\(\int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx\) [209]

Optimal result

Integrand size = 17, antiderivative size = 121 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{8 b^{3/2}}+\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)} \]

[Out]

1/8*(a^2-4*a*b-8*b^2)*arctanh(b^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))/b^(3/2)+arctanh((a+b)^(1/2)*tanh(x)/(a+b*
tanh(x)^2)^(1/2))*(a+b)^(1/2)-1/8*(a+4*b)*(a+b*tanh(x)^2)^(1/2)*tanh(x)/b-1/4*(a+b*tanh(x)^2)^(1/2)*tanh(x)^3

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3751, 489, 596, 537, 223, 212, 385} \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{8 b^{3/2}}+\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)} \]

[In]

Int[Tanh[x]^4*Sqrt[a + b*Tanh[x]^2],x]

[Out]

((a^2 - 4*a*b - 8*b^2)*ArcTanh[(Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(8*b^(3/2)) + Sqrt[a + b]*ArcTanh[(Sq
rt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]] - ((a + 4*b)*Tanh[x]*Sqrt[a + b*Tanh[x]^2])/(8*b) - (Tanh[x]^3*Sqrt[
a + b*Tanh[x]^2])/4

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rule 385

Int[((a_) + (b_.)*(x_)^(n_))^(p_)/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Subst[Int[1/(c - (b*c - a*d)*x^n), x]
, x, x/(a + b*x^n)^(1/n)] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[n*p + 1, 0] && IntegerQ[n]

Rule 489

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

Rule 537

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

Rule 596

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

Rule 3751

Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol]
:> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[c*(ff/f), Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2
+ ff^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ
[n, 2] || EqQ[n, 4] || (IntegerQ[p] && RationalQ[n]))

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {x^4 \sqrt {a+b x^2}}{1-x^2} \, dx,x,\tanh (x)\right ) \\ & = -\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}+\frac {1}{4} \text {Subst}\left (\int \frac {x^2 \left (3 a+(a+4 b) x^2\right )}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right ) \\ & = -\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}+\frac {\text {Subst}\left (\int \frac {a (a+4 b)+\left (-a^2+4 a b+8 b^2\right ) x^2}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{8 b} \\ & = -\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}+(a+b) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )+\frac {\left (a^2-4 a b-8 b^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tanh (x)\right )}{8 b} \\ & = -\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}+(a+b) \text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )+\frac {\left (a^2-4 a b-8 b^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{8 b} \\ & = \frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{8 b^{3/2}}+\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 6.21 (sec) , antiderivative size = 580, normalized size of antiderivative = 4.79 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {-\frac {b \left (a^2-4 b^2\right ) \sqrt {\frac {a-b+(a+b) \cosh (2 x)}{1+\cosh (2 x)}} \sqrt {-\frac {a \coth ^2(x)}{b}} \sqrt {-\frac {a (1+\cosh (2 x)) \text {csch}^2(x)}{b}} \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \text {csch}(2 x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right ) \sinh ^4(x)}{a (a-b+(a+b) \cosh (2 x))}-\frac {4 i b \left (4 a b+4 b^2\right ) \sqrt {1+\cosh (2 x)} \sqrt {\frac {a-b+(a+b) \cosh (2 x)}{1+\cosh (2 x)}} \left (-\frac {i \sqrt {-\frac {a \coth ^2(x)}{b}} \sqrt {-\frac {a (1+\cosh (2 x)) \text {csch}^2(x)}{b}} \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \text {csch}(2 x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right ) \sinh ^4(x)}{4 a \sqrt {1+\cosh (2 x)} \sqrt {a-b+(a+b) \cosh (2 x)}}+\frac {i \sqrt {-\frac {a \coth ^2(x)}{b}} \sqrt {-\frac {a (1+\cosh (2 x)) \text {csch}^2(x)}{b}} \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \text {csch}(2 x) \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right ) \sinh ^4(x)}{2 (a+b) \sqrt {1+\cosh (2 x)} \sqrt {a-b+(a+b) \cosh (2 x)}}\right )}{\sqrt {a-b+(a+b) \cosh (2 x)}}}{4 b}+\sqrt {\frac {a-b+a \cosh (2 x)+b \cosh (2 x)}{1+\cosh (2 x)}} \left (\frac {\text {sech}(x) (-a \sinh (x)-6 b \sinh (x))}{8 b}+\frac {1}{4} \text {sech}^2(x) \tanh (x)\right ) \]

