Integrand size = 25, antiderivative size = 68 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {a^2}{7 f \left (a \cosh ^2(e+f x)\right )^{7/2}}+\frac {2 a}{5 f \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac {1}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
-1/7*a^2/f/(a*cosh(f*x+e)^2)^(7/2)+2/5*a/f/(a*cosh(f*x+e)^2)^(5/2)-1/3/f/( a*cosh(f*x+e)^2)^(3/2)
Time = 0.20 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.75 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {\left (-15+42 \cosh ^2(e+f x)-35 \cosh ^4(e+f x)\right ) \text {sech}^4(e+f x)}{105 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
((-15 + 42*Cosh[e + f*x]^2 - 35*Cosh[e + f*x]^4)*Sech[e + f*x]^4)/(105*f*( a*Cosh[e + f*x]^2)^(3/2))
Time = 0.41 (sec) , antiderivative size = 74, 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.
| \(\displaystyle \int \frac {\tanh ^5(e+f x)}{\left (a \sinh ^2(e+f x)+a\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i \tan (i e+i f x)^5}{\left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {\tan (i e+i f x)^5}{\left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3655 |
| \(\displaystyle -i \int \frac {i \tanh ^5(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \int \frac {\tanh ^5(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {i}{\tan \left (i e+i f x+\frac {\pi }{2}\right )^5 \left (a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle i \int \frac {1}{\left (a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2\right )^{3/2} \tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^5}dx\) |
| \(\Big \downarrow \) 3684 |
| \(\displaystyle \frac {\int \frac {\left (1-\cosh ^2(e+f x)\right )^2 \text {sech}^6(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}d\cosh ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 8 |
| \(\displaystyle \frac {a^3 \int \frac {\left (1-\cosh ^2(e+f x)\right )^2}{\left (a \cosh ^2(e+f x)\right )^{9/2}}d\cosh ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {a^3 \int \left (\frac {1}{\left (a \cosh ^2(e+f x)\right )^{9/2}}-\frac {2}{\left (a \cosh ^2(e+f x)\right )^{7/2} a}+\frac {1}{\left (a \cosh ^2(e+f x)\right )^{5/2} a^2}\right )d\cosh ^2(e+f x)}{2 f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 \left (-\frac {2}{3 a^3 \left (a \cosh ^2(e+f x)\right )^{3/2}}+\frac {4}{5 a^2 \left (a \cosh ^2(e+f x)\right )^{5/2}}-\frac {2}{7 a \left (a \cosh ^2(e+f x)\right )^{7/2}}\right )}{2 f}\) |
(a^3*(-2/(7*a*(a*Cosh[e + f*x]^2)^(7/2)) + 4/(5*a^2*(a*Cosh[e + f*x]^2)^(5 /2)) - 2/(3*a^3*(a*Cosh[e + f*x]^2)^(3/2))))/(2*f)
3.5.47.3.1 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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(\frac {\operatorname {`\,int/indef0`\,}\left (\frac {\sinh \left (f x +e \right )^{5}}{\cosh \left (f x +e \right )^{8} a \sqrt {a \cosh \left (f x +e \right )^{2}}}, \sinh \left (f x +e \right )\right )}{f}\) | \(44\) |
| risch | \(-\frac {8 \left (35 \,{\mathrm e}^{8 f x +8 e}-28 \,{\mathrm e}^{6 f x +6 e}+114 \,{\mathrm e}^{4 f x +4 e}-28 \,{\mathrm e}^{2 f x +2 e}+35\right ) {\mathrm e}^{2 f x +2 e}}{105 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left ({\mathrm e}^{2 f x +2 e}+1\right )^{6} a}\) | \(103\) |
Leaf count of result is larger than twice the leaf count of optimal. 2507 vs. \(2 (56) = 112\).
