### 3.957 $$\int e^{c+d x} \coth ^2(a+b x) \, dx$$

Optimal. Leaf size=94 $-\frac{4 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{4 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{e^{c+d x}}{d}$

[Out]

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

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Rubi [A]  time = 0.114463, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.188, Rules used = {5485, 2194, 2251} $-\frac{4 e^{c+d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{4 e^{c+d x} \, _2F_1\left (2,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}+\frac{e^{c+d x}}{d}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(c + d*x)*Coth[a + b*x]^2,x]

[Out]

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

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]

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])

Rubi steps

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

Mathematica [A]  time = 0.87354, size = 145, normalized size = 1.54 $-\frac{2 d e^{2 a+c} \left (\frac{e^{d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{e^{x (2 b+d)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{\left (e^{2 a}-1\right ) b}+\frac{\text{csch}(a) \sinh (b x) e^{c+d x} \text{csch}(a+b x)}{b}+\frac{e^{c+d x}}{d}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [F]  time = 0.101, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \left ( \cosh \left ( bx+a \right ) \right ) ^{2} \left ({\rm csch} \left (bx+a\right ) \right ) ^{2}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(exp(d*x+c)*cosh(b*x+a)^2*csch(b*x+a)^2,x)

[Out]

int(exp(d*x+c)*cosh(b*x+a)^2*csch(b*x+a)^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 16 \, b d \int -\frac{e^{\left (d x + c\right )}}{8 \, b^{2} - 6 \, b d + d^{2} -{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (6 \, b x + 6 \, a\right )} + 3 \,{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 3 \,{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (2 \, b x + 2 \, a\right )}}\,{d x} + \frac{{\left (8 \, b^{2} e^{c} + 10 \, b d e^{c} + d^{2} e^{c} +{\left (8 \, b^{2} e^{c} - 6 \, b d e^{c} + d^{2} e^{c}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (8 \, b^{2} e^{c} + 2 \, b d e^{c} - d^{2} e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )}\right )} e^{\left (d x\right )}}{8 \, b^{2} d - 6 \, b d^{2} + d^{3} +{\left (8 \, b^{2} d - 6 \, b d^{2} + d^{3}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (8 \, b^{2} d - 6 \, b d^{2} + d^{3}\right )} e^{\left (2 \, b x + 2 \, a\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x+c)*cosh(b*x+a)**2*csch(b*x+a)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \cosh \left (b x + a\right )^{2} \operatorname{csch}\left (b x + a\right )^{2} e^{\left (d x + c\right )}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(d*x+c)*cosh(b*x+a)^2*csch(b*x+a)^2,x, algorithm="giac")

[Out]

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