\(\int F^{c (a+b x)} \text {csch}(d+e x) \, dx\) [325]

Optimal result

Integrand size = 16, antiderivative size = 66 \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=-\frac {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),e^{2 (d+e x)}\right )}{e+b c \log (F)} \]

[Out]

-2*exp(e*x+d)*F^(c*(b*x+a))*hypergeom([1, 1/2*(e+b*c*ln(F))/e],[3/2+1/2*b*c*ln(F)/e],exp(2*e*x+2*d))/(e+b*c*ln
(F))

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 66, 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} \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=-\frac {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (\frac {b c \log (F)}{e}+3\right ),e^{2 (d+e x)}\right )}{b c \log (F)+e} \]

[In]

Int[F^(c*(a + b*x))*Csch[d + e*x],x]

[Out]

(-2*E^(d + e*x)*F^(c*(a + b*x))*Hypergeometric2F1[1, (e + b*c*Log[F])/(2*e), (3 + (b*c*Log[F])/e)/2, E^(2*(d +
e*x))])/(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 {2 e^{d+e x} F^{c (a+b x)} \operatorname {Hypergeometric2F1}\left (1,\frac {e+b c \log (F)}{2 e},\frac {1}{2} \left (3+\frac {b c \log (F)}{e}\right ),e^{2 (d+e x)}\right )}{e+b c \log (F)} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.62 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.20 \[ \int F^{c (a+b x)} \text {csch}(d+e x) \, dx=\frac {F^{c (a+b x)} \left (\operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e},1+\frac {b c \log (F)}{e},-e^{d+e x}\right )-\operatorname {Hypergeometric2F1}\left (1,\frac {b c \log (F)}{e},1+\frac {b c \log (F)}{e},e^{d+e x}\right )\right )}{b c \log (F)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int(F^(c*(a + b*x))/sinh(d + e*x),x)

[Out]

int(F^(c*(a + b*x))/sinh(d + e*x), x)