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

Optimal result

Integrand size = 18, antiderivative size = 113 \[ \int e^{c (a+b x)} \coth ^2(d+e x) \, dx=\frac {e^{c (a+b x)}}{b c}-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )}{b c} \]

[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

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {5593, 2225, 2283} \[ \int e^{c (a+b x)} \coth ^2(d+e x) \, dx=-\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {b c}{2 e},\frac {b c}{2 e}+1,e^{2 (d+e x)}\right )}{b c}+\frac {4 e^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b c}{2 e},\frac {b c}{2 e}+1,e^{2 (d+e x)}\right )}{b c}+\frac {e^{c (a+b x)}}{b c} \]

[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*} \text {integral}& = \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)} \operatorname {Hypergeometric2F1}\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)} \operatorname {Hypergeometric2F1}\left (2,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )}{b c} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.45 \[ \int e^{c (a+b x)} \coth ^2(d+e x) \, dx=\frac {e^{c (a+b x)} \left (2 b^2 c^2 e^{2 (d+e x)} \operatorname {Hypergeometric2F1}\left (1,1+\frac {b c}{2 e},2+\frac {b c}{2 e},e^{2 (d+e x)}\right )-(b c+2 e) \left (2 b c e^{2 d} \operatorname {Hypergeometric2F1}\left (1,\frac {b c}{2 e},1+\frac {b c}{2 e},e^{2 (d+e x)}\right )-\left (-1+e^{2 d}\right ) (e+b c \text {csch}(d) \text {csch}(d+e x) \sinh (e x))\right )\right )}{b c e (b c+2 e) \left (-1+e^{2 d}\right )} \]

[In]

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

[Out]

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

Maple [F]

\[\int {\mathrm e}^{c \left (b x +a \right )} \coth \left (e x +d \right )^{2}d x\]

[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)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[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)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int e^{c (a+b x)} \coth ^2(d+e x) \, dx=\int {\mathrm {coth}\left (d+e\,x\right )}^2\,{\mathrm {e}}^{c\,\left (a+b\,x\right )} \,d x \]

[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)