3.436 \(\int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx\)

Optimal. Leaf size=124 \[ \frac {\tanh (e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {\text {csch}^5(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{5 f}-\frac {\text {csch}^3(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {3 \text {csch}(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f} \]

[Out]

-3*csch(f*x+e)*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f-csch(f*x+e)^3*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f-1/5*c
sch(f*x+e)^5*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f+(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f

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Rubi [A]  time = 0.12, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3176, 3207, 2590, 270} \[ \frac {\tanh (e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {\text {csch}^5(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{5 f}-\frac {\text {csch}^3(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f}-\frac {3 \text {csch}(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{f} \]

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^6*Sqrt[a + a*Sinh[e + f*x]^2],x]

[Out]

(-3*Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]*Sech[e + f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^3*Sech[e +
f*x])/f - (Sqrt[a*Cosh[e + f*x]^2]*Csch[e + f*x]^5*Sech[e + f*x])/(5*f) + (Sqrt[a*Cosh[e + f*x]^2]*Tanh[e + f
*x])/f

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2590

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[f^(-1), Subst[Int[(1 - x^2
)^((m + n - 1)/2)/x^n, x], x, Cos[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]

Rule 3176

Int[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[ActivateTrig[u*(a*cos[e + f*x]^2)^p]
, x] /; FreeQ[{a, b, e, f, p}, x] && EqQ[a + b, 0]

Rule 3207

Int[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Di
st[((b*ff^n)^IntPart[p]*(b*Sin[e + f*x]^n)^FracPart[p])/(Sin[e + f*x]/ff)^(n*FracPart[p]), Int[ActivateTrig[u]
*(Sin[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] &&  !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] |
| MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig
]])

Rubi steps

\begin {align*} \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx &=\int \sqrt {a \cosh ^2(e+f x)} \coth ^6(e+f x) \, dx\\ &=\left (\sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \int \cosh (e+f x) \coth ^6(e+f x) \, dx\\ &=-\frac {\left (i \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^3}{x^6} \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac {\left (i \sqrt {a \cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \operatorname {Subst}\left (\int \left (-1+\frac {1}{x^6}-\frac {3}{x^4}+\frac {3}{x^2}\right ) \, dx,x,-i \sinh (e+f x)\right )}{f}\\ &=-\frac {3 \sqrt {a \cosh ^2(e+f x)} \text {csch}(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^3(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^5(e+f x) \text {sech}(e+f x)}{5 f}+\frac {\sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{f}\\ \end {align*}

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Mathematica [A]  time = 0.20, size = 67, normalized size = 0.54 \[ \frac {(235 \cosh (2 (e+f x))-90 \cosh (4 (e+f x))+5 \cosh (6 (e+f x))-182) \text {csch}^5(e+f x) \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)}}{160 f} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^6*Sqrt[a + a*Sinh[e + f*x]^2],x]

[Out]

(Sqrt[a*Cosh[e + f*x]^2]*(-182 + 235*Cosh[2*(e + f*x)] - 90*Cosh[4*(e + f*x)] + 5*Cosh[6*(e + f*x)])*Csch[e +
f*x]^5*Sech[e + f*x])/(160*f)

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fricas [B]  time = 0.62, size = 1696, normalized size = 13.68 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

