3.3.33 \(\int e^{c (a+b x)} \coth ^3(d+e x) \, dx\) [233]

Optimal. Leaf size=161 \[ \frac {e^{c (a+b x)}}{b c}-\frac {6 e^{c (a+b x)} \, _2F_1\left (1,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c}+\frac {12 e^{c (a+b x)} \, _2F_1\left (2,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c}-\frac {8 e^{c (a+b x)} \, _2F_1\left (3,\frac {b c}{2 e};1+\frac {b c}{2 e};e^{2 (d+e x)}\right )}{b c} \]

[Out]

exp(c*(b*x+a))/b/c-6*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+12*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-8*exp(c*(b*x+a))*hypergeom([3, 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 = 0.12, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {6 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 {12 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 {8 e^{c (a+b x)} \, _2F_1\left (3,\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]^3,x]

[Out]

E^(c*(a + b*x))/(b*c) - (6*E^(c*(a + b*x))*Hypergeometric2F1[1, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))]
)/(b*c) + (12*E^(c*(a + b*x))*Hypergeometric2F1[2, (b*c)/(2*e), 1 + (b*c)/(2*e), E^(2*(d + e*x))])/(b*c) - (8*
E^(c*(a + b*x))*Hypergeometric2F1[3, (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 ^3(d+e x) \, dx &=\int \left (e^{c (a+b x)}+\frac {8 e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^3}+\frac {12 e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^2}+\frac {6 e^{c (a+b x)}}{-1+e^{2 (d+e x)}}\right ) \, dx\\ &=6 \int \frac {e^{c (a+b x)}}{-1+e^{2 (d+e x)}} \, dx+8 \int \frac {e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^3} \, dx+12 \int \frac {e^{c (a+b x)}}{\left (-1+e^{2 (d+e x)}\right )^2} \, dx+\int e^{c (a+b x)} \, dx\\ &=\frac {e^{c (a+b x)}}{b c}-\frac {6 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 {12 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}-\frac {8 e^{c (a+b x)} \, _2F_1\left (3,\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 = 2.15, size = 185, normalized size = 1.15 \begin {gather*} \frac {1}{2} e^{c (a+b x)} \left (\frac {2 \coth (d)}{b c}-\frac {\text {csch}^2(d+e x)}{e}+\frac {2 \left (b^2 c^2+2 e^2\right ) 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 )}{b c e^2 (b c+2 e) \left (-1+e^{2 d}\right )}+\frac {b c \text {csch}(d) \text {csch}(d+e x) \sinh (e x)}{e^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[E^(c*(a + b*x))*Coth[d + e*x]^3,x]

[Out]

(E^(c*(a + b*x))*((2*Coth[d])/(b*c) - Csch[d + e*x]^2/e + (2*(b^2*c^2 + 2*e^2)*E^(2*d)*(b*c*E^(2*e*x)*Hypergeo
metric2F1[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))]))/(b*c*e^2*(b*c + 2*e)*(-1 + E^(2*d))) + (b*c*Csch[d]*Csch[d + e*x]*Sinh[e
*x])/e^2))/2

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Maple [F]
time = 1.30, size = 0, normalized size = 0.00 \[\int {\mathrm e}^{c \left (b x +a \right )} \left (\coth ^{3}\left (e x +d \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(c*(b*x+a))*coth(e*x+d)^3,x)

[Out]

int(exp(c*(b*x+a))*coth(e*x+d)^3,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)^3,x, algorithm="maxima")

[Out]

-48*(b^2*c^2*e^(a*c + 1) + 2*e^(a*c + 3))*integrate(-e^(b*c*x)/(b^3*c^3 - 12*b^2*c^2*e + 44*b*c*e^2 + (b^3*c^3
*e^(8*d) - 12*b^2*c^2*e^(8*d + 1) + 44*b*c*e^(8*d + 2) - 48*e^(8*d + 3))*e^(8*x*e) - 4*(b^3*c^3*e^(6*d) - 12*b
^2*c^2*e^(6*d + 1) + 44*b*c*e^(6*d + 2) - 48*e^(6*d + 3))*e^(6*x*e) + 6*(b^3*c^3*e^(4*d) - 12*b^2*c^2*e^(4*d +
1) + 44*b*c*e^(4*d + 2) - 48*e^(4*d + 3))*e^(4*x*e) - 4*(b^3*c^3*e^(2*d) - 12*b^2*c^2*e^(2*d + 1) + 44*b*c*e^
(2*d + 2) - 48*e^(2*d + 3))*e^(2*x*e) - 48*e^3), x) - (b^3*c^3*e^(a*c) + 36*b^2*c^2*e^(a*c + 1) + 44*b*c*e^(a*
c + 2) + (b^3*c^3*e^(a*c + 6*d) - 12*b^2*c^2*e^(a*c + 6*d + 1) + 44*b*c*e^(a*c + 6*d + 2) - 48*e^(a*c + 6*d +
3))*e^(6*x*e) + 3*(b^3*c^3*e^(a*c + 4*d) - 8*b^2*c^2*e^(a*c + 4*d + 1) + 4*b*c*e^(a*c + 4*d + 2) + 48*e^(a*c +
4*d + 3))*e^(4*x*e) + 3*(b^3*c^3*e^(a*c + 2*d) - 28*b*c*e^(a*c + 2*d + 2) - 48*e^(a*c + 2*d + 3))*e^(2*x*e) +
48*e^(a*c + 3))*e^(b*c*x)/(b^4*c^4 - 12*b^3*c^3*e + 44*b^2*c^2*e^2 - 48*b*c*e^3 - (b^4*c^4*e^(6*d) - 12*b^3*c
^3*e^(6*d + 1) + 44*b^2*c^2*e^(6*d + 2) - 48*b*c*e^(6*d + 3))*e^(6*x*e) + 3*(b^4*c^4*e^(4*d) - 12*b^3*c^3*e^(4
*d + 1) + 44*b^2*c^2*e^(4*d + 2) - 48*b*c*e^(4*d + 3))*e^(4*x*e) - 3*(b^4*c^4*e^(2*d) - 12*b^3*c^3*e^(2*d + 1)
+ 44*b^2*c^2*e^(2*d + 2) - 48*b*c*e^(2*d + 3))*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)^3,x, algorithm="fricas")

[Out]

integral(coth(x*e + d)^3*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 ^{3}{\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)**3,x)

[Out]

exp(a*c)*Integral(exp(b*c*x)*coth(d + e*x)**3, 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)^3,x, algorithm="giac")

[Out]

integrate(coth(e*x + d)^3*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 )}^3\,{\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)^3*exp(c*(a + b*x)),x)

[Out]

int(coth(d + e*x)^3*exp(c*(a + b*x)), x)

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