Integrand size = 25, antiderivative size = 66 \[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {a^2}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}+\frac {2 a}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}}-\frac {1}{f \sqrt {a \cosh ^2(e+f x)}} \]
-1/5*a^2/f/(a*cosh(f*x+e)^2)^(5/2)+2/3*a/f/(a*cosh(f*x+e)^2)^(3/2)-1/f/(a* cosh(f*x+e)^2)^(1/2)
Time = 0.17 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.65 \[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\frac {-15+10 \text {sech}^2(e+f x)-3 \text {sech}^4(e+f x)}{15 f \sqrt {a \cosh ^2(e+f x)}} \]
Time = 0.38 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3042, 26, 3655, 26, 3042, 26, 3684, 8, 53, 2009}
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.
\(\displaystyle \int \frac {\tanh ^5(e+f x)}{\sqrt {a \sinh ^2(e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {i \tan (i e+i f x)^5}{\sqrt {a-a \sin (i e+i f x)^2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \frac {\tan (i e+i f x)^5}{\sqrt {a-a \sin (i e+i f x)^2}}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle -i \int \frac {i \tanh ^5(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \int \frac {\tanh ^5(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i}{\tan \left (i e+i f x+\frac {\pi }{2}\right )^5 \sqrt {a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {1}{\sqrt {a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2} \tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^5}dx\) |
\(\Big \downarrow \) 3684 |
\(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(e+f x)\right )^2 \text {sech}^6(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}d\cosh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 8 |
\(\displaystyle \frac {a^3 \int \frac {\left (1-\cosh ^2(e+f x)\right )^2}{\left (a \cosh ^2(e+f x)\right )^{7/2}}d\cosh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {a^3 \int \left (\frac {1}{\left (a \cosh ^2(e+f x)\right )^{7/2}}-\frac {2}{\left (a \cosh ^2(e+f x)\right )^{5/2} a}+\frac {1}{\left (a \cosh ^2(e+f x)\right )^{3/2} a^2}\right )d\cosh ^2(e+f x)}{2 f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \left (-\frac {2}{a^3 \sqrt {a \cosh ^2(e+f x)}}+\frac {4}{3 a^2 \left (a \cosh ^2(e+f x)\right )^{3/2}}-\frac {2}{5 a \left (a \cosh ^2(e+f x)\right )^{5/2}}\right )}{2 f}\) |
(a^3*(-2/(5*a*(a*Cosh[e + f*x]^2)^(5/2)) + 4/(3*a^2*(a*Cosh[e + f*x]^2)^(3 /2)) - 2/(a^3*Sqrt[a*Cosh[e + f*x]^2])))/(2*f)
3.5.37.3.1 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] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 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 = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.73
method | result | size |
default | \(-\frac {15 \cosh \left (f x +e \right )^{4}-10 \cosh \left (f x +e \right )^{2}+3}{15 \cosh \left (f x +e \right )^{4} \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(48\) |
risch | \(-\frac {2 \left (15 \,{\mathrm e}^{8 f x +8 e}+20 \,{\mathrm e}^{6 f x +6 e}+58 \,{\mathrm e}^{4 f x +4 e}+20 \,{\mathrm e}^{2 f x +2 e}+15\right )}{15 \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left ({\mathrm e}^{2 f x +2 e}+1\right )^{4} f}\) | \(91\) |
Leaf count of result is larger than twice the leaf count of optimal. 1387 vs. \(2 (56) = 112\).
Time = 0.27 (sec) , antiderivative size = 1387, normalized size of antiderivative = 21.02 \[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
-2/15*(135*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^8 + 15*e^(f*x + e)*sinh (f*x + e)^9 + 20*(27*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^7 + 14 0*(9*cosh(f*x + e)^3 + cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^6 + 2*(945 *cosh(f*x + e)^4 + 210*cosh(f*x + e)^2 + 29)*e^(f*x + e)*sinh(f*x + e)^5 + 10*(189*cosh(f*x + e)^5 + 70*cosh(f*x + e)^3 + 29*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 20*(63*cosh(f*x + e)^6 + 35*cosh(f*x + e)^4 + 29*cos h(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^3 + 20*(27*cosh(f*x + e)^7 + 2 1*cosh(f*x + e)^5 + 29*cosh(f*x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh (f*x + e)^2 + 5*(27*cosh(f*x + e)^8 + 28*cosh(f*x + e)^6 + 58*cosh(f*x + e )^4 + 12*cosh(f*x + e)^2 + 3)*e^(f*x + e)*sinh(f*x + e) + (15*cosh(f*x + e )^9 + 20*cosh(f*x + e)^7 + 58*cosh(f*x + e)^5 + 20*cosh(f*x + e)^3 + 15*co sh(f*x + e))*e^(f*x + e))*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)^10 + (a*f*e^(2*f*x + 2*e) + a*f)*sinh(f* x + e)^10 + 5*a*f*cosh(f*x + e)^8 + 10*(a*f*cosh(f*x + e)*e^(2*f*x + 2*e) + a*f*cosh(f*x + e))*sinh(f*x + e)^9 + 5*(9*a*f*cosh(f*x + e)^2 + a*f + (9 *a*f*cosh(f*x + e)^2 + a*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^8 + 10*a*f*cosh (f*x + e)^6 + 40*(3*a*f*cosh(f*x + e)^3 + a*f*cosh(f*x + e) + (3*a*f*cosh( f*x + e)^3 + a*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^7 + 10*(21* a*f*cosh(f*x + e)^4 + 14*a*f*cosh(f*x + e)^2 + a*f + (21*a*f*cosh(f*x + e) ^4 + 14*a*f*cosh(f*x + e)^2 + a*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^6 + 1...
\[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\int \frac {\tanh ^{5}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 446 vs. \(2 (56) = 112\).