Time = 0.31 (sec) , antiderivative size = 2507, normalized size of antiderivative = 36.87 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-8/105*(385*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^10 + 35*e^(f*x + e)*si nh(f*x + e)^11 + 7*(275*cosh(f*x + e)^2 - 4)*e^(f*x + e)*sinh(f*x + e)^9 + 21*(275*cosh(f*x + e)^3 - 12*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^8 + 6*(1925*cosh(f*x + e)^4 - 168*cosh(f*x + e)^2 + 19)*e^(f*x + e)*sinh(f*x + e)^7 + 42*(385*cosh(f*x + e)^5 - 56*cosh(f*x + e)^3 + 19*cosh(f*x + e))* e^(f*x + e)*sinh(f*x + e)^6 + 14*(1155*cosh(f*x + e)^6 - 252*cosh(f*x + e) ^4 + 171*cosh(f*x + e)^2 - 2)*e^(f*x + e)*sinh(f*x + e)^5 + 14*(825*cosh(f *x + e)^7 - 252*cosh(f*x + e)^5 + 285*cosh(f*x + e)^3 - 10*cosh(f*x + e))* e^(f*x + e)*sinh(f*x + e)^4 + 7*(825*cosh(f*x + e)^8 - 336*cosh(f*x + e)^6 + 570*cosh(f*x + e)^4 - 40*cosh(f*x + e)^2 + 5)*e^(f*x + e)*sinh(f*x + e) ^3 + 7*(275*cosh(f*x + e)^9 - 144*cosh(f*x + e)^7 + 342*cosh(f*x + e)^5 - 40*cosh(f*x + e)^3 + 15*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + 7*(55 *cosh(f*x + e)^10 - 36*cosh(f*x + e)^8 + 114*cosh(f*x + e)^6 - 20*cosh(f*x + e)^4 + 15*cosh(f*x + e)^2)*e^(f*x + e)*sinh(f*x + e) + (35*cosh(f*x + e )^11 - 28*cosh(f*x + e)^9 + 114*cosh(f*x + e)^7 - 28*cosh(f*x + e)^5 + 35* cosh(f*x + e)^3)*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^2*f*cosh(f*x + e)^14 + 7*a^2*f*cosh(f*x + e)^12 + (a ^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^14 + 14*(a^2*f*cosh(f*x + e)*e ^(2*f*x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e)^13 + 21*a^2*f*cosh(f*x + e)^10 + 7*(13*a^2*f*cosh(f*x + e)^2 + a^2*f + (13*a^2*f*cosh(f*x + e...
\[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\tanh ^{5}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 586 vs. \(2 (56) = 112\).
Time = 0.34 (sec) , antiderivative size = 586, normalized size of antiderivative = 8.62 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {8 \, e^{\left (-3 \, f x - 3 \, e\right )}}{3 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {32 \, e^{\left (-5 \, f x - 5 \, e\right )}}{15 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {304 \, e^{\left (-7 \, f x - 7 \, e\right )}}{35 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {32 \, e^{\left (-9 \, f x - 9 \, e\right )}}{15 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {8 \, e^{\left (-11 \, f x - 11 \, e\right )}}{3 \, {\left (7 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + 35 \, a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + 21 \, a^{\frac {3}{2}} e^{\left (-10 \, f x - 10 \, e\right )} + 7 \, a^{\frac {3}{2}} e^{\left (-12 \, f x - 12 \, e\right )} + a^{\frac {3}{2}} e^{\left (-14 \, f x - 14 \, e\right )} + a^{\frac {3}{2}}\right )} f} \]
-8/3*e^(-3*f*x - 3*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e) + 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^ (3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f *x - 14*e) + a^(3/2))*f) + 32/15*e^(-5*f*x - 5*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e) + 35*a^(3 /2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f* x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) + a^(3/2))*f) - 304/35*e^(-7*f*x - 7*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/ 2)*e^(-6*f*x - 6*e) + 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) + a^(3 /2))*f) + 32/15*e^(-9*f*x - 9*e)/((7*a^(3/2)*e^(-2*f*x - 2*e) + 21*a^(3/2) *e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e) + 35*a^(3/2)*e^(-8*f*x - 8 *e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/ 2)*e^(-14*f*x - 14*e) + a^(3/2))*f) - 8/3*e^(-11*f*x - 11*e)/((7*a^(3/2)*e ^(-2*f*x - 2*e) + 21*a^(3/2)*e^(-4*f*x - 4*e) + 35*a^(3/2)*e^(-6*f*x - 6*e ) + 35*a^(3/2)*e^(-8*f*x - 8*e) + 21*a^(3/2)*e^(-10*f*x - 10*e) + 7*a^(3/2 )*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) + a^(3/2))*f)
Exception generated. \[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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 = 0.17 (sec) , antiderivative size = 457, normalized size of antiderivative = 6.72 \[ \int \frac {\tanh ^5(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {464\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {3072\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}+\frac {4736\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{35\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {768\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^6\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}+\frac {256\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{7\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )}^7\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \]
(464*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2)) /(15*a^2*f*(exp(2*e + 2*f*x) + 1)^3*(exp(e + f*x) + exp(3*e + 3*f*x))) - ( 16*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/( 3*a^2*f*(exp(2*e + 2*f*x) + 1)^2*(exp(e + f*x) + exp(3*e + 3*f*x))) - (307 2*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(3 5*a^2*f*(exp(2*e + 2*f*x) + 1)^4*(exp(e + f*x) + exp(3*e + 3*f*x))) + (473 6*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(3 5*a^2*f*(exp(2*e + 2*f*x) + 1)^5*(exp(e + f*x) + exp(3*e + 3*f*x))) - (768 *exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(7* a^2*f*(exp(2*e + 2*f*x) + 1)^6*(exp(e + f*x) + exp(3*e + 3*f*x))) + (256*e xp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(7*a^ 2*f*(exp(2*e + 2*f*x) + 1)^7*(exp(e + f*x) + exp(3*e + 3*f*x)))