Integrand size = 17, antiderivative size = 121 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{8 b^{3/2}}+\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{8 b}-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)} \] Output:
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)+(a+b)^(1/2)*arctanh((a+b)^(1/2)*tanh(x)/(a+b*tanh(x)^2)^(1/2))-1/8*(a+4 *b)*tanh(x)*(a+b*tanh(x)^2)^(1/2)/b-1/4*tanh(x)^3*(a+b*tanh(x)^2)^(1/2)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.56 (sec) , antiderivative size = 242, normalized size of antiderivative = 2.00 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\frac {\left (4 \sqrt {2} a (a+4 b) \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )-32 \sqrt {2} a b \sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}} \operatorname {EllipticPi}\left (\frac {b}{a+b},\arcsin \left (\frac {\sqrt {\frac {(a-b+(a+b) \cosh (2 x)) \text {csch}^2(x)}{b}}}{\sqrt {2}}\right ),1\right )-\left (3 a^2+9 a b+2 b^2+4 \left (a^2+4 a b-2 b^2\right ) \cosh (2 x)+\left (a^2+7 a b+6 b^2\right ) \cosh (4 x)\right ) \text {sech}^4(x)\right ) \tanh (x)}{32 \sqrt {2} b \sqrt {(a-b+(a+b) \cosh (2 x)) \text {sech}^2(x)}} \] Input:
Integrate[Tanh[x]^4*Sqrt[a + b*Tanh[x]^2],x]
Output:
((4*Sqrt[2]*a*(a + 4*b)*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*El lipticF[ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt[2]], 1 ] - 32*Sqrt[2]*a*b*Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]*Ellipti cPi[b/(a + b), ArcSin[Sqrt[((a - b + (a + b)*Cosh[2*x])*Csch[x]^2)/b]/Sqrt [2]], 1] - (3*a^2 + 9*a*b + 2*b^2 + 4*(a^2 + 4*a*b - 2*b^2)*Cosh[2*x] + (a ^2 + 7*a*b + 6*b^2)*Cosh[4*x])*Sech[x]^4)*Tanh[x])/(32*Sqrt[2]*b*Sqrt[(a - b + (a + b)*Cosh[2*x])*Sech[x]^2])
Time = 0.40 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.10, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 4153, 380, 444, 398, 224, 219, 291, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan (i x)^4 \sqrt {a-b \tan (i x)^2}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle \int \frac {\tanh ^4(x) \sqrt {a+b \tanh ^2(x)}}{1-\tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 380 |
\(\displaystyle \frac {1}{4} \int \frac {\tanh ^2(x) \left ((a+4 b) \tanh ^2(x)+3 a\right )}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
\(\Big \downarrow \) 444 |
\(\displaystyle \frac {1}{4} \left (\frac {\int \frac {a (a+4 b)-\left (a^2-4 b a-8 b^2\right ) \tanh ^2(x)}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{2 b}-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\right )-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
\(\Big \downarrow \) 398 |
\(\displaystyle \frac {1}{4} \left (\frac {\left (a^2-4 a b-8 b^2\right ) \int \frac {1}{\sqrt {b \tanh ^2(x)+a}}d\tanh (x)+8 b (a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{2 b}-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\right )-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
\(\Big \downarrow \) 224 |
\(\displaystyle \frac {1}{4} \left (\frac {\left (a^2-4 a b-8 b^2\right ) \int \frac {1}{1-\frac {b \tanh ^2(x)}{b \tanh ^2(x)+a}}d\frac {\tanh (x)}{\sqrt {b \tanh ^2(x)+a}}+8 b (a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)}{2 b}-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\right )-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {8 b (a+b) \int \frac {1}{\left (1-\tanh ^2(x)\right ) \sqrt {b \tanh ^2(x)+a}}d\tanh (x)+\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {b}}}{2 b}-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\right )-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
\(\Big \downarrow \) 291 |
\(\displaystyle \frac {1}{4} \left (\frac {8 b (a+b) \int \frac {1}{1-\frac {(a+b) \tanh ^2(x)}{b \tanh ^2(x)+a}}d\frac {\tanh (x)}{\sqrt {b \tanh ^2(x)+a}}+\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {b}}}{2 b}-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\right )-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{4} \left (\frac {\frac {\left (a^2-4 a b-8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{\sqrt {b}}+8 b \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b} \tanh (x)}{\sqrt {a+b \tanh ^2(x)}}\right )}{2 b}-\frac {(a+4 b) \tanh (x) \sqrt {a+b \tanh ^2(x)}}{2 b}\right )-\frac {1}{4} \tanh ^3(x) \sqrt {a+b \tanh ^2(x)}\) |
Input:
Int[Tanh[x]^4*Sqrt[a + b*Tanh[x]^2],x]
Output:
-1/4*(Tanh[x]^3*Sqrt[a + b*Tanh[x]^2]) + ((((a^2 - 4*a*b - 8*b^2)*ArcTanh[ (Sqrt[b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/Sqrt[b] + 8*b*Sqrt[a + b]*ArcTan h[(Sqrt[a + b]*Tanh[x])/Sqrt[a + b*Tanh[x]^2]])/(2*b) - ((a + 4*b)*Tanh[x] *Sqrt[a + b*Tanh[x]^2])/(2*b))/4
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] && (Gt Q[a, 0] || LtQ[b, 0])
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]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b* (m + 2*(p + q) + 1))), x] - Simp[e^2/(b*(m + 2*(p + q) + 1)) Int[(e*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[a*c*(m - 1) + (a*d*(m - 1) - 2 *q*(b*c - a*d))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && GtQ[q, 0] && GtQ[m, 1] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*g*(g*x)^(m - 1)*(a + b*x^2)^ (p + 1)*((c + d*x^2)^(q + 1)/(b*d*(m + 2*(p + q + 1) + 1))), x] - Simp[g^2/ (b*d*(m + 2*(p + q + 1) + 1)) Int[(g*x)^(m - 2)*(a + b*x^2)^p*(c + d*x^2) ^q*Simp[a*f*c*(m - 1) + (a*f*d*(m + 2*q + 1) + b*(f*c*(m + 2*p + 1) - e*d*( m + 2*(p + q + 1) + 1)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && GtQ[m, 1]
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]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^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] && Ratio nalQ[n]))
Leaf count of result is larger than twice the leaf count of optimal. \(336\) vs. \(2(99)=198\).
