Integrand size = 17, antiderivative size = 87 \[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\sqrt {a+b \tanh ^2(x)}+\frac {(a-b) \left (a+b \tanh ^2(x)\right )^{3/2}}{3 b^2}-\frac {\left (a+b \tanh ^2(x)\right )^{5/2}}{5 b^2} \]
arctanh((a+b*tanh(x)^2)^(1/2)/(a+b)^(1/2))*(a+b)^(1/2)-(a+b*tanh(x)^2)^(1/ 2)+1/3*(a-b)*(a+b*tanh(x)^2)^(3/2)/b^2-1/5*(a+b*tanh(x)^2)^(5/2)/b^2
Time = 0.58 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=\sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )+\frac {\sqrt {a+b \tanh ^2(x)} \left (2 a^2-5 a b-15 b^2-b (a+5 b) \tanh ^2(x)-3 b^2 \tanh ^4(x)\right )}{15 b^2} \]
Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] + (Sqrt[a + b*Tanh[ x]^2]*(2*a^2 - 5*a*b - 15*b^2 - b*(a + 5*b)*Tanh[x]^2 - 3*b^2*Tanh[x]^4))/ (15*b^2)
Time = 0.33 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.412, Rules used = {3042, 26, 4153, 26, 354, 99, 2009}
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.
\(\displaystyle \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan (i x)^5 \sqrt {a-b \tan (i x)^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \tan (i x)^5 \sqrt {a-b \tan (i x)^2}dx\) |
\(\Big \downarrow \) 4153 |
\(\displaystyle -i \int \frac {i \tanh ^5(x) \sqrt {b \tanh ^2(x)+a}}{1-\tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\tanh ^5(x) \sqrt {a+b \tanh ^2(x)}}{1-\tanh ^2(x)}d\tanh (x)\) |
\(\Big \downarrow \) 354 |
\(\displaystyle \frac {1}{2} \int \frac {\tanh ^4(x) \sqrt {b \tanh ^2(x)+a}}{1-\tanh ^2(x)}d\tanh ^2(x)\) |
\(\Big \downarrow \) 99 |
\(\displaystyle \frac {1}{2} \int \left (-\frac {\left (b \tanh ^2(x)+a\right )^{3/2}}{b}+\frac {(a-b) \sqrt {b \tanh ^2(x)+a}}{b}+\frac {\sqrt {b \tanh ^2(x)+a}}{1-\tanh ^2(x)}\right )d\tanh ^2(x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (2 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \tanh ^2(x)}}{\sqrt {a+b}}\right )-\frac {2 \left (a+b \tanh ^2(x)\right )^{5/2}}{5 b^2}+\frac {2 (a-b) \left (a+b \tanh ^2(x)\right )^{3/2}}{3 b^2}-2 \sqrt {a+b \tanh ^2(x)}\right )\) |
(2*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Tanh[x]^2]/Sqrt[a + b]] - 2*Sqrt[a + b*T anh[x]^2] + (2*(a - b)*(a + b*Tanh[x]^2)^(3/2))/(3*b^2) - (2*(a + b*Tanh[x ]^2)^(5/2))/(5*b^2))/2
3.3.8.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
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. \(287\) vs. \(2(71)=142\).
