Integrand size = 17, antiderivative size = 76 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-a \sqrt {a+b \text {sech}^2(x)}-\frac {1}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}+\frac {\left (a+b \text {sech}^2(x)\right )^{5/2}}{5 b} \]
a^(3/2)*arctanh((a+b*sech(x)^2)^(1/2)/a^(1/2))-1/3*(a+b*sech(x)^2)^(3/2)+1 /5*(a+b*sech(x)^2)^(5/2)/b-a*(a+b*sech(x)^2)^(1/2)
Time = 0.52 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.05 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=\frac {a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-a b \sqrt {a+b \text {sech}^2(x)}-\frac {1}{3} b \left (a+b \text {sech}^2(x)\right )^{3/2}+\frac {1}{5} \left (a+b \text {sech}^2(x)\right )^{5/2}}{b} \]
(a^(3/2)*b*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - a*b*Sqrt[a + b*Sech[x] ^2] - (b*(a + b*Sech[x]^2)^(3/2))/3 + (a + b*Sech[x]^2)^(5/2)/5)/b
Time = 0.30 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {3042, 26, 4627, 25, 354, 90, 60, 60, 73, 221}
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.
\(\displaystyle \int \tanh ^3(x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \tan (i x)^3 \left (a+b \sec (i x)^2\right )^{3/2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \left (b \sec (i x)^2+a\right )^{3/2} \tan (i x)^3dx\) |
\(\Big \downarrow \) 4627 |
\(\displaystyle \int -\cosh (x) \left (1-\text {sech}^2(x)\right ) \left (a+b \text {sech}^2(x)\right )^{3/2}d\text {sech}(x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \cosh (x) \left (1-\text {sech}^2(x)\right ) \left (b \text {sech}^2(x)+a\right )^{3/2}d\text {sech}(x)\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{2} \int \cosh (x) \left (1-\text {sech}^2(x)\right ) \left (b \text {sech}^2(x)+a\right )^{3/2}d\text {sech}^2(x)\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (a+b \text {sech}^2(x)\right )^{5/2}}{5 b}-\int \cosh (x) \left (b \text {sech}^2(x)+a\right )^{3/2}d\text {sech}^2(x)\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \int \cosh (x) \sqrt {b \text {sech}^2(x)+a}d\text {sech}^2(x)+\frac {2 \left (a+b \text {sech}^2(x)\right )^{5/2}}{5 b}-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{2} \left (-a \left (a \int \frac {\cosh (x)}{\sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)+2 \sqrt {a+b \text {sech}^2(x)}\right )+\frac {2 \left (a+b \text {sech}^2(x)\right )^{5/2}}{5 b}-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-a \left (\frac {2 a \int \frac {1}{\frac {\text {sech}^4(x)}{b}-\frac {a}{b}}d\sqrt {b \text {sech}^2(x)+a}}{b}+2 \sqrt {a+b \text {sech}^2(x)}\right )+\frac {2 \left (a+b \text {sech}^2(x)\right )^{5/2}}{5 b}-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (-a \left (2 \sqrt {a+b \text {sech}^2(x)}-2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )\right )+\frac {2 \left (a+b \text {sech}^2(x)\right )^{5/2}}{5 b}-\frac {2}{3} \left (a+b \text {sech}^2(x)\right )^{3/2}\right )\) |
((-2*(a + b*Sech[x]^2)^(3/2))/3 + (2*(a + b*Sech[x]^2)^(5/2))/(5*b) - a*(- 2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] + 2*Sqrt[a + b*Sech[x]^2] ))/2
3.2.87.3.1 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_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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[((a_) + (b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_.)*tan[(e_.) + ( f_.)*(x_)]^(m_.), x_Symbol] :> With[{ff = FreeFactors[Sec[e + f*x], x]}, Si mp[1/f Subst[Int[(-1 + ff^2*x^2)^((m - 1)/2)*((a + b*(c*ff*x)^n)^p/x), x] , x, Sec[e + f*x]/ff], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && IntegerQ[( m - 1)/2] && (GtQ[m, 0] || EqQ[n, 2] || EqQ[n, 4] || IGtQ[p, 0] || Integers Q[2*n, p])
\[\int \left (a +\operatorname {sech}\left (x \right )^{2} b \right )^{\frac {3}{2}} \tanh \left (x \right )^{3}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 1745 vs. \(2 (60) = 120\).
