### 3.882 $$\int e^{c+d x} \text{csch}^2(a+b x) \, dx$$

Optimal. Leaf size=54 $\frac{4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}$

[Out]

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

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Rubi [A]  time = 0.0294736, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.062, Rules used = {5493} $\frac{4 e^{2 (a+b x)+c+d x} \, _2F_1\left (2,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}$

Antiderivative was successfully veriﬁed.

[In]

Int[E^(c + d*x)*Csch[a + b*x]^2,x]

[Out]

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

Rule 5493

Int[Csch[(d_.) + (e_.)*(x_)]^(n_.)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> Simp[((-2)^n*E^(n*(d + e*x)
)*F^(c*(a + b*x))*Hypergeometric2F1[n, n/2 + (b*c*Log[F])/(2*e), 1 + n/2 + (b*c*Log[F])/(2*e), E^(2*(d + e*x))
])/(e*n + b*c*Log[F]), x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps

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

Mathematica [B]  time = 3.22928, size = 131, normalized size = 2.43 $\frac{e^c \left (\text{csch}(a) e^{d x} \sinh (b x) \text{csch}(a+b x)-\frac{2 e^{2 a} d \left (\frac{e^{d x} \, _2F_1\left (1,\frac{d}{2 b};\frac{d}{2 b}+1;e^{2 (a+b x)}\right )}{d}-\frac{e^{x (2 b+d)} \, _2F_1\left (1,\frac{d}{2 b}+1;\frac{d}{2 b}+2;e^{2 (a+b x)}\right )}{2 b+d}\right )}{e^{2 a}-1}\right )}{b}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [F]  time = 0.031, size = 0, normalized size = 0. \begin{align*} \int{{\rm e}^{dx+c}} \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)*csch(b*x+a)^2,x)

[Out]

int(exp(d*x+c)*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{4 \,{\left ({\left (4 \, b e^{c} - d e^{c}\right )} e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b e^{c}\right )} e^{\left (d x\right )}}{8 \, b^{2} - 6 \, b d + d^{2} +{\left (8 \, b^{2} - 6 \, b d + d^{2}\right )} e^{\left (4 \, b x + 4 \, a\right )} - 2 \,{\left (8 \, b^{2} - 6 \, b d + d^{2}\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)*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) - 4*((4*b*e^c - d*e^c)*e^(2*b*x + 2*a) -
4*b*e^c)*e^(d*x)/(8*b^2 - 6*b*d + d^2 + (8*b^2 - 6*b*d + d^2)*e^(4*b*x + 4*a) - 2*(8*b^2 - 6*b*d + d^2)*e^(2*
b*x + 2*a))

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\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)*csch(b*x+a)^2,x, algorithm="fricas")

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \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)*csch(b*x+a)^2,x, algorithm="giac")

[Out]

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