Integrand size = 23, antiderivative size = 62 \[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=-\frac {\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{f}+\frac {\sqrt {a+b \sinh ^2(e+f x)}}{f} \]
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.05 \[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=\frac {-\sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a-b+b \cosh ^2(e+f x)}}{\sqrt {a-b}}\right )+\sqrt {a-b+b \cosh ^2(e+f x)}}{f} \]
(-(Sqrt[a - b]*ArcTanh[Sqrt[a - b + b*Cosh[e + f*x]^2]/Sqrt[a - b]]) + Sqr t[a - b + b*Cosh[e + f*x]^2])/f
Time = 0.25 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3042, 26, 3673, 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.
\(\displaystyle \int \tanh (e+f x) \sqrt {a+b \sinh ^2(e+f x)} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan (i e+i f x) \sqrt {a-b \sin (i e+i f x)^2}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sqrt {a-b \sin (i e+i f x)^2} \tan (i e+i f x)dx\) |
\(\Big \downarrow \) 3673 |
\(\displaystyle \frac {\int \frac {\sqrt {b \sinh ^2(e+f x)+a}}{\sinh ^2(e+f x)+1}d\sinh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {(a-b) \int \frac {1}{\left (\sinh ^2(e+f x)+1\right ) \sqrt {b \sinh ^2(e+f x)+a}}d\sinh ^2(e+f x)+2 \sqrt {a+b \sinh ^2(e+f x)}}{2 f}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {2 (a-b) \int \frac {1}{\frac {\sinh ^4(e+f x)}{b}-\frac {a}{b}+1}d\sqrt {b \sinh ^2(e+f x)+a}}{b}+2 \sqrt {a+b \sinh ^2(e+f x)}}{2 f}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \sqrt {a+b \sinh ^2(e+f x)}-2 \sqrt {a-b} \text {arctanh}\left (\frac {\sqrt {a+b \sinh ^2(e+f x)}}{\sqrt {a-b}}\right )}{2 f}\) |
(-2*Sqrt[a - b]*ArcTanh[Sqrt[a + b*Sinh[e + f*x]^2]/Sqrt[a - b]] + 2*Sqrt[ a + b*Sinh[e + f*x]^2])/(2*f)
3.5.59.3.1 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[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(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)*((a + b*ff*x)^p/(1 - ff*x)^((m + 1 )/2)), x], x, Sin[e + f*x]^2/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && Integ erQ[(m - 1)/2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.66
method | result | size |
default | \(\frac {\operatorname {`\,int/indef0`\,}\left (\frac {\sqrt {a +b \sinh \left (f x +e \right )^{2}}\, \sinh \left (f x +e \right )}{\cosh \left (f x +e \right )^{2}}, \sinh \left (f x +e \right )\right )}{f}\) | \(41\) |
Leaf count of result is larger than twice the leaf count of optimal. 215 vs. \(2 (54) = 108\).
Time = 0.59 (sec) , antiderivative size = 624, normalized size of antiderivative = 10.06 \[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=\left [\frac {\sqrt {a - b} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} \log \left (\frac {b \cosh \left (f x + e\right )^{4} + 4 \, b \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + b \sinh \left (f x + e\right )^{4} + 2 \, {\left (4 \, a - 3 \, b\right )} \cosh \left (f x + e\right )^{2} + 2 \, {\left (3 \, b \cosh \left (f x + e\right )^{2} + 4 \, a - 3 \, b\right )} \sinh \left (f x + e\right )^{2} - 4 \, \sqrt {2} \sqrt {a - b} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} + 4 \, {\left (b \cosh \left (f x + e\right )^{3} + {\left (4 \, a - 3 \, b\right )} \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + b}{\cosh \left (f x + e\right )^{4} + 4 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right )^{3} + \sinh \left (f x + e\right )^{4} + 2 \, {\left (3 \, \cosh \left (f x + e\right )^{2} + 1\right )} \sinh \left (f x + e\right )^{2} + 2 \, \cosh \left (f x + e\right )^{2} + 4 \, {\left (\cosh \left (f x + e\right )^{3} + \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right ) + 1}\right ) + \sqrt {2} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left (f \cosh \left (f x + e\right ) + f \sinh \left (f x + e\right )\right )}}, -\frac {2 \, \sqrt {-a + b} {\left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {-a + b} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left ({\left (a - b\right )} \cosh \left (f x + e\right ) + {\left (a - b\right )} \sinh \left (f x + e\right )\right )}}\right ) - \sqrt {2} \sqrt {\frac {b \cosh \left (f x + e\right )^{2} + b \sinh \left (f x + e\right )^{2} + 2 \, a - b}{\cosh \left (f x + e\right )^{2} - 2 \, \cosh \left (f x + e\right ) \sinh \left (f x + e\right ) + \sinh \left (f x + e\right )^{2}}}}{2 \, {\left (f \cosh \left (f x + e\right ) + f \sinh \left (f x + e\right )\right )}}\right ] \]
[1/2*(sqrt(a - b)*(cosh(f*x + e) + sinh(f*x + e))*log((b*cosh(f*x + e)^4 + 4*b*cosh(f*x + e)*sinh(f*x + e)^3 + b*sinh(f*x + e)^4 + 2*(4*a - 3*b)*cos h(f*x + e)^2 + 2*(3*b*cosh(f*x + e)^2 + 4*a - 3*b)*sinh(f*x + e)^2 - 4*sqr t(2)*sqrt(a - b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(c osh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2))*(cosh(f *x + e) + sinh(f*x + e)) + 4*(b*cosh(f*x + e)^3 + (4*a - 3*b)*cosh(f*x + e ))*sinh(f*x + e) + b)/(cosh(f*x + e)^4 + 4*cosh(f*x + e)*sinh(f*x + e)^3 + sinh(f*x + e)^4 + 2*(3*cosh(f*x + e)^2 + 1)*sinh(f*x + e)^2 + 2*cosh(f*x + e)^2 + 4*(cosh(f*x + e)^3 + cosh(f*x + e))*sinh(f*x + e) + 1)) + sqrt(2) *sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(f*cosh(f*x + e) + f*s inh(f*x + e)), -1/2*(2*sqrt(-a + b)*(cosh(f*x + e) + sinh(f*x + e))*arctan (-1/2*sqrt(2)*sqrt(-a + b)*sqrt((b*cosh(f*x + e)^2 + b*sinh(f*x + e)^2 + 2 *a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2 ))/((a - b)*cosh(f*x + e) + (a - b)*sinh(f*x + e))) - sqrt(2)*sqrt((b*cosh (f*x + e)^2 + b*sinh(f*x + e)^2 + 2*a - b)/(cosh(f*x + e)^2 - 2*cosh(f*x + e)*sinh(f*x + e) + sinh(f*x + e)^2)))/(f*cosh(f*x + e) + f*sinh(f*x + e)) ]
\[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=\int \sqrt {a + b \sinh ^{2}{\left (e + f x \right )}} \tanh {\left (e + f x \right )}\, dx \]
\[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=\int { \sqrt {b \sinh \left (f x + e\right )^{2} + a} \tanh \left (f x + e\right ) \,d x } \]
Exception generated. \[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Error: Bad Argument Type
Timed out. \[ \int \sqrt {a+b \sinh ^2(e+f x)} \tanh (e+f x) \, dx=\int \mathrm {tanh}\left (e+f\,x\right )\,\sqrt {b\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a} \,d x \]