1/10*(60*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^11 + 5*e^(f*x + e)*sinh(f*x + e)^12 + 30*(11*cosh(f*x + e)^2
- 3)*e^(f*x + e)*sinh(f*x + e)^10 + 100*(11*cosh(f*x + e)^3 - 9*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^9 + 5
*(495*cosh(f*x + e)^4 - 810*cosh(f*x + e)^2 + 47)*e^(f*x + e)*sinh(f*x + e)^8 + 40*(99*cosh(f*x + e)^5 - 270*c
osh(f*x + e)^3 + 47*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^7 + 28*(165*cosh(f*x + e)^6 - 675*cosh(f*x + e)^4
+ 235*cosh(f*x + e)^2 - 13)*e^(f*x + e)*sinh(f*x + e)^6 + 8*(495*cosh(f*x + e)^7 - 2835*cosh(f*x + e)^5 + 164
5*cosh(f*x + e)^3 - 273*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^5 + 5*(495*cosh(f*x + e)^8 - 3780*cosh(f*x +
e)^6 + 3290*cosh(f*x + e)^4 - 1092*cosh(f*x + e)^2 + 47)*e^(f*x + e)*sinh(f*x + e)^4 + 20*(55*cosh(f*x + e)^9
- 540*cosh(f*x + e)^7 + 658*cosh(f*x + e)^5 - 364*cosh(f*x + e)^3 + 47*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e
)^3 + 10*(33*cosh(f*x + e)^10 - 405*cosh(f*x + e)^8 + 658*cosh(f*x + e)^6 - 546*cosh(f*x + e)^4 + 141*cosh(f*x
+ e)^2 - 9)*e^(f*x + e)*sinh(f*x + e)^2 + 4*(15*cosh(f*x + e)^11 - 225*cosh(f*x + e)^9 + 470*cosh(f*x + e)^7
- 546*cosh(f*x + e)^5 + 235*cosh(f*x + e)^3 - 45*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) + (5*cosh(f*x + e)^1
2 - 90*cosh(f*x + e)^10 + 235*cosh(f*x + e)^8 - 364*cosh(f*x + e)^6 + 235*cosh(f*x + e)^4 - 90*cosh(f*x + e)^2
+ 5)*e^(f*x + e))*sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/(f*cosh(f*x + e)^11 + (f*e^(
2*f*x + 2*e) + f)*sinh(f*x + e)^11 + 11*(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e))*sinh(f*x + e)^10 -
5*f*cosh(f*x + e)^9 + 5*(11*f*cosh(f*x + e)^2 + (11*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(f*x + e)
^9 + 15*(11*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e) + (11*f*cosh(f*x + e)^3 - 3*f*cosh(f*x + e))*e^(2*f*x + 2*e)
)*sinh(f*x + e)^8 + 10*f*cosh(f*x + e)^7 + 10*(33*f*cosh(f*x + e)^4 - 18*f*cosh(f*x + e)^2 + (33*f*cosh(f*x +
e)^4 - 18*f*cosh(f*x + e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^7 + 14*(33*f*cosh(f*x + e)^5 - 30*f*cosh(f
*x + e)^3 + 5*f*cosh(f*x + e) + (33*f*cosh(f*x + e)^5 - 30*f*cosh(f*x + e)^3 + 5*f*cosh(f*x + e))*e^(2*f*x + 2
*e))*sinh(f*x + e)^6 - 10*f*cosh(f*x + e)^5 + 2*(231*f*cosh(f*x + e)^6 - 315*f*cosh(f*x + e)^4 + 105*f*cosh(f*
x + e)^2 + (231*f*cosh(f*x + e)^6 - 315*f*cosh(f*x + e)^4 + 105*f*cosh(f*x + e)^2 - 5*f)*e^(2*f*x + 2*e) - 5*f
)*sinh(f*x + e)^5 + 10*(33*f*cosh(f*x + e)^7 - 63*f*cosh(f*x + e)^5 + 35*f*cosh(f*x + e)^3 - 5*f*cosh(f*x + e)
+ (33*f*cosh(f*x + e)^7 - 63*f*cosh(f*x + e)^5 + 35*f*cosh(f*x + e)^3 - 5*f*cosh(f*x + e))*e^(2*f*x + 2*e))*s
inh(f*x + e)^4 + 5*f*cosh(f*x + e)^3 + 5*(33*f*cosh(f*x + e)^8 - 84*f*cosh(f*x + e)^6 + 70*f*cosh(f*x + e)^4 -
20*f*cosh(f*x + e)^2 + (33*f*cosh(f*x + e)^8 - 84*f*cosh(f*x + e)^6 + 70*f*cosh(f*x + e)^4 - 20*f*cosh(f*x +
e)^2 + f)*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^3 + 5*(11*f*cosh(f*x + e)^9 - 36*f*cosh(f*x + e)^7 + 42*f*cosh(f*
x + e)^5 - 20*f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e) + (11*f*cosh(f*x + e)^9 - 36*f*cosh(f*x + e)^7 + 42*f*cosh
(f*x + e)^5 - 20*f*cosh(f*x + e)^3 + 3*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^2 - f*cosh(f*x + e) + (
f*cosh(f*x + e)^11 - 5*f*cosh(f*x + e)^9 + 10*f*cosh(f*x + e)^7 - 10*f*cosh(f*x + e)^5 + 5*f*cosh(f*x + e)^3 -
f*cosh(f*x + e))*e^(2*f*x + 2*e) + (11*f*cosh(f*x + e)^10 - 45*f*cosh(f*x + e)^8 + 70*f*cosh(f*x + e)^6 - 50*
f*cosh(f*x + e)^4 + 15*f*cosh(f*x + e)^2 + (11*f*cosh(f*x + e)^10 - 45*f*cosh(f*x + e)^8 + 70*f*cosh(f*x + e)^
6 - 50*f*cosh(f*x + e)^4 + 15*f*cosh(f*x + e)^2 - f)*e^(2*f*x + 2*e) - f)*sinh(f*x + e))