[In]

Integrate[Tanh[x]^4*Sqrt[a + b*Tanh[x]^2],x]

[Out]

(-((b*(a^2 - 4*b^2)*Sqrt[(a - b + (a + b)*Cosh[2*x])/(1 + Cosh[2*x])]*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 +
Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticF[ArcSin[Sqrt[((a
- b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/(a*(a - b + (a + b)*Cosh[2*x]))) - ((4*I)*b*(4*
a*b + 4*b^2)*Sqrt[1 + Cosh[2*x]]*Sqrt[(a - b + (a + b)*Cosh[2*x])/(1 + Cosh[2*x])]*(((-1/4*I)*Sqrt[-((a*Coth[x
]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*El
lipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/(a*Sqrt[1 + Cosh[2*x]]*
Sqrt[a - b + (a + b)*Cosh[2*x]]) + ((I/2)*Sqrt[-((a*Coth[x]^2)/b)]*Sqrt[-((a*(1 + Cosh[2*x])*Csch[x]^2)/b)]*Sq
rt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Csch[2*x]*EllipticPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Co
sh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1]*Sinh[x]^4)/((a + b)*Sqrt[1 + Cosh[2*x]]*Sqrt[a - b + (a + b)*Cosh[2*x]])))
/Sqrt[a - b + (a + b)*Cosh[2*x]])/(4*b) + Sqrt[(a - b + a*Cosh[2*x] + b*Cosh[2*x])/(1 + Cosh[2*x])]*((Sech[x]*
(-(a*Sinh[x]) - 6*b*Sinh[x]))/(8*b) + (Sech[x]^2*Tanh[x])/4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(336\) vs. \(2(99)=198\).

Time = 0.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.79

method	result	size
derivativedivides	\(-\frac {\sqrt {a +b \tanh \left (x \right )^{2}}\, \tanh \left (x \right )}{2}-\frac {a \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{2 \sqrt {b}}-\frac {\tanh \left (x \right ) \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{8 b}+\frac {a^{2} \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{8 b^{\frac {3}{2}}}-\frac {\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )}{2}\)	\(337\)
default	\(-\frac {\sqrt {a +b \tanh \left (x \right )^{2}}\, \tanh \left (x \right )}{2}-\frac {a \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{2 \sqrt {b}}-\frac {\tanh \left (x \right ) \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{8 b}+\frac {a^{2} \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{8 b^{\frac {3}{2}}}-\frac {\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )}{2}\)	\(337\)

[In]

int((a+b*tanh(x)^2)^(1/2)*tanh(x)^4,x,method=_RETURNVERBOSE)

[Out]

-1/2*(a+b*tanh(x)^2)^(1/2)*tanh(x)-1/2*a/b^(1/2)*ln(b^(1/2)*tanh(x)+(a+b*tanh(x)^2)^(1/2))-1/4*tanh(x)*(a+b*ta
nh(x)^2)^(3/2)/b+1/8*a/b*tanh(x)*(a+b*tanh(x)^2)^(1/2)+1/8*a^2/b^(3/2)*ln(b^(1/2)*tanh(x)+(a+b*tanh(x)^2)^(1/2
))-1/2*(b*(tanh(x)-1)^2+2*b*(tanh(x)-1)+a+b)^(1/2)-1/2*b^(1/2)*ln((b*(tanh(x)-1)+b)/b^(1/2)+(b*(tanh(x)-1)^2+2
*b*(tanh(x)-1)+a+b)^(1/2))+1/2*(a+b)^(1/2)*ln((2*a+2*b+2*b*(tanh(x)-1)+2*(a+b)^(1/2)*(b*(tanh(x)-1)^2+2*b*(tan
h(x)-1)+a+b)^(1/2))/(tanh(x)-1))+1/2*(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2)-1/2*b^(1/2)*ln((b*(1+tanh(x))
-b)/b^(1/2)+(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))-1/2*(a+b)^(1/2)*ln((2*a+2*b-2*b*(1+tanh(x))+2*(a+b)^(
1/2)*(b*(1+tanh(x))^2-2*b*(1+tanh(x))+a+b)^(1/2))/(1+tanh(x)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2030 vs. \(2 (99) = 198\).

