\(\int e^{c+d x} \text {csch}^2(a+b x) \, dx\) [882]

Optimal result

Integrand size = 16, antiderivative size = 54 \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\frac {4 e^{c+d x+2 (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {d}{2 b},2+\frac {d}{2 b},e^{2 (a+b x)}\right )}{2 b+d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {5601} \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=\frac {4 e^{2 (a+b x)+c+d x} \operatorname {Hypergeometric2F1}\left (2,\frac {d}{2 b}+1,\frac {d}{2 b}+2,e^{2 (a+b x)}\right )}{2 b+d} \]

[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 5601

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))/(e*n + b*c*Log[F]))*Hypergeometric2F1[n, n/2 + b*c*(Log[F]/(2*e)), 1 + n/2 + b*c*(Log[F]/(2*
e)), E^(2*(d + e*x))], x] /; FreeQ[{F, a, b, c, d, e}, x] && IntegerQ[n]

Rubi steps \begin{align*} \text {integral}& = \frac {4 e^{c+d x+2 (a+b x)} \operatorname {Hypergeometric2F1}\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] (verified)

Leaf count is larger than twice the leaf count of optimal. \(137\) vs. \(2(54)=108\).

Time = 1.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.54 \[ \int e^{c+d x} \text {csch}^2(a+b x) \, dx=-\frac {2 d \left (\frac {e^{2 a+c+d x} \operatorname {Hypergeometric2F1}\left (1,\frac {d}{2 b},1+\frac {d}{2 b},e^{2 (a+b x)}\right )}{d}-\frac {e^{2 a+c+(2 b+d) x} \operatorname {Hypergeometric2F1}\left (1,1+\frac {d}{2 b},2+\frac {d}{2 b},e^{2 (a+b x)}\right )}{2 b+d}\right )}{b \left (-1+e^{2 a}\right )}+\frac {e^{c+d x} \text {csch}(a) \text {csch}(a+b x) \sinh (b x)}{b} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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))

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

int(exp(c + d*x)/sinh(a + b*x)^2,x)

[Out]

int(exp(c + d*x)/sinh(a + b*x)^2, x)