Time = 0.18 (sec) , antiderivative size = 288, normalized size of antiderivative = 3.31
method | result | size |
derivativedivides | \(-\frac {\left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}-\frac {\tanh \left (x \right )^{2} \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{5 b}+\frac {2 a \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{15 b^{2}}-\frac {\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )}{2}-\frac {\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2}\) | \(288\) |
default | \(-\frac {\left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{3 b}-\frac {\tanh \left (x \right )^{2} \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{5 b}+\frac {2 a \left (a +b \tanh \left (x \right )^{2}\right )^{\frac {3}{2}}}{15 b^{2}}-\frac {\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{2}+\frac {\sqrt {b}\, \ln \left (\frac {b \left (1+\tanh \left (x \right )\right )-b}{\sqrt {b}}+\sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b -2 b \left (1+\tanh \left (x \right )\right )+2 \sqrt {a +b}\, \sqrt {b \left (1+\tanh \left (x \right )\right )^{2}-2 b \left (1+\tanh \left (x \right )\right )+a +b}}{1+\tanh \left (x \right )}\right )}{2}-\frac {\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{2}-\frac {\sqrt {b}\, \ln \left (\frac {b \left (\tanh \left (x \right )-1\right )+b}{\sqrt {b}}+\sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}\right )}{2}+\frac {\sqrt {a +b}\, \ln \left (\frac {2 a +2 b +2 b \left (\tanh \left (x \right )-1\right )+2 \sqrt {a +b}\, \sqrt {b \left (\tanh \left (x \right )-1\right )^{2}+2 b \left (\tanh \left (x \right )-1\right )+a +b}}{\tanh \left (x \right )-1}\right )}{2}\) | \(288\) |
-1/3*(a+b*tanh(x)^2)^(3/2)/b-1/5*tanh(x)^2*(a+b*tanh(x)^2)^(3/2)/b+2/15*a/ b^2*(a+b*tanh(x)^2)^(3/2)-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)))-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*(t anh(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*(t anh(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. 1982 vs. \(2 (71) = 142\).
Time = 0.49 (sec) , antiderivative size = 4529, normalized size of antiderivative = 52.06 \[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \]
[1/60*(15*(b^2*cosh(x)^10 + 10*b^2*cosh(x)*sinh(x)^9 + b^2*sinh(x)^10 + 5* b^2*cosh(x)^8 + 5*(9*b^2*cosh(x)^2 + b^2)*sinh(x)^8 + 10*b^2*cosh(x)^6 + 4 0*(3*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x)^7 + 10*(21*b^2*cosh(x)^4 + 14*b^ 2*cosh(x)^2 + b^2)*sinh(x)^6 + 10*b^2*cosh(x)^4 + 4*(63*b^2*cosh(x)^5 + 70 *b^2*cosh(x)^3 + 15*b^2*cosh(x))*sinh(x)^5 + 10*(21*b^2*cosh(x)^6 + 35*b^2 *cosh(x)^4 + 15*b^2*cosh(x)^2 + b^2)*sinh(x)^4 + 5*b^2*cosh(x)^2 + 40*(3*b ^2*cosh(x)^7 + 7*b^2*cosh(x)^5 + 5*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x)^3 + 5*(9*b^2*cosh(x)^8 + 28*b^2*cosh(x)^6 + 30*b^2*cosh(x)^4 + 12*b^2*cosh(x )^2 + b^2)*sinh(x)^2 + b^2 + 10*(b^2*cosh(x)^9 + 4*b^2*cosh(x)^7 + 6*b^2*c osh(x)^5 + 4*b^2*cosh(x)^3 + b^2*cosh(x))*sinh(x))*sqrt(a + b)*log(((a^3 + a^2*b)*cosh(x)^8 + 8*(a^3 + a^2*b)*cosh(x)*sinh(x)^7 + (a^3 + a^2*b)*sinh (x)^8 + 2*(2*a^3 + a^2*b)*cosh(x)^6 + 2*(2*a^3 + a^2*b + 14*(a^3 + a^2*b)* cosh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + a^2*b)*cosh(x)^3 + 3*(2*a^3 + a^2*b)*c osh(x))*sinh(x)^5 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^4 + (70*(a^3 + a^2*b)*cosh(x)^4 + 6*a^3 + 4*a^2*b - a*b^2 + b^3 + 30*(2*a^3 + a^2*b)*cos h(x)^2)*sinh(x)^4 + 4*(14*(a^3 + a^2*b)*cosh(x)^5 + 10*(2*a^3 + a^2*b)*cos h(x)^3 + (6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x))*sinh(x)^3 + a^3 + 3*a^2* b + 3*a*b^2 + b^3 + 2*(2*a^3 + 3*a^2*b - b^3)*cosh(x)^2 + 2*(14*(a^3 + a^2 *b)*cosh(x)^6 + 15*(2*a^3 + a^2*b)*cosh(x)^4 + 2*a^3 + 3*a^2*b - b^3 + 3*( 6*a^3 + 4*a^2*b - a*b^2 + b^3)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*(a^2*cosh...