Time = 0.51 (sec) , antiderivative size = 4226, normalized size of antiderivative = 55.61 \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=\text {Too large to display} \]
[1/60*(15*(a*b*cosh(x)^10 + 10*a*b*cosh(x)*sinh(x)^9 + a*b*sinh(x)^10 + 5* a*b*cosh(x)^8 + 5*(9*a*b*cosh(x)^2 + a*b)*sinh(x)^8 + 10*a*b*cosh(x)^6 + 4 0*(3*a*b*cosh(x)^3 + a*b*cosh(x))*sinh(x)^7 + 10*(21*a*b*cosh(x)^4 + 14*a* b*cosh(x)^2 + a*b)*sinh(x)^6 + 10*a*b*cosh(x)^4 + 4*(63*a*b*cosh(x)^5 + 70 *a*b*cosh(x)^3 + 15*a*b*cosh(x))*sinh(x)^5 + 10*(21*a*b*cosh(x)^6 + 35*a*b *cosh(x)^4 + 15*a*b*cosh(x)^2 + a*b)*sinh(x)^4 + 5*a*b*cosh(x)^2 + 40*(3*a *b*cosh(x)^7 + 7*a*b*cosh(x)^5 + 5*a*b*cosh(x)^3 + a*b*cosh(x))*sinh(x)^3 + 5*(9*a*b*cosh(x)^8 + 28*a*b*cosh(x)^6 + 30*a*b*cosh(x)^4 + 12*a*b*cosh(x )^2 + a*b)*sinh(x)^2 + a*b + 10*(a*b*cosh(x)^9 + 4*a*b*cosh(x)^7 + 6*a*b*c osh(x)^5 + 4*a*b*cosh(x)^3 + a*b*cosh(x))*sinh(x))*sqrt(a)*log(((a^3 + 2*a ^2*b + a*b^2)*cosh(x)^8 + 8*(a^3 + 2*a^2*b + a*b^2)*cosh(x)*sinh(x)^7 + (a ^3 + 2*a^2*b + a*b^2)*sinh(x)^8 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh (x)^6 + 2*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3 + 14*(a^3 + 2*a^2*b + a*b^2)*co sh(x)^2)*sinh(x)^6 + 4*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^3 + 3*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x))*sinh(x)^5 + (6*a^3 + 14*a^2*b + 9*a*b^2) *cosh(x)^4 + (70*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^4 + 6*a^3 + 14*a^2*b + 9* a*b^2 + 30*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^2)*sinh(x)^4 + 4*(14* (a^3 + 2*a^2*b + a*b^2)*cosh(x)^5 + 10*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*c osh(x)^3 + (6*a^3 + 14*a^2*b + 9*a*b^2)*cosh(x))*sinh(x)^3 + a^3 + 2*(2*a^ 3 + 3*a^2*b)*cosh(x)^2 + 2*(14*(a^3 + 2*a^2*b + a*b^2)*cosh(x)^6 + 15*(...
\[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (x \right )}\right )^{\frac {3}{2}} \tanh ^{3}{\left (x \right )}\, dx \]
\[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=\int { {\left (b \operatorname {sech}\left (x\right )^{2} + a\right )}^{\frac {3}{2}} \tanh \left (x\right )^{3} \,d x } \]
Exception generated. \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \left (a+b \text {sech}^2(x)\right )^{3/2} \tanh ^3(x) \, dx=\int {\mathrm {tanh}\left (x\right )}^3\,{\left (a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2} \,d x \]