Integrand size = 17, antiderivative size = 83 \[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-\frac {(2 a+b) \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{2 \sqrt {a+b}}-\frac {1}{2} \coth ^2(x) \sqrt {a+b \text {sech}^2(x)} \]
arctanh((a+b*sech(x)^2)^(1/2)/a^(1/2))*a^(1/2)-1/2*(2*a+b)*arctanh((a+b*se ch(x)^2)^(1/2)/(a+b)^(1/2))/(a+b)^(1/2)-1/2*coth(x)^2*(a+b*sech(x)^2)^(1/2 )
Time = 0.58 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.88 \[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=-\frac {\left (\sqrt {2} (2 a+b) \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a+b} \cosh (x)}{\sqrt {a+2 b+a \cosh (2 x)}}\right ) \cosh (x)+\sqrt {a+b} \left (\sqrt {a+2 b+a \cosh (2 x)} \coth ^2(x)-2 \sqrt {2} \sqrt {a} \cosh (x) \log \left (\sqrt {2} \sqrt {a} \cosh (x)+\sqrt {a+2 b+a \cosh (2 x)}\right )\right )\right ) \sqrt {a+b \text {sech}^2(x)}}{2 \sqrt {a+b} \sqrt {a+2 b+a \cosh (2 x)}} \]
-1/2*((Sqrt[2]*(2*a + b)*ArcTanh[(Sqrt[2]*Sqrt[a + b]*Cosh[x])/Sqrt[a + 2* b + a*Cosh[2*x]]]*Cosh[x] + Sqrt[a + b]*(Sqrt[a + 2*b + a*Cosh[2*x]]*Coth[ x]^2 - 2*Sqrt[2]*Sqrt[a]*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]*Sqrt[a + 2*b + a*C osh[2*x]])
Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {3042, 26, 4627, 354, 110, 27, 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.
\(\displaystyle \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \sqrt {a+b \sec (i x)^2}}{\tan (i x)^3}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\sqrt {b \sec (i x)^2+a}}{\tan (i x)^3}dx\) |
\(\Big \downarrow \) 4627 |
\(\displaystyle -\int \frac {\cosh (x) \sqrt {b \text {sech}^2(x)+a}}{\left (1-\text {sech}^2(x)\right )^2}d\text {sech}(x)\) |
\(\Big \downarrow \) 354 |
\(\displaystyle -\frac {1}{2} \int \frac {\cosh (x) \sqrt {b \text {sech}^2(x)+a}}{\left (1-\text {sech}^2(x)\right )^2}d\text {sech}^2(x)\) |
\(\Big \downarrow \) 110 |
\(\displaystyle \frac {1}{2} \left (\int -\frac {\cosh (x) \left (b \text {sech}^2(x)+2 a\right )}{2 \left (1-\text {sech}^2(x)\right ) \sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)-\frac {\sqrt {a+b \text {sech}^2(x)}}{1-\text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{2} \left (-\frac {1}{2} \int \frac {\cosh (x) \left (b \text {sech}^2(x)+2 a\right )}{\left (1-\text {sech}^2(x)\right ) \sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)-\frac {\sqrt {a+b \text {sech}^2(x)}}{1-\text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-(2 a+b) \int \frac {1}{\left (1-\text {sech}^2(x)\right ) \sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)-2 a \int \frac {\cosh (x)}{\sqrt {b \text {sech}^2(x)+a}}d\text {sech}^2(x)\right )-\frac {\sqrt {a+b \text {sech}^2(x)}}{1-\text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (-\frac {2 (2 a+b) \int \frac {1}{\frac {a+b}{b}-\frac {\text {sech}^4(x)}{b}}d\sqrt {b \text {sech}^2(x)+a}}{b}-\frac {4 a \int \frac {1}{\frac {\text {sech}^4(x)}{b}-\frac {a}{b}}d\sqrt {b \text {sech}^2(x)+a}}{b}\right )-\frac {\sqrt {a+b \text {sech}^2(x)}}{1-\text {sech}^2(x)}\right )\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {1}{2} \left (\frac {1}{2} \left (4 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a}}\right )-\frac {2 (2 a+b) \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}^2(x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}\right )-\frac {\sqrt {a+b \text {sech}^2(x)}}{1-\text {sech}^2(x)}\right )\) |
((4*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[x]^2]/Sqrt[a]] - (2*(2*a + b)*ArcTanh[ Sqrt[a + b*Sech[x]^2]/Sqrt[a + b]])/Sqrt[a + b])/2 - Sqrt[a + b*Sech[x]^2] /(1 - Sech[x]^2))/2
3.2.84.3.1 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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
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])
\[\int \coth \left (x \right )^{3} \sqrt {a +\operatorname {sech}\left (x \right )^{2} b}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 906 vs. \(2 (65) = 130\).
Time = 0.40 (sec) , antiderivative size = 5247, normalized size of antiderivative = 63.22 \[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=\text {Too large to display} \]
\[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=\int \sqrt {a + b \operatorname {sech}^{2}{\left (x \right )}} \coth ^{3}{\left (x \right )}\, dx \]
\[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=\int { \sqrt {b \operatorname {sech}\left (x\right )^{2} + a} \coth \left (x\right )^{3} \,d x } \]
Exception generated. \[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \coth ^3(x) \sqrt {a+b \text {sech}^2(x)} \, dx=\int {\mathrm {coth}\left (x\right )}^3\,\sqrt {a+\frac {b}{{\mathrm {cosh}\left (x\right )}^2}} \,d x \]