Time = 0.58 (sec) , antiderivative size = 9360, normalized size of antiderivative = 77.36 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tanh(x)^2)^(1/2)*tanh(x)^4,x, algorithm="fricas")

[Out]

Too large to include

Sympy [F]

\[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int \sqrt {a + b \tanh ^{2}{\left (x \right )}} \tanh ^{4}{\left (x \right )}\, dx \]

[In]

integrate((a+b*tanh(x)**2)**(1/2)*tanh(x)**4,x)

[Out]

Integral(sqrt(a + b*tanh(x)**2)*tanh(x)**4, x)

Maxima [F]

\[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int { \sqrt {b \tanh \left (x\right )^{2} + a} \tanh \left (x\right )^{4} \,d x } \]

[In]

integrate((a+b*tanh(x)^2)^(1/2)*tanh(x)^4,x, algorithm="maxima")

[Out]

integrate(sqrt(b*tanh(x)^2 + a)*tanh(x)^4, x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (99) = 198\).

Time = 0.99 (sec) , antiderivative size = 938, normalized size of antiderivative = 7.75 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*tanh(x)^2)^(1/2)*tanh(x)^4,x, algorithm="giac")

[Out]

-1/2*sqrt(a + b)*log(abs(-(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a +
b))*(a + b) - sqrt(a + b)*(a - b))) - 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a*e^(4*x) + b*e^(4*x
) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b))) + 1/2*sqrt(a + b)*log(abs(-sqrt(a + b)*e^(2*x) + sqrt(a
*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) - sqrt(a + b))) + 1/4*(a^2 - 4*a*b - 8*b^2)*arctan(-
1/2*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b) + sqrt(a + b))/sqrt
(-b))/(sqrt(-b)*b) - 1/2*((a^2 + 12*a*b + 16*b^2)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2
*x) - 2*b*e^(2*x) + a + b))^7 + (7*a^2 + 52*a*b + 16*b^2)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) +
2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^6*sqrt(a + b) + (21*a^3 + 109*a^2*b + 28*a*b^2 - 48*b^3)*(sqrt(a + b)*e^(2
*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^5 + (35*a^3 + 115*a^2*b - 156*a*b^2 - 1
76*b^3)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^4*sqrt(a + b)
+ (35*a^4 + 130*a^3*b - 317*a^2*b^2 - 156*a*b^3 + 304*b^4)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) +
2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^3 + (21*a^4 + 94*a^3*b - 379*a^2*b^2 + 476*a*b^3 + 48*b^4)*(sqrt(a + b)*e
^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^2*sqrt(a + b) + (7*a^5 + 53*a^4*b -
135*a^3*b^2 + 271*a^2*b^3 - 140*a*b^4 - 272*b^5)*(sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*
x) - 2*b*e^(2*x) + a + b)) + (a^5 + 11*a^4*b - 17*a^3*b^2 + 65*a^2*b^3 - 116*a*b^4 + 112*b^5)*sqrt(a + b))/(((
sqrt(a + b)*e^(2*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))^2 + 2*(sqrt(a + b)*e^(2
*x) - sqrt(a*e^(4*x) + b*e^(4*x) + 2*a*e^(2*x) - 2*b*e^(2*x) + a + b))*sqrt(a + b) + a - 3*b)^4*b)

Mupad [F(-1)]

Timed out. \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int {\mathrm {tanh}\left (x\right )}^4\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a} \,d x \]

[In]

int(tanh(x)^4*(a + b*tanh(x)^2)^(1/2),x)

[Out]

int(tanh(x)^4*(a + b*tanh(x)^2)^(1/2), x)