3.5.89 \(\int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx\) [489]

Optimal. Leaf size=285 \[ -\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {2 (2 a-b) E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-b) F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^2 f} \]

[Out]

-2/3*(2*a-b)*coth(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/f-1/3*coth(f*x+e)*csch(f*x+e)^2*(a+b*sinh(f*x+e)^2)^(1/
2)/a/f-2/3*(2*a-b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)/(1+sinh(f*x+e)^2)
^(1/2),(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+
1/3*(3*a-b)*(1/(1+sinh(f*x+e)^2))^(1/2)*(1+sinh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)/(1+sinh(f*x+e)^2)^(1/2),
(1-b/a)^(1/2))*sech(f*x+e)*(a+b*sinh(f*x+e)^2)^(1/2)/a^2/f/(sech(f*x+e)^2*(a+b*sinh(f*x+e)^2)/a)^(1/2)+2/3*(2*
a-b)*(a+b*sinh(f*x+e)^2)^(1/2)*tanh(f*x+e)/a^2/f

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Rubi [A]
time = 0.21, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3275, 485, 597, 545, 429, 506, 422} \begin {gather*} \frac {(3 a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} F\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}-\frac {2 (2 a-b) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)} E\left (\text {ArcTan}(\sinh (e+f x))\left |1-\frac {b}{a}\right .\right )}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 a-b) \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Coth[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]

[Out]

(-2*(2*a - b)*Coth[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f) - (Coth[e + f*x]*Csch[e + f*x]^2*Sqrt[a + b
*Sinh[e + f*x]^2])/(3*a*f) - (2*(2*a - b)*EllipticE[ArcTan[Sinh[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*S
inh[e + f*x]^2])/(3*a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e + f*x]^2))/a]) + ((3*a - b)*EllipticF[ArcTan[Sin
h[e + f*x]], 1 - b/a]*Sech[e + f*x]*Sqrt[a + b*Sinh[e + f*x]^2])/(3*a^2*f*Sqrt[(Sech[e + f*x]^2*(a + b*Sinh[e
+ f*x]^2))/a]) + (2*(2*a - b)*Sqrt[a + b*Sinh[e + f*x]^2]*Tanh[e + f*x])/(3*a^2*f)

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 485

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[c*(e*x)^(
m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q - 1)/(a*e*(m + 1))), x] - Dist[1/(a*e^n*(m + 1)), Int[(e*x)^(m + n)
*(a + b*x^n)^p*(c + d*x^n)^(q - 2)*Simp[c*(c*b - a*d)*(m + 1) + c*n*(b*c*(p + 1) + a*d*(q - 1)) + d*((c*b - a*
d)*(m + 1) + c*b*n*(p + q))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0]
&& GtQ[q, 1] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rule 545

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

Rule 597

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

Rule 3275

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_), x_Symbol] :> With[{ff = FreeF
actors[Sin[e + f*x], x]}, Dist[ff^(m + 1)*(Sqrt[Cos[e + f*x]^2]/(f*Cos[e + f*x])), Subst[Int[x^m*((a + b*ff^2*
x^2)^p/(1 - ff^2*x^2)^((m + 1)/2)), x], x, Sin[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2]
&&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\coth ^4(e+f x)}{\sqrt {a+b \sinh ^2(e+f x)}} \, dx &=\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\left (1+x^2\right )^{3/2}}{x^4 \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{f}\\ &=-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {2 (2 a-b)+(3 a-b) x^2}{x^2 \sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}\\ &=-\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {\left (\sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {-a (3 a-b)-2 (2 a-b) b x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 f}\\ &=-\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {\left ((3 a-b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a f}+\frac {\left (2 (2 a-b) b \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} \sqrt {a+b x^2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 f}\\ &=-\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}+\frac {(3 a-b) F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^2 f}-\frac {\left (2 (2 a-b) \sqrt {\cosh ^2(e+f x)} \text {sech}(e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1+x^2\right )^{3/2}} \, dx,x,\sinh (e+f x)\right )}{3 a^2 f}\\ &=-\frac {2 (2 a-b) \coth (e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f}-\frac {\coth (e+f x) \text {csch}^2(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a f}-\frac {2 (2 a-b) E\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {(3 a-b) F\left (\tan ^{-1}(\sinh (e+f x))|1-\frac {b}{a}\right ) \text {sech}(e+f x) \sqrt {a+b \sinh ^2(e+f x)}}{3 a^2 f \sqrt {\frac {\text {sech}^2(e+f x) \left (a+b \sinh ^2(e+f x)\right )}{a}}}+\frac {2 (2 a-b) \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x)}{3 a^2 f}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.88, size = 208, normalized size = 0.73 \begin {gather*} \frac {-\frac {\left (2 \left (4 a^2-5 a b+2 b^2\right ) \cosh (2 (e+f x))-(2 a-b) (2 a-3 b-b \cosh (4 (e+f x)))\right ) \coth (e+f x) \text {csch}^2(e+f x)}{\sqrt {2}}-4 i a (2 a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} E\left (i (e+f x)\left |\frac {b}{a}\right .\right )+2 i a (a-b) \sqrt {\frac {2 a-b+b \cosh (2 (e+f x))}{a}} F\left (i (e+f x)\left |\frac {b}{a}\right .\right )}{6 a^2 f \sqrt {2 a-b+b \cosh (2 (e+f x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Coth[e + f*x]^4/Sqrt[a + b*Sinh[e + f*x]^2],x]

