3.328 \(\int F^{c (a+b x)} \text {csch}^4(d+e x) \, dx\)

Optimal. Leaf size=131 \[ -\frac {2 e^{2 (d+e x)} F^{c (a+b x)} (2 e-b c \log (F)) \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;e^{2 (d+e x)}\right )}{3 e^2}-\frac {b c \log (F) \text {csch}^2(d+e x) F^{c (a+b x)}}{6 e^2}-\frac {\coth (d+e x) \text {csch}^2(d+e x) F^{c (a+b x)}}{3 e} \]

[Out]

-1/3*F^(c*(b*x+a))*coth(e*x+d)*csch(e*x+d)^2/e-1/6*b*c*F^(c*(b*x+a))*csch(e*x+d)^2*ln(F)/e^2-2/3*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))*(2*e-b*c*ln(F))/e^2

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Rubi [A]  time = 0.06, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5491, 5493} \[ -\frac {2 e^{2 (d+e x)} F^{c (a+b x)} (2 e-b c \log (F)) \, _2F_1\left (2,\frac {b c \log (F)}{2 e}+1;\frac {b c \log (F)}{2 e}+2;e^{2 (d+e x)}\right )}{3 e^2}-\frac {b c \log (F) \text {csch}^2(d+e x) F^{c (a+b x)}}{6 e^2}-\frac {\coth (d+e x) \text {csch}^2(d+e x) F^{c (a+b x)}}{3 e} \]

Antiderivative was successfully verified.

[In]

Int[F^(c*(a + b*x))*Csch[d + e*x]^4,x]

[Out]

-(F^(c*(a + b*x))*Coth[d + e*x]*Csch[d + e*x]^2)/(3*e) - (b*c*F^(c*(a + b*x))*Csch[d + e*x]^2*Log[F])/(6*e^2)
- (2*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]))/(3*e^2)

Rule 5491

Int[Csch[(d_.) + (e_.)*(x_)]^(n_)*(F_)^((c_.)*((a_.) + (b_.)*(x_))), x_Symbol] :> -Simp[(b*c*Log[F]*F^(c*(a +
b*x))*Csch[d + e*x]^(n - 2))/(e^2*(n - 1)*(n - 2)), x] + (-Dist[(e^2*(n - 2)^2 - b^2*c^2*Log[F]^2)/(e^2*(n - 1
)*(n - 2)), Int[F^(c*(a + b*x))*Csch[d + e*x]^(n - 2), x], x] - Simp[(F^(c*(a + b*x))*Csch[d + e*x]^(n - 1)*Co
sh[d + e*x])/(e*(n - 1)), x]) /; FreeQ[{F, a, b, c, d, e}, x] && NeQ[e^2*(n - 2)^2 - b^2*c^2*Log[F]^2, 0] && G
tQ[n, 1] && NeQ[n, 2]

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 F^{c (a+b x)} \text {csch}^4(d+e x) \, dx &=-\frac {F^{c (a+b x)} \coth (d+e x) \text {csch}^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \text {csch}^2(d+e x) \log (F)}{6 e^2}-\frac {1}{6} \left (4-\frac {b^2 c^2 \log ^2(F)}{e^2}\right ) \int F^{c (a+b x)} \text {csch}^2(d+e x) \, dx\\ &=-\frac {F^{c (a+b x)} \coth (d+e x) \text {csch}^2(d+e x)}{3 e}-\frac {b c F^{c (a+b x)} \text {csch}^2(d+e x) \log (F)}{6 e^2}-\frac {2 e^{2 (d+e x)} F^{c (a+b x)} \, _2F_1\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))}{3 e^2}\\ \end {align*}

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Mathematica [A]  time = 7.06, size = 202, normalized size = 1.54 \[ \frac {F^{c (a+b x)} \left (4 e^2-b^2 c^2 \log ^2(F)\right ) \left (2 \, _2F_1\left (1,\frac {b c \log (F)}{2 e};\frac {b c \log (F)}{2 e}+1;\cosh (2 (d+e x))+\sinh (2 (d+e x))\right )+\coth (d)-1\right )}{6 e^3}-\frac {\text {csch}(d) \sinh (e x) \text {csch}(d+e x) F^{a c+b c x} \left (4 e^2-b^2 c^2 \log ^2(F)\right )}{6 e^3}-\frac {\text {csch}(d) \text {csch}^2(d+e x) F^{a c+b c x} (b c \sinh (d) \log (F)+2 e \cosh (d))}{6 e^2}+\frac {\text {csch}(d) \sinh (e x) \text {csch}^3(d+e x) F^{a c+b c x}}{3 e} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(c*(a + b*x))*Csch[d + e*x]^4,x]

[Out]

(F^(c*(a + b*x))*(-1 + Coth[d] + 2*Hypergeometric2F1[1, (b*c*Log[F])/(2*e), 1 + (b*c*Log[F])/(2*e), Cosh[2*(d
+ e*x)] + Sinh[2*(d + e*x)]])*(4*e^2 - b^2*c^2*Log[F]^2))/(6*e^3) - (F^(a*c + b*c*x)*Csch[d]*Csch[d + e*x]^2*(
2*e*Cosh[d] + b*c*Log[F]*Sinh[d]))/(6*e^2) + (F^(a*c + b*c*x)*Csch[d]*Csch[d + e*x]^3*Sinh[e*x])/(3*e) - (F^(a
*c + b*c*x)*Csch[d]*Csch[d + e*x]*(4*e^2 - b^2*c^2*Log[F]^2)*Sinh[e*x])/(6*e^3)

