]

16*b*c*integrate(-e^(b*c*x + a*c + 1)/(b^2*c^2 - 6*b*c*e - (b^2*c^2*e^(6*d) - 6*b*c*e^(6*d + 1) + 8*e^(6*d + 2
))*e^(6*x*e) + 3*(b^2*c^2*e^(4*d) - 6*b*c*e^(4*d + 1) + 8*e^(4*d + 2))*e^(4*x*e) - 3*(b^2*c^2*e^(2*d) - 6*b*c*
e^(2*d + 1) + 8*e^(2*d + 2))*e^(2*x*e) + 8*e^2), x) + (b^2*c^2*e^(a*c) + 10*b*c*e^(a*c + 1) + (b^2*c^2*e^(a*c
+ 4*d) - 6*b*c*e^(a*c + 4*d + 1) + 8*e^(a*c + 4*d + 2))*e^(4*x*e) + 2*(b^2*c^2*e^(a*c + 2*d) - 2*b*c*e^(a*c +
2*d + 1) - 8*e^(a*c + 2*d + 2))*e^(2*x*e) + 8*e^(a*c + 2))*e^(b*c*x)/(b^3*c^3 - 6*b^2*c^2*e + 8*b*c*e^2 + (b^3
*c^3*e^(4*d) - 6*b^2*c^2*e^(4*d + 1) + 8*b*c*e^(4*d + 2))*e^(4*x*e) - 2*(b^3*c^3*e^(2*d) - 6*b^2*c^2*e^(2*d +
1) + 8*b*c*e^(2*d + 2))*e^(2*x*e))

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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(c*(b*x+a))*coth(e*x+d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{a c} \int e^{b c x} \coth ^{2}{\left (d + e x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(c*(b*x+a))*coth(e*x+d)**2,x)

[Out]

exp(a*c)*Integral(exp(b*c*x)*coth(d + e*x)**2, 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(c*(b*x+a))*coth(e*x+d)^2,x, algorithm="giac")

[Out]

integrate(coth(e*x + d)^2*e^((b*x + a)*c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\mathrm {coth}\left (d+e\,x\right )}^2\,{\mathrm {e}}^{c\,\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(d + e*x)^2*exp(c*(a + b*x)),x)

[Out]

int(coth(d + e*x)^2*exp(c*(a + b*x)), x)

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