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giac [A]  time = 0.20, size = 104, normalized size = 0.84 \[ -\frac {\sqrt {a} {\left (\frac {4 \, {\left (15 \, e^{\left (9 \, f x + 9 \, e\right )} - 40 \, e^{\left (7 \, f x + 7 \, e\right )} + 66 \, e^{\left (5 \, f x + 5 \, e\right )} - 40 \, e^{\left (3 \, f x + 3 \, e\right )} + 15 \, e^{\left (f x + e\right )}\right )}}{{\left (e^{\left (2 \, f x + 2 \, e\right )} - 1\right )}^{5}} - 5 \, e^{\left (f x + e\right )} + 5 \, e^{\left (-f x - e\right )}\right )}}{10 \, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

-1/10*sqrt(a)*(4*(15*e^(9*f*x + 9*e) - 40*e^(7*f*x + 7*e) + 66*e^(5*f*x + 5*e) - 40*e^(3*f*x + 3*e) + 15*e^(f*
x + e))/(e^(2*f*x + 2*e) - 1)^5 - 5*e^(f*x + e) + 5*e^(-f*x - e))/f

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maple [A]  time = 0.21, size = 65, normalized size = 0.52 \[ \frac {\cosh \left (f x +e \right ) a \left (5 \left (\sinh ^{6}\left (f x +e \right )\right )-15 \left (\sinh ^{4}\left (f x +e \right )\right )-5 \left (\sinh ^{2}\left (f x +e \right )\right )-1\right )}{5 \sinh \left (f x +e \right )^{5} \sqrt {a \left (\cosh ^{2}\left (f x +e \right )\right )}\, f} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x)

[Out]

1/5*cosh(f*x+e)*a*(5*sinh(f*x+e)^6-15*sinh(f*x+e)^4-5*sinh(f*x+e)^2-1)/sinh(f*x+e)^5/(a*cosh(f*x+e)^2)^(1/2)/f

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maxima [B]  time = 0.74, size = 1050, normalized size = 8.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^6*(a+a*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