[Out]

(-(((2*(4*a^2 - 5*a*b + 2*b^2)*Cosh[2*(e + f*x)] - (2*a - b)*(2*a - 3*b - b*Cosh[4*(e + f*x)]))*Coth[e + f*x]*
Csch[e + f*x]^2)/Sqrt[2]) - (4*I)*a*(2*a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticE[I*(e + f*x), b
/a] + (2*I)*a*(a - b)*Sqrt[(2*a - b + b*Cosh[2*(e + f*x)])/a]*EllipticF[I*(e + f*x), b/a])/(6*a^2*f*Sqrt[2*a -
b + b*Cosh[2*(e + f*x)]])

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Maple [A]
time = 1.92, size = 522, normalized size = 1.83

method	result	size
default	\(\frac {-4 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{6}\left (f x +e \right )\right )+2 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{6}\left (f x +e \right )\right )+3 a^{2} \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) \left (\sinh ^{3}\left (f x +e \right )\right )-5 b \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a \left (\sinh ^{3}\left (f x +e \right )\right )+2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticF \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )+4 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) a b \left (\sinh ^{3}\left (f x +e \right )\right )-2 \sqrt {\frac {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}{a}}\, \sqrt {\frac {\cosh \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \EllipticE \left (\sinh \left (f x +e \right ) \sqrt {-\frac {b}{a}}, \sqrt {\frac {a}{b}}\right ) b^{2} \left (\sinh ^{3}\left (f x +e \right )\right )-4 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-3 \sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{4}\left (f x +e \right )\right )+2 \sqrt {-\frac {b}{a}}\, b^{2} \left (\sinh ^{4}\left (f x +e \right )\right )-5 \sqrt {-\frac {b}{a}}\, a^{2} \left (\sinh ^{2}\left (f x +e \right )\right )+\sqrt {-\frac {b}{a}}\, a b \left (\sinh ^{2}\left (f x +e \right )\right )-\sqrt {-\frac {b}{a}}\, a^{2}}{3 \sqrt {-\frac {b}{a}}\, a^{2} \sinh \left (f x +e \right )^{3} \cosh \left (f x +e \right ) \sqrt {a +b \left (\sinh ^{2}\left (f x +e \right )\right )}\, f}\)	\(522\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/3*(-4*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^6+2*(-1/a*b)^(1/2)*b^2*sinh(f*x+e)^6+3*a^2*((a+b*sinh(f*x+e)^2)/a)^(1/2
)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*sinh(f*x+e)^3-5*((a+b*sinh(f*x+e)^2)
/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b*a*sinh(f*x+e)^3+2*((a+b*si
nh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticF(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*b^2*sinh(f*x+e)^
3+4*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)^(1/2))*a*b*
sinh(f*x+e)^3-2*((a+b*sinh(f*x+e)^2)/a)^(1/2)*(cosh(f*x+e)^2)^(1/2)*EllipticE(sinh(f*x+e)*(-1/a*b)^(1/2),(a/b)
^(1/2))*b^2*sinh(f*x+e)^3-4*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^4-3*(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^4+2*(-1/a*b)^(1/
2)*b^2*sinh(f*x+e)^4-5*(-1/a*b)^(1/2)*a^2*sinh(f*x+e)^2+(-1/a*b)^(1/2)*a*b*sinh(f*x+e)^2-(-1/a*b)^(1/2)*a^2)/(
-1/a*b)^(1/2)/a^2/sinh(f*x+e)^3/cosh(f*x+e)/(a+b*sinh(f*x+e)^2)^(1/2)/f