Time = 0.33 (sec) , antiderivative size = 446, normalized size of antiderivative = 6.76 \[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {2 \, e^{\left (-f x - e\right )}}{{\left (5 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + \sqrt {a} e^{\left (-10 \, f x - 10 \, e\right )} + \sqrt {a}\right )} f} - \frac {8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (5 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + \sqrt {a} e^{\left (-10 \, f x - 10 \, e\right )} + \sqrt {a}\right )} f} - \frac {116 \, e^{\left (-5 \, f x - 5 \, e\right )}}{15 \, {\left (5 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + \sqrt {a} e^{\left (-10 \, f x - 10 \, e\right )} + \sqrt {a}\right )} f} - \frac {8 \, e^{\left (-7 \, f x - 7 \, e\right )}}{3 \, {\left (5 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + \sqrt {a} e^{\left (-10 \, f x - 10 \, e\right )} + \sqrt {a}\right )} f} - \frac {2 \, e^{\left (-9 \, f x - 9 \, e\right )}}{{\left (5 \, \sqrt {a} e^{\left (-2 \, f x - 2 \, e\right )} + 10 \, \sqrt {a} e^{\left (-4 \, f x - 4 \, e\right )} + 10 \, \sqrt {a} e^{\left (-6 \, f x - 6 \, e\right )} + 5 \, \sqrt {a} e^{\left (-8 \, f x - 8 \, e\right )} + \sqrt {a} e^{\left (-10 \, f x - 10 \, e\right )} + \sqrt {a}\right )} f} \]
-2*e^(-f*x - e)/((5*sqrt(a)*e^(-2*f*x - 2*e) + 10*sqrt(a)*e^(-4*f*x - 4*e) + 10*sqrt(a)*e^(-6*f*x - 6*e) + 5*sqrt(a)*e^(-8*f*x - 8*e) + sqrt(a)*e^(- 10*f*x - 10*e) + sqrt(a))*f) - 8/3*e^(-3*f*x - 3*e)/((5*sqrt(a)*e^(-2*f*x - 2*e) + 10*sqrt(a)*e^(-4*f*x - 4*e) + 10*sqrt(a)*e^(-6*f*x - 6*e) + 5*sqr t(a)*e^(-8*f*x - 8*e) + sqrt(a)*e^(-10*f*x - 10*e) + sqrt(a))*f) - 116/15* e^(-5*f*x - 5*e)/((5*sqrt(a)*e^(-2*f*x - 2*e) + 10*sqrt(a)*e^(-4*f*x - 4*e ) + 10*sqrt(a)*e^(-6*f*x - 6*e) + 5*sqrt(a)*e^(-8*f*x - 8*e) + sqrt(a)*e^( -10*f*x - 10*e) + sqrt(a))*f) - 8/3*e^(-7*f*x - 7*e)/((5*sqrt(a)*e^(-2*f*x - 2*e) + 10*sqrt(a)*e^(-4*f*x - 4*e) + 10*sqrt(a)*e^(-6*f*x - 6*e) + 5*sq rt(a)*e^(-8*f*x - 8*e) + sqrt(a)*e^(-10*f*x - 10*e) + sqrt(a))*f) - 2*e^(- 9*f*x - 9*e)/((5*sqrt(a)*e^(-2*f*x - 2*e) + 10*sqrt(a)*e^(-4*f*x - 4*e) + 10*sqrt(a)*e^(-6*f*x - 6*e) + 5*sqrt(a)*e^(-8*f*x - 8*e) + sqrt(a)*e^(-10* f*x - 10*e) + sqrt(a))*f)
Exception generated. \[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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
Time = 1.64 (sec) , antiderivative size = 381, normalized size of antiderivative = 5.77 \[ \int \frac {\tanh ^5(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\frac {32\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {4\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{a\,f\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {352\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}+\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \]
(32*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/ (3*a*f*(exp(2*e + 2*f*x) + 1)^2*(exp(e + f*x) + exp(3*e + 3*f*x))) - (4*ex p(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(a*f*( exp(2*e + 2*f*x) + 1)*(exp(e + f*x) + exp(3*e + 3*f*x))) - (352*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(15*a*f*(exp(2 *e + 2*f*x) + 1)^3*(exp(e + f*x) + exp(3*e + 3*f*x))) + (128*exp(3*e + 3*f *x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(5*a*f*(exp(2*e + 2*f*x) + 1)^4*(exp(e + f*x) + exp(3*e + 3*f*x))) - (64*exp(3*e + 3*f*x)*( a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(5*a*f*(exp(2*e + 2*f* x) + 1)^5*(exp(e + f*x) + exp(3*e + 3*f*x)))