\(\int F^{c (a+b x)} \text {csch}^2(d+e x) \, dx\) [326]

Optimal result

Integrand size = 18, antiderivative size = 68 \[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \, dx=\frac {4 e^{2 (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b c \log (F)}{2 e},2+\frac {b c \log (F)}{2 e},e^{2 (d+e x)}\right )}{2 e+b c \log (F)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {5601} \[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \, dx=\frac {4 e^{2 (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,\frac {b c \log (F)}{2 e}+1,\frac {b c \log (F)}{2 e}+2,e^{2 (d+e x)}\right )}{b c \log (F)+2 e} \]

[In]

Int[F^(c*(a + b*x))*Csch[d + e*x]^2,x]

[Out]

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

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^{2 (d+e x)} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (2,1+\frac {b c \log (F)}{2 e},2+\frac {b c \log (F)}{2 e},e^{2 (d+e x)}\right )}{2 e+b c \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.91 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.28 \[ \int F^{c (a+b x)} \text {csch}^2(d+e x) \, dx=-\frac {2 F^{c (a+b x)} \left (\left (-1+e^{2 d}\right ) \operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{2 e},1+\frac {b c \log (F)}{2 e},e^{2 (d+e x)}\right )+\text {csch}(d+e x) \sinh (d) (\cosh (e x)-\sinh (e x))\right )}{e \left (-1+e^{2 d}\right )} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(F^(c*(a + b*x))/sinh(d + e*x)^2,x)

[Out]

int(F^(c*(a + b*x))/sinh(d + e*x)^2, x)