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="maxima")

[Out]

integrate(coth(f*x + e)^4/sqrt(b*sinh(f*x + e)^2 + a), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2584 vs. \(2 (289) = 578\).
time = 0.13, size = 2584, normalized size = 9.07 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="fricas")

[Out]

2/3*(((4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^6 + 6*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)*sinh(f*x + e)^5 +
(4*a^2*b - 4*a*b^2 + b^3)*sinh(f*x + e)^6 - 3*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^4 - 3*(4*a^2*b - 4*a*b^2
+ b^3 - 5*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 + 4*(5*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*
x + e)^3 - 3*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - 4*a^2*b + 4*a*b^2 - b^3 + 3*(4*a^2*b -
4*a*b^2 + b^3)*cosh(f*x + e)^2 + 3*(5*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^4 + 4*a^2*b - 4*a*b^2 + b^3 - 6
*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^2 + 6*((4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^5 - 2
*(4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e)^3 + (4*a^2*b - 4*a*b^2 + b^3)*cosh(f*x + e))*sinh(f*x + e) - 2*((2*a*
b^2 - b^3)*cosh(f*x + e)^6 + 6*(2*a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (2*a*b^2 - b^3)*sinh(f*x + e)^6
- 3*(2*a*b^2 - b^3)*cosh(f*x + e)^4 - 3*(2*a*b^2 - b^3 - 5*(2*a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^4 +
4*(5*(2*a*b^2 - b^3)*cosh(f*x + e)^3 - 3*(2*a*b^2 - b^3)*cosh(f*x + e))*sinh(f*x + e)^3 - 2*a*b^2 + b^3 + 3*(
2*a*b^2 - b^3)*cosh(f*x + e)^2 + 3*(5*(2*a*b^2 - b^3)*cosh(f*x + e)^4 + 2*a*b^2 - b^3 - 6*(2*a*b^2 - b^3)*cosh
(f*x + e)^2)*sinh(f*x + e)^2 + 6*((2*a*b^2 - b^3)*cosh(f*x + e)^5 - 2*(2*a*b^2 - b^3)*cosh(f*x + e)^3 + (2*a*b
^2 - b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a +
b)/b)*elliptic_e(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*x + e) + sinh(f*x + e))), (8*a^
2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - ((6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^6 + 6
*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)*sinh(f*x + e)^5 + (6*a^3 - 5*a^2*b + a*b^2)*sinh(f*x + e)^6 - 3*(6*a^
3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^4 - 3*(6*a^3 - 5*a^2*b + a*b^2 - 5*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^
2)*sinh(f*x + e)^4 + 4*(5*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^3 - 3*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e
))*sinh(f*x + e)^3 - 6*a^3 + 5*a^2*b - a*b^2 + 3*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^2 + 3*(5*(6*a^3 - 5*a
^2*b + a*b^2)*cosh(f*x + e)^4 + 6*a^3 - 5*a^2*b + a*b^2 - 6*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^2)*sinh(f*
x + e)^2 + 6*((6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^5 - 2*(6*a^3 - 5*a^2*b + a*b^2)*cosh(f*x + e)^3 + (6*a^3
- 5*a^2*b + a*b^2)*cosh(f*x + e))*sinh(f*x + e) + 2*((3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^6 + 6*(3*a^2*b
