Integrand size = 25, antiderivative size = 66 \[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 \sqrt {a} f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^2(e+f x)}{2 a f} \]
-1/2*arctanh((a*cosh(f*x+e)^2)^(1/2)/a^(1/2))/f/a^(1/2)-1/2*csch(f*x+e)^2* (a*cosh(f*x+e)^2)^(1/2)/a/f
Time = 0.19 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.85 \[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {\text {arctanh}\left (\sqrt {\cosh ^2(e+f x)}\right ) \sqrt {\cosh ^2(e+f x)}+\coth ^2(e+f x)}{2 f \sqrt {a \cosh ^2(e+f x)}} \]
-1/2*(ArcTanh[Sqrt[Cosh[e + f*x]^2]]*Sqrt[Cosh[e + f*x]^2] + Coth[e + f*x] ^2)/(f*Sqrt[a*Cosh[e + f*x]^2])
Time = 0.36 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.03, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {3042, 26, 3655, 26, 3042, 26, 3684, 8, 51, 73, 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.
\(\displaystyle \int \frac {\coth ^3(e+f x)}{\sqrt {a \sinh ^2(e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i}{\tan (i e+i f x)^3 \sqrt {a-a \sin (i e+i f x)^2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {1}{\sqrt {a-a \sin (i e+i f x)^2} \tan (i e+i f x)^3}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle -i \int \frac {i \coth ^3(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\coth ^3(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \tan \left (i e+i f x+\frac {\pi }{2}\right )^3}{\sqrt {a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^3}{\sqrt {a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2}}dx\) |
\(\Big \downarrow \) 3684 |
\(\displaystyle \frac {\int \frac {\cosh ^2(e+f x)}{\sqrt {a \cosh ^2(e+f x)} \left (1-\cosh ^2(e+f x)\right )^2}d\cosh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 8 |
\(\displaystyle \frac {\int \frac {\sqrt {a \cosh ^2(e+f x)}}{\left (1-\cosh ^2(e+f x)\right )^2}d\cosh ^2(e+f x)}{2 a f}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {\sqrt {a \cosh ^2(e+f x)}}{1-\cosh ^2(e+f x)}-\frac {1}{2} a \int \frac {1}{\sqrt {a \cosh ^2(e+f x)} \left (1-\cosh ^2(e+f x)\right )}d\cosh ^2(e+f x)}{2 a f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {\sqrt {a \cosh ^2(e+f x)}}{1-\cosh ^2(e+f x)}-\int \frac {1}{1-\frac {\cosh ^4(e+f x)}{a}}d\sqrt {a \cosh ^2(e+f x)}}{2 a f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\frac {\sqrt {a \cosh ^2(e+f x)}}{1-\cosh ^2(e+f x)}-\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{2 a f}\) |
(-(Sqrt[a]*ArcTanh[Sqrt[a*Cosh[e + f*x]^2]/Sqrt[a]]) + Sqrt[a*Cosh[e + f*x ]^2]/(1 - Cosh[e + f*x]^2))/(2*a*f)
3.5.41.3.1 Defintions of rubi rules used
Int[(u_.)*(x_)^(m_.)*((a_.)*(x_))^(p_), x_Symbol] :> Simp[1/a^m Int[u*(a* x)^(m + p), x], x] /; FreeQ[{a, m, p}, x] && IntegerQ[m]
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 + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
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_)^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[(u_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Int[A ctivateTrig[u*(a*cos[e + f*x]^2)^p], x] /; FreeQ[{a, b, e, f, p}, x] && EqQ [a + b, 0]
Int[((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*tan[(e_.) + (f_.)*(x_)]^(m_. ), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x]^2, x]}, Simp[ff^((m + 1 )/2)/(2*f) Subst[Int[x^((m - 1)/2)*((b*ff^(n/2)*x^(n/2))^p/(1 - ff*x)^((m + 1)/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{b, e, f, p}, x] && Inte gerQ[(m - 1)/2] && IntegerQ[n/2]
Time = 0.22 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.26
method | result | size |
default | \(\frac {\cosh \left (f x +e \right ) \left (-2 \cosh \left (f x +e \right )+\left (\ln \left (\cosh \left (f x +e \right )-1\right )-\ln \left (\cosh \left (f x +e \right )+1\right )\right ) \sinh \left (f x +e \right )^{2}\right )}{4 \sqrt {a \cosh \left (f x +e \right )^{2}}\, \left (\cosh \left (f x +e \right )-1\right ) \left (\cosh \left (f x +e \right )+1\right ) f}\) | \(83\) |
risch | \(-\frac {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2}}{\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f \left ({\mathrm e}^{2 f x +2 e}-1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{f x}+{\mathrm e}^{-e}\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{-f x -e}}{2 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}+\frac {\ln \left ({\mathrm e}^{f x}-{\mathrm e}^{-e}\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{-f x -e}}{2 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}\) | \(183\) |
1/4*cosh(f*x+e)*(-2*cosh(f*x+e)+(ln(cosh(f*x+e)-1)-ln(cosh(f*x+e)+1))*sinh (f*x+e)^2)/(a*cosh(f*x+e)^2)^(1/2)/(cosh(f*x+e)-1)/(cosh(f*x+e)+1)/f
Leaf count of result is larger than twice the leaf count of optimal. 529 vs. \(2 (54) = 108\).