-1/320*(105*sqrt(a)*log(e^(-f*x - e) + 1) - 105*sqrt(a)*log(e^(-f*x - e) - 1) - 2*(375*sqrt(a)*e^(-f*x - e) -
790*sqrt(a)*e^(-3*f*x - 3*e) + 896*sqrt(a)*e^(-5*f*x - 5*e) - 490*sqrt(a)*e^(-7*f*x - 7*e) + 105*sqrt(a)*e^(-9
*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x
- 10*e) - 1))/f + 1/320*(105*sqrt(a)*log(e^(-f*x - e) + 1) - 105*sqrt(a)*log(e^(-f*x - e) - 1) + 2*(105*sqrt(
a)*e^(-f*x - e) - 490*sqrt(a)*e^(-3*f*x - 3*e) + 896*sqrt(a)*e^(-5*f*x - 5*e) - 790*sqrt(a)*e^(-7*f*x - 7*e) +
375*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x -
8*e) + e^(-10*f*x - 10*e) - 1))/f + 1/256*(15*sqrt(a)*log(e^(-f*x - e) + 1) - 15*sqrt(a)*log(e^(-f*x - e) - 1
) + 2*(15*sqrt(a)*e^(-f*x - e) + 250*sqrt(a)*e^(-3*f*x - 3*e) - 128*sqrt(a)*e^(-5*f*x - 5*e) + 70*sqrt(a)*e^(-
7*f*x - 7*e) - 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) -
5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f - 1/256*(15*sqrt(a)*log(e^(-f*x - e) + 1) - 15*sqrt(a)*log(e^(
-f*x - e) - 1) + 2*(15*sqrt(a)*e^(-f*x - e) - 70*sqrt(a)*e^(-3*f*x - 3*e) + 128*sqrt(a)*e^(-5*f*x - 5*e) - 250
*sqrt(a)*e^(-7*f*x - 7*e) - 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*
f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f + 2*sqrt(a)*e^(-5*f*x - 5*e)/(f*(5*e^(-2*f*x - 2*
e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1)) - 1/640*(2895*s
qrt(a)*e^(-2*f*x - 2*e) - 7110*sqrt(a)*e^(-4*f*x - 4*e) + 8064*sqrt(a)*e^(-6*f*x - 6*e) - 4410*sqrt(a)*e^(-8*f
*x - 8*e) + 945*sqrt(a)*e^(-10*f*x - 10*e) - 320*sqrt(a))/(f*(e^(-f*x - e) - 5*e^(-3*f*x - 3*e) + 10*e^(-5*f*x
- 5*e) - 10*e^(-7*f*x - 7*e) + 5*e^(-9*f*x - 9*e) - e^(-11*f*x - 11*e))) + 1/640*(945*sqrt(a)*e^(-f*x - e) -
4410*sqrt(a)*e^(-3*f*x - 3*e) + 8064*sqrt(a)*e^(-5*f*x - 5*e) - 7110*sqrt(a)*e^(-7*f*x - 7*e) + 2895*sqrt(a)*e
^(-9*f*x - 9*e) - 320*sqrt(a)*e^(-11*f*x - 11*e))/(f*(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x
- 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))

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mupad [B]  time = 0.91, size = 427, normalized size = 3.44 \[ -\frac {\left (\frac {1}{f}-\frac {{\mathrm {e}}^{2\,e+2\,f\,x}}{f}\right )\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{{\mathrm {e}}^{2\,e+2\,f\,x}+1}-\frac {12\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{f\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {144\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(e + f*x)^6*(a + a*sinh(e + f*x)^2)^(1/2),x)

[Out]

- ((1/f - exp(2*e + 2*f*x)/f)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(exp(2*e + 2*f*x) + 1) - (1
2*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(f*(exp(2*e + 2*f*x) - 1)*(exp(e + f*x
) + exp(3*e + 3*f*x))) - (16*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(f*(exp(2*e
+ 2*f*x) - 1)^2*(exp(e + f*x) + exp(3*e + 3*f*x))) - (144*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e -
f*x)/2)^2)^(1/2))/(5*f*(exp(2*e + 2*f*x) - 1)^3*(exp(e + f*x) + exp(3*e + 3*f*x))) - (128*exp(3*e + 3*f*x)*(a
+ a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(5*f*(exp(2*e + 2*f*x) - 1)^4*(exp(e + f*x) + exp(3*e + 3*f
*x))) - (64*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(5*f*(exp(2*e + 2*f*x) - 1)^
5*(exp(e + f*x) + exp(3*e + 3*f*x)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)**6*(a+a*sinh(f*x+e)**2)**(1/2),x)

[Out]

Timed out

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