- 5*a*b^2 + 2*b^3)*cosh(f*x + e)*sinh(f*x + e)^5 + (3*a^2*b - 5*a*b^2 + 2*b^3)*sinh(f*x + e)^6 - 3*(3*a^2*b -
5*a*b^2 + 2*b^3)*cosh(f*x + e)^4 - 3*(3*a^2*b - 5*a*b^2 + 2*b^3 - 5*(3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)
^2)*sinh(f*x + e)^4 + 4*(5*(3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^3 - 3*(3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*
x + e))*sinh(f*x + e)^3 - 3*a^2*b + 5*a*b^2 - 2*b^3 + 3*(3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^2 + 3*(5*(3*
a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^4 + 3*a^2*b - 5*a*b^2 + 2*b^3 - 6*(3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x
+ e)^2)*sinh(f*x + e)^2 + 6*((3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e)^5 - 2*(3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(
f*x + e)^3 + (3*a^2*b - 5*a*b^2 + 2*b^3)*cosh(f*x + e))*sinh(f*x + e))*sqrt((a^2 - a*b)/b^2))*sqrt(b)*sqrt((2*
b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*elliptic_f(arcsin(sqrt((2*b*sqrt((a^2 - a*b)/b^2) - 2*a + b)/b)*(cosh(f*
x + e) + sinh(f*x + e))), (8*a^2 - 8*a*b + b^2 + 4*(2*a*b - b^2)*sqrt((a^2 - a*b)/b^2))/b^2) - sqrt(2)*((2*a*b
^2 - b^3)*cosh(f*x + e)^5 + 5*(2*a*b^2 - b^3)*cosh(f*x + e)*sinh(f*x + e)^4 + (2*a*b^2 - b^3)*sinh(f*x + e)^5
- (3*a*b^2 - 2*b^3)*cosh(f*x + e)^3 - (3*a*b^2 - 2*b^3 - 10*(2*a*b^2 - b^3)*cosh(f*x + e)^2)*sinh(f*x + e)^3 +
(10*(2*a*b^2 - b^3)*cosh(f*x + e)^3 - 3*(3*a*b^2 - 2*b^3)*cosh(f*x + e))*sinh(f*x + e)^2 + (3*a*b^2 - b^3)*co
sh(f*x + e) + (5*(2*a*b^2 - b^3)*cosh(f*x + e)^4 + 3*a*b^2 - b^3 - 3*(3*a*b^2 - 2*b^3)*cosh(f*x + e)^2)*sinh(f
*x + e))*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x +
e) + sinh(f*x + e)^2)))/(a^2*b^2*f*cosh(f*x + e)^6 + 6*a^2*b^2*f*cosh(f*x + e)*sinh(f*x + e)^5 + a^2*b^2*f*sin
h(f*x + e)^6 - 3*a^2*b^2*f*cosh(f*x + e)^4 + 3*a^2*b^2*f*cosh(f*x + e)^2 - a^2*b^2*f + 3*(5*a^2*b^2*f*cosh(f*x
+ e)^2 - a^2*b^2*f)*sinh(f*x + e)^4 + 4*(5*a^2*b^2*f*cosh(f*x + e)^3 - 3*a^2*b^2*f*cosh(f*x + e))*sinh(f*x +
e)^3 + 3*(5*a^2*b^2*f*cosh(f*x + e)^4 - 6*a^2*b^2*f*cosh(f*x + e)^2 + a^2*b^2*f)*sinh(f*x + e)^2 + 6*(a^2*b^2*
f*cosh(f*x + e)^5 - 2*a^2*b^2*f*cosh(f*x + e)^3 + a^2*b^2*f*cosh(f*x + e))*sinh(f*x + e))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{4}{\left (e + f x \right )}}{\sqrt {a + b \sinh ^{2}{\left (e + f x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)**4/(a+b*sinh(f*x+e)**2)**(1/2),x)

[Out]

Integral(coth(e + f*x)**4/sqrt(a + b*sinh(e + f*x)**2), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(f*x+e)^4/(a+b*sinh(f*x+e)^2)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Unable to divide, perhaps due to rounding error%%%{1024,[8,10,8]%%%}+%%%{%%%{-4096,[1]%%%},[8,10,7]%%%}+%%%
{%%%{6144,[

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\mathrm {coth}\left (e+f\,x\right )}^4}{\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(1/2),x)

[Out]

int(coth(e + f*x)^4/(a + b*sinh(e + f*x)^2)^(1/2), x)

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