Integrand size = 15, antiderivative size = 70 \[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-(a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )+b \sqrt {a+b \text {sech}^2(x)} \]
a^(3/2)*arctanh((a+b*sech(x)^2)^(1/2)/a^(1/2))-(a+b)^(3/2)*arctanh((a+b*se ch(x)^2)^(1/2)/(a+b)^(1/2))+b*(a+b*sech(x)^2)^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(159\) vs. \(2(70)=140\).
Time = 0.52 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.27 \[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=-\frac {2 \left (b+a \cosh ^2(x)\right ) \left (\sqrt {2} (a+b)^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right ) \cosh (x)-\sqrt {a+b} \left (b \sqrt {a+2 b+a \cosh (2 x)}+\sqrt {2} a^{3/2} \cosh (x) \log \left (\sqrt {2} \sqrt {a} \cosh (x)+\sqrt {a+2 b+a \cosh (2 x)}\right )\right )\right ) \sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b} (a+2 b+a \cosh (2 x))^{3/2}} \]
(-2*(b + a*Cosh[x]^2)*(Sqrt[2]*(a + b)^2*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh [x])/Sqrt[a + 2*b + a*Cosh[2*x]]]*Cosh[x] - Sqrt[a + b]*(b*Sqrt[a + 2*b + a*Cosh[2*x]] + Sqrt[2]*a^(3/2)*Cosh[x]*Log[Sqrt[2]*Sqrt[a]*Cosh[x] + Sqrt[ a + 2*b + a*Cosh[2*x]]]))*Sqrt[a + b*Sech[x]^2])/(Sqrt[a + b]*(a + 2*b + a *Cosh[2*x])^(3/2))
Time = 0.32 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 26, 4627, 25, 354, 95, 25, 174, 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.
\(\displaystyle \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \left (a+b \sec (i x)^2\right )^{3/2}}{\tan (i x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\left (b \sec (i x)^2+a\right )^{3/2}}{\tan (i x)}dx\) |
\(\Big \downarrow \) 4627 |
\(\displaystyle \int -\frac {\cosh (x) \left (a+b \text {sech}^2(x)\right )^{3/2}}{1-\text {sech}^2(x)}d\text {sech}(x)\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cosh (x) \left (b \text {sech}^2(x)+a\right )^{3/2}}{1-\text {sech}^2(x)}d\text {sech}(x)\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{2} \int \frac {\cosh (x) \left (b \text {sech}^2(x)+a\right )^{3/2}}{1-\text {sech}^2(x)}d\text {sech}^2(x)\) |
\(\Big \downarrow \) 95 |
\(\displaystyle \frac {1}{2} \left (\int -\frac {\cosh (x) \left (a^2+b (2 a+b) \text {sech}^2(x)\right )}{\left (1-\text {sech}^2(x)\right ) \sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)+2 b \sqrt {a+b \text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (2 b \sqrt {a+b \text {sech}^2(x)}-\int \frac {\cosh (x) \left (a^2+b (2 a+b) \text {sech}^2(x)\right )}{\left (1-\text {sech}^2(x)\right ) \sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)\right )\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{2} \left (a^2 \left (-\int \frac {\cosh (x)}{\sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)\right )-(a+b)^2 \int \frac {1}{\left (1-\text {sech}^2(x)\right ) \sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)+2 b \sqrt {a+b \text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 a^2 \int \frac {1}{\frac {\text {sech}^4(x)}{b}-\frac {a}{b}}d\sqrt {b \text {sech}^2(x)+a}}{b}-\frac {2 (a+b)^2 \int \frac {1}{\frac {a+b}{b}-\frac {\text {sech}^4(x)}{b}}d\sqrt {b \text {sech}^2(x)+a}}{b}+2 b \sqrt {a+b \text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-2 (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )+2 b \sqrt {a+b \text {sech}^2(x)}\right )\) |
(2*a^(3/2)*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - 2*(a + b)^(3/2)*ArcTan h[Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]] + 2*b*Sqrt[a + b*Sech[x]^2])/2
3.2.91.3.1 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] :> 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[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Simp[f*((e + f*x)^(p - 1)/(b*d*(p - 1))), x] + Simp[1/(b*d) Int[(b *d*e^2 - a*c*f^2 + f*(2*b*d*e - b*c*f - a*d*f)*x)*((e + f*x)^(p - 2)/((a + b*x)*(c + d*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[p, 1]
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
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])
\[\int \coth \left (x \right ) \left (a +\operatorname {sech}\left (x \right )^{2} b \right )^{\frac {3}{2}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 626 vs. \(2 (56) = 112\).
Time = 0.38 (sec) , antiderivative size = 4123, normalized size of antiderivative = 58.90 \[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=\text {Too large to display} \]
[1/4*((a*cosh(x)^2 + 2*a*cosh(x)*sinh(x) + a*sinh(x)^2 + a)*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)*cosh(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)*cosh(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*(2*a^3 + 5*a^2*b + 4*a*b^2 + b^3)*cosh(x)^4 + 2*a^3 + 3*a^2*b + 3*(6* a^3 + 14*a^2*b + 9*a*b^2)*cosh(x)^2)*sinh(x)^2 + sqrt(2)*((a^2 + 2*a*b + b ^2)*cosh(x)^6 + 6*(a^2 + 2*a*b + b^2)*cosh(x)*sinh(x)^5 + (a^2 + 2*a*b + b ^2)*sinh(x)^6 + 3*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 3*(5*(a^2 + 2*a*b + b^2) *cosh(x)^2 + a^2 + 2*a*b + b^2)*sinh(x)^4 + 4*(5*(a^2 + 2*a*b + b^2)*cosh( x)^3 + 3*(a^2 + 2*a*b + b^2)*cosh(x))*sinh(x)^3 + (3*a^2 + 4*a*b)*cosh(x)^ 2 + (15*(a^2 + 2*a*b + b^2)*cosh(x)^4 + 18*(a^2 + 2*a*b + b^2)*cosh(x)^2 + 3*a^2 + 4*a*b)*sinh(x)^2 + a^2 + 2*(3*(a^2 + 2*a*b + b^2)*cosh(x)^5 + 6*( a^2 + 2*a*b + b^2)*cosh(x)^3 + (3*a^2 + 4*a*b)*cosh(x))*sinh(x))*sqrt(a...
Timed out. \[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=\text {Timed out} \]
\[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=\int { {\left (b \operatorname {sech}\left (x\right )^{2} + a\right )}^{\frac {3}{2}} \coth \left (x\right ) \,d x } \]
Exception generated. \[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{2048,[4,6]%%%}+%%%{%%%{6144,[1]%%%},[4,5]%%%}+%%%{%%%{6144 ,[2]%%%},
Timed out. \[ \int \coth (x) \left (a+b \text {sech}^2(x)\right )^{3/2} \, dx=\int \mathrm {coth}\left (x\right )\,{\left (a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}\right )}^{3/2} \,d x \]