Time = 0.26 (sec) , antiderivative size = 529, normalized size of antiderivative = 8.02 \[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {{\left (6 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 2 \, e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + 2 \, {\left (\cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} - {\left (4 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} - \cosh \left (f x + e\right )\right )} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + {\left (\cosh \left (f x + e\right )^{4} - 2 \, \cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )}\right )} \log \left (\frac {\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1}{\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1}\right )\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{2 \, {\left (a f \cosh \left (f x + e\right )^{4} + {\left (a f e^{\left (2 \, f x + 2 \, e\right )} + a f\right )} \sinh \left (f x + e\right )^{4} - 2 \, a f \cosh \left (f x + e\right )^{2} + 4 \, {\left (a f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )^{3} + 2 \, {\left (3 \, a f \cosh \left (f x + e\right )^{2} - a f + {\left (3 \, a f \cosh \left (f x + e\right )^{2} - a f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + a f + {\left (a f \cosh \left (f x + e\right )^{4} - 2 \, a f \cosh \left (f x + e\right )^{2} + a f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 4 \, {\left (a f \cosh \left (f x + e\right )^{3} - a f \cosh \left (f x + e\right ) + {\left (a f \cosh \left (f x + e\right )^{3} - a f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )\right )}} \]
-1/2*(6*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^2 + 2*e^(f*x + e)*sinh(f*x + e)^3 + 2*(3*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e) + 2*(cosh(f* x + e)^3 + cosh(f*x + e))*e^(f*x + e) - (4*cosh(f*x + e)*e^(f*x + e)*sinh( f*x + e)^3 + e^(f*x + e)*sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 - 1)*e^(f* x + e)*sinh(f*x + e)^2 + 4*(cosh(f*x + e)^3 - cosh(f*x + e))*e^(f*x + e)*s inh(f*x + e) + (cosh(f*x + e)^4 - 2*cosh(f*x + e)^2 + 1)*e^(f*x + e))*log( (cosh(f*x + e) + sinh(f*x + e) - 1)/(cosh(f*x + e) + sinh(f*x + e) + 1)))* sqrt(a*e^(4*f*x + 4*e) + 2*a*e^(2*f*x + 2*e) + a)*e^(-f*x - e)/(a*f*cosh(f *x + e)^4 + (a*f*e^(2*f*x + 2*e) + a*f)*sinh(f*x + e)^4 - 2*a*f*cosh(f*x + e)^2 + 4*(a*f*cosh(f*x + e)*e^(2*f*x + 2*e) + a*f*cosh(f*x + e))*sinh(f*x + e)^3 + 2*(3*a*f*cosh(f*x + e)^2 - a*f + (3*a*f*cosh(f*x + e)^2 - a*f)*e ^(2*f*x + 2*e))*sinh(f*x + e)^2 + a*f + (a*f*cosh(f*x + e)^4 - 2*a*f*cosh( f*x + e)^2 + a*f)*e^(2*f*x + 2*e) + 4*(a*f*cosh(f*x + e)^3 - a*f*cosh(f*x + e) + (a*f*cosh(f*x + e)^3 - a*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e))
\[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\int \frac {\coth ^{3}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \]
Time = 0.32 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.52 \[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {\log \left (e^{\left (-f x - e\right )} + 1\right )}{2 \, \sqrt {a} f} + \frac {\log \left (e^{\left (-f x - e\right )} - 1\right )}{2 \, \sqrt {a} f} + \frac {e^{\left (-f x - e\right )} + e^{\left (-3 \, f x - 3 \, e\right )}}{{\left (2 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} - \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} - \sqrt {a}\right )} f} \]
-1/2*log(e^(-f*x - e) + 1)/(sqrt(a)*f) + 1/2*log(e^(-f*x - e) - 1)/(sqrt(a )*f) + (e^(-f*x - e) + e^(-3*f*x - 3*e))/((2*sqrt(a)*e^(-2*f*x - 2*e) - sq rt(a)*e^(-4*f*x - 4*e) - sqrt(a))*f)
Exception generated. \[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\coth ^3(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\int \frac {{\mathrm {coth}\left (e+f\,x\right )}^3}{\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a}} \,d x \]