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fricas [F]  time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (F^{b c x + a c} \operatorname {csch}\left (e x + d\right )^{4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*csch(e*x+d)^4,x, algorithm="fricas")

[Out]

integral(F^(b*c*x + a*c)*csch(e*x + d)^4, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{{\left (b x + a\right )} c} \operatorname {csch}\left (e x + d\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*csch(e*x+d)^4,x, algorithm="giac")

[Out]

integrate(F^((b*x + a)*c)*csch(e*x + d)^4, x)

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maple [F]  time = 0.06, size = 0, normalized size = 0.00 \[ \int F^{c \left (b x +a \right )} \mathrm {csch}\left (e x +d \right )^{4}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(b*x+a))*csch(e*x+d)^4,x)

[Out]

int(F^(c*(b*x+a))*csch(e*x+d)^4,x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(c*(b*x+a))*csch(e*x+d)^4,x, algorithm="maxima")

[Out]

128*(F^(a*c)*b^2*c^2*e*log(F)^2 + 2*F^(a*c)*b*c*e^2*log(F))*integrate(-F^(b*c*x)/(b^3*c^3*log(F)^3 - 18*b^2*c^
2*e*log(F)^2 + 104*b*c*e^2*log(F) - 192*e^3 - (b^3*c^3*e^(10*d)*log(F)^3 - 18*b^2*c^2*e*e^(10*d)*log(F)^2 + 10
4*b*c*e^2*e^(10*d)*log(F) - 192*e^3*e^(10*d))*e^(10*e*x) + 5*(b^3*c^3*e^(8*d)*log(F)^3 - 18*b^2*c^2*e*e^(8*d)*
log(F)^2 + 104*b*c*e^2*e^(8*d)*log(F) - 192*e^3*e^(8*d))*e^(8*e*x) - 10*(b^3*c^3*e^(6*d)*log(F)^3 - 18*b^2*c^2
*e*e^(6*d)*log(F)^2 + 104*b*c*e^2*e^(6*d)*log(F) - 192*e^3*e^(6*d))*e^(6*e*x) + 10*(b^3*c^3*e^(4*d)*log(F)^3 -
18*b^2*c^2*e*e^(4*d)*log(F)^2 + 104*b*c*e^2*e^(4*d)*log(F) - 192*e^3*e^(4*d))*e^(4*e*x) - 5*(b^3*c^3*e^(2*d)*
log(F)^3 - 18*b^2*c^2*e*e^(2*d)*log(F)^2 + 104*b*c*e^2*e^(2*d)*log(F) - 192*e^3*e^(2*d))*e^(2*e*x)), x) + 16*(
8*F^(a*c)*b*c*e*log(F) + 16*F^(a*c)*e^2 + (F^(a*c)*b^2*c^2*e^(4*d)*log(F)^2 - 14*F^(a*c)*b*c*e*e^(4*d)*log(F)
+ 48*F^(a*c)*e^2*e^(4*d))*e^(4*e*x) + 8*(F^(a*c)*b*c*e*e^(2*d)*log(F) - 8*F^(a*c)*e^2*e^(2*d))*e^(2*e*x))*F^(b
*c*x)/(b^3*c^3*log(F)^3 - 18*b^2*c^2*e*log(F)^2 + 104*b*c*e^2*log(F) - 192*e^3 + (b^3*c^3*e^(8*d)*log(F)^3 - 1
8*b^2*c^2*e*e^(8*d)*log(F)^2 + 104*b*c*e^2*e^(8*d)*log(F) - 192*e^3*e^(8*d))*e^(8*e*x) - 4*(b^3*c^3*e^(6*d)*lo
g(F)^3 - 18*b^2*c^2*e*e^(6*d)*log(F)^2 + 104*b*c*e^2*e^(6*d)*log(F) - 192*e^3*e^(6*d))*e^(6*e*x) + 6*(b^3*c^3*
e^(4*d)*log(F)^3 - 18*b^2*c^2*e*e^(4*d)*log(F)^2 + 104*b*c*e^2*e^(4*d)*log(F) - 192*e^3*e^(4*d))*e^(4*e*x) - 4
*(b^3*c^3*e^(2*d)*log(F)^3 - 18*b^2*c^2*e*e^(2*d)*log(F)^2 + 104*b*c*e^2*e^(2*d)*log(F) - 192*e^3*e^(2*d))*e^(
2*e*x))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {F^{c\,\left (a+b\,x\right )}}{{\mathrm {sinh}\left (d+e\,x\right )}^4} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(c*(a + b*x))/sinh(d + e*x)^4,x)

[Out]

int(F^(c*(a + b*x))/sinh(d + e*x)^4, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int F^{c \left (a + b x\right )} \operatorname {csch}^{4}{\left (d + e x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(c*(b*x+a))*csch(e*x+d)**4,x)

[Out]

Integral(F**(c*(a + b*x))*csch(d + e*x)**4, x)

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