Time = 0.07 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.79
method | result | size |
derivativedivides | \(-\frac {\tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{2}-\frac {a \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{2 \sqrt {b}}-\frac {\tanh \left (x \right ) \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{8 b}+\frac {a^{2} \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{8 b^{\frac {3}{2}}}-\frac {\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (\tanh \left (x \right )+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{\tanh \left (x \right )+1}\right )}{2}\) | \(337\) |
default | \(-\frac {\tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{2}-\frac {a \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{2 \sqrt {b}}-\frac {\tanh \left (x \right ) \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{4 b}+\frac {a \tanh \left (x \right ) \sqrt {a +b \tanh \left (x \right )^{2}}}{8 b}+\frac {a^{2} \ln \left (\sqrt {b}\, \tanh \left (x \right )+\sqrt {a +b \tanh \left (x \right )^{2}}\right )}{8 b^{\frac {3}{2}}}-\frac {\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2}+\frac {\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )+1\right )-b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}\right )}{2}-\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (\tanh \left (x \right )+1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )+1\right )^{2}-2 b \left (\tanh \left (x \right )+1\right )+a +b}}{\tanh \left (x \right )+1}\right )}{2}\) | \(337\) |
Input:
int(tanh(x)^4*(a+b*tanh(x)^2)^(1/2),x,method=_RETURNVERBOSE)
Output:
-1/2*tanh(x)*(a+b*tanh(x)^2)^(1/2)-1/2*a/b^(1/2)*ln(b^(1/2)*tanh(x)+(a+b*t anh(x)^2)^(1/2))-1/4*tanh(x)*(a+b*tanh(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*(tanh(x) -1)+a+b)^(1/2))/(tanh(x)-1))+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*(tanh(x)+1)+a+b)^(1/2))/(tanh(x)+1))
Leaf count of result is larger than twice the leaf count of optimal. 1960 vs. \(2 (99) = 198\).
Time = 0.42 (sec) , antiderivative size = 9220, normalized size of antiderivative = 76.20 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^4*(a+b*tanh(x)^2)^(1/2),x, algorithm="fricas")
Output:
Too large to include
\[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int \sqrt {a + b \tanh ^{2}{\left (x \right )}} \tanh ^{4}{\left (x \right )}\, dx \] Input:
integrate(tanh(x)**4*(a+b*tanh(x)**2)**(1/2),x)
Output:
Integral(sqrt(a + b*tanh(x)**2)*tanh(x)**4, x)
\[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int { \sqrt {b \tanh \left (x\right )^{2} + a} \tanh \left (x\right )^{4} \,d x } \] Input:
integrate(tanh(x)^4*(a+b*tanh(x)^2)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*tanh(x)^2 + a)*tanh(x)^4, x)
Leaf count of result is larger than twice the leaf count of optimal. 938 vs. \(2 (99) = 198\).
Time = 0.81 (sec) , antiderivative size = 938, normalized size of antiderivative = 7.75 \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \] Input:
integrate(tanh(x)^4*(a+b*tanh(x)^2)^(1/2),x, algorithm="giac")
Output:
-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) - sq rt(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 - 176*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) - sq rt(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) - sqr t(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...
Timed out. \[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int {\mathrm {tanh}\left (x\right )}^4\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a} \,d x \] Input:
int(tanh(x)^4*(a + b*tanh(x)^2)^(1/2),x)
Output:
int(tanh(x)^4*(a + b*tanh(x)^2)^(1/2), x)
\[ \int \tanh ^4(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int \sqrt {\tanh \left (x \right )^{2} b +a}\, \tanh \left (x \right )^{4}d x \] Input:
int(tanh(x)^4*(a+b*tanh(x)^2)^(1/2),x)
Output:
int(sqrt(tanh(x)**2*b + a)*tanh(x)**4,x)