Time = 2.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.45 \[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=- \begin {cases} \frac {2 \left (\frac {b^{3} \sqrt {a + b \tanh ^{2}{\left (x \right )}}}{2} + \frac {b^{3} \left (a + b\right ) \operatorname {atan}{\left (\frac {\sqrt {a + b \tanh ^{2}{\left (x \right )}}}{\sqrt {- a - b}} \right )}}{2 \sqrt {- a - b}} + \frac {b \left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {5}{2}}}{10} + \frac {\left (a + b \tanh ^{2}{\left (x \right )}\right )^{\frac {3}{2}} \left (- \frac {a b}{2} + \frac {b^{2}}{2}\right )}{3}\right )}{b^{3}} & \text {for}\: b \neq 0 \\\sqrt {a} \left (\frac {\log {\left (\tanh ^{2}{\left (x \right )} - 1 \right )}}{2} + \frac {\tanh ^{4}{\left (x \right )}}{4} + \frac {\tanh ^{2}{\left (x \right )}}{2}\right ) & \text {otherwise} \end {cases} \]
-Piecewise((2*(b**3*sqrt(a + b*tanh(x)**2)/2 + b**3*(a + b)*atan(sqrt(a + b*tanh(x)**2)/sqrt(-a - b))/(2*sqrt(-a - b)) + b*(a + b*tanh(x)**2)**(5/2) /10 + (a + b*tanh(x)**2)**(3/2)*(-a*b/2 + b**2/2)/3)/b**3, Ne(b, 0)), (sqr t(a)*(log(tanh(x)**2 - 1)/2 + tanh(x)**4/4 + tanh(x)**2/2), True))
\[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=\int { \sqrt {b \tanh \left (x\right )^{2} + a} \tanh \left (x\right )^{5} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 980 vs. \(2 (71) = 142\).
Time = 1.13 (sec) , antiderivative size = 980, normalized size of antiderivative = 11.26 \[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=\text {Too large to display} \]
-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))) - 4/15*(15*(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))^9*(2*a + 3*b) + 15*(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))^8*(10*a + 9*b)*sqrt(a + b) + 20*(18*a^2 + 23*a*b + 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 + 20*(30*a^2 - 7*a*b - 65*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) + 2*(330*a^3 - 705*a^2*b - 1480*a*b^2 + 19*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 + 10*(18* a^3 - 279*a^2*b + 68*a*b^2 + 349*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) - 20*(30 *a^4 + 81*a^3*b - 149*a^2*b^2 - 245*a*b^3 + 19*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 - 20*( 42*a^4 - 33*a^3*b - 139*a^2*b^2 + 69*a*b^3 + 325*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*s...
Time = 9.06 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.37 \[ \int \tanh ^5(x) \sqrt {a+b \tanh ^2(x)} \, dx=-\frac {{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{5/2}}{5\,b^2}-2\,\mathrm {atan}\left (\frac {2\,\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a}\,\sqrt {-\frac {a}{4}-\frac {b}{4}}}{a+b}\right )\,\sqrt {-\frac {a}{4}-\frac {b}{4}}-\sqrt {b\,{\mathrm {tanh}\left (x\right )}^2+a}\,\left (\left (a+b\right )\,\left (\frac {a+b}{b^2}-\frac {2\,a}{b^2}\right )+\frac {a^2}{b^2}\right )-\left (\frac {a+b}{3\,b^2}-\frac {2\,a}{3\,b^2}\right )\,{\left (b\,{\mathrm {tanh}\left (x\right )}^2+a\right )}^{3/2} \]