Integrand size = 23, antiderivative size = 50 \[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\sqrt {a \cosh ^2(e+f x)}}{f} \]
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=-\frac {\sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )-\sqrt {a \cosh ^2(e+f x)}}{f} \]
Time = 0.32 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.04, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {3042, 26, 3655, 26, 3042, 26, 3684, 60, 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.
\(\displaystyle \int \coth (e+f x) \sqrt {a \sinh ^2(e+f x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \sqrt {a-a \sin (i e+i f x)^2}}{\tan (i e+i f x)}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\sqrt {a-a \sin (i e+i f x)^2}}{\tan (i e+i f x)}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle i \int -i \sqrt {a \cosh ^2(e+f x)} \coth (e+f x)dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \coth (e+f x) \sqrt {a \cosh ^2(e+f x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan \left (i e+i f x+\frac {\pi }{2}\right ) \sqrt {a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sqrt {a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2} \tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )dx\) |
\(\Big \downarrow \) 3684 |
\(\displaystyle -\frac {\int \frac {\sqrt {a \cosh ^2(e+f x)}}{1-\cosh ^2(e+f x)}d\cosh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle -\frac {a \int \frac {1}{\sqrt {a \cosh ^2(e+f x)} \left (1-\cosh ^2(e+f x)\right )}d\cosh ^2(e+f x)-2 \sqrt {a \cosh ^2(e+f x)}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {2 \int \frac {1}{1-\frac {\cosh ^4(e+f x)}{a}}d\sqrt {a \cosh ^2(e+f x)}-2 \sqrt {a \cosh ^2(e+f x)}}{2 f}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a \cosh ^2(e+f x)}}{\sqrt {a}}\right )-2 \sqrt {a \cosh ^2(e+f x)}}{2 f}\) |
3.5.29.3.1 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_)^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 = 0.20 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.08
method | result | size |
default | \(\frac {a \cosh \left (f x +e \right ) \left (2 \cosh \left (f x +e \right )+\ln \left (\cosh \left (f x +e \right )-1\right )-\ln \left (\cosh \left (f x +e \right )+1\right )\right )}{2 \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(54\) |
risch | \(\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{2 f x +2 e}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}}{2 f \left ({\mathrm e}^{2 f x +2 e}+1\right )}+\frac {\ln \left ({\mathrm e}^{f x}-{\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{f \left ({\mathrm e}^{2 f x +2 e}+1\right )}-\frac {\ln \left ({\mathrm e}^{f x}+{\mathrm e}^{-e}\right ) \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, {\mathrm e}^{f x +e}}{f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\) | \(220\) |
1/2*a*cosh(f*x+e)*(2*cosh(f*x+e)+ln(cosh(f*x+e)-1)-ln(cosh(f*x+e)+1))/(a*c osh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 200 vs. \(2 (42) = 84\).
Time = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 4.00 \[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\frac {{\left (2 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + {\left (\cosh \left (f x + e\right )^{2} + 1\right )} e^{\left (f x + e\right )} + 2 \, {\left (\cosh \left (f x + e\right ) e^{\left (f x + e\right )} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )\right )} \log \left (\frac {\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) - 1}{\cosh \left (f x + e\right ) + \sinh \left (f x + e\right ) + 1}\right )\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{2 \, {\left (f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + f \cosh \left (f x + e\right ) + {\left (f e^{\left (2 \, f x + 2 \, e\right )} + f\right )} \sinh \left (f x + e\right )\right )}} \]
1/2*(2*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e) + e^(f*x + e)*sinh(f*x + e) ^2 + (cosh(f*x + e)^2 + 1)*e^(f*x + e) + 2*(cosh(f*x + e)*e^(f*x + e) + e^ (f*x + e)*sinh(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)/(f*cosh(f*x + e)*e^(2*f*x + 2*e) + f*cosh(f*x + e) + (f* e^(2*f*x + 2*e) + f)*sinh(f*x + e))
\[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\int \sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )} \coth {\left (e + f x \right )}\, dx \]
Time = 0.31 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.36 \[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\frac {{\left (\sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + \sqrt {a}\right )} e^{\left (f x + e\right )}}{2 \, f} - \frac {\sqrt {a} \log \left (e^{\left (-f x - e\right )} + 1\right )}{f} + \frac {\sqrt {a} \log \left (e^{\left (-f x - e\right )} - 1\right )}{f} \]
1/2*(sqrt(a)*e^(-2*f*x - 2*e) + sqrt(a))*e^(f*x + e)/f - sqrt(a)*log(e^(-f *x - e) + 1)/f + sqrt(a)*log(e^(-f*x - e) - 1)/f
Time = 0.27 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.94 \[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\frac {\sqrt {a} {\left (e^{\left (f x + e\right )} + e^{\left (-f x - e\right )} - 2 \, \log \left (e^{\left (f x + e\right )} + 1\right ) + 2 \, \log \left ({\left | e^{\left (f x + e\right )} - 1 \right |}\right )\right )}}{2 \, f} \]
1/2*sqrt(a)*(e^(f*x + e) + e^(-f*x - e) - 2*log(e^(f*x + e) + 1) + 2*log(a bs(e^(f*x + e) - 1)))/f
Timed out. \[ \int \coth (e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\int \mathrm {coth}\left (e+f\,x\right )\,\sqrt {a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]