Integrand size = 25, antiderivative size = 106 \[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {\arctan (\sinh (e+f x)) \cosh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}+\frac {\tanh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\text {sech}^2(e+f x) \tanh (e+f x)}{4 a f \sqrt {a \cosh ^2(e+f x)}} \]
1/8*arctan(sinh(f*x+e))*cosh(f*x+e)/a/f/(a*cosh(f*x+e)^2)^(1/2)+1/8*tanh(f *x+e)/a/f/(a*cosh(f*x+e)^2)^(1/2)-1/4*sech(f*x+e)^2*tanh(f*x+e)/a/f/(a*cos h(f*x+e)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.55 \[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\frac {\arctan (\sinh (e+f x)) \cosh (e+f x)+\left (1-2 \text {sech}^2(e+f x)\right ) \tanh (e+f x)}{8 a f \sqrt {a \cosh ^2(e+f x)}} \]
(ArcTan[Sinh[e + f*x]]*Cosh[e + f*x] + (1 - 2*Sech[e + f*x]^2)*Tanh[e + f* x])/(8*a*f*Sqrt[a*Cosh[e + f*x]^2])
Time = 0.60 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.80, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 25, 3655, 25, 3042, 25, 3686, 25, 3042, 25, 3091, 3042, 4255, 3042, 4257}
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.
| \(\displaystyle \int \frac {\tanh ^2(e+f x)}{\left (a \sinh ^2(e+f x)+a\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\tan (i e+i f x)^2}{\left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\tan (i e+i f x)^2}{\left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3655 |
| \(\displaystyle -\int -\frac {\tanh ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\tanh ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan \left (i e+i f x+\frac {\pi }{2}\right )^2 \left (a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2\right )^{3/2} \tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle -\frac {\cosh (e+f x) \int -\text {sech}^3(e+f x) \tanh ^2(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\cosh (e+f x) \int \text {sech}^3(e+f x) \tanh ^2(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh (e+f x) \int -\sec (i e+i f x)^3 \tan (i e+i f x)^2dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\cosh (e+f x) \int \sec (i e+i f x)^3 \tan (i e+i f x)^2dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 3091 |
| \(\displaystyle -\frac {\cosh (e+f x) \left (\frac {\tanh (e+f x) \text {sech}^3(e+f x)}{4 f}-\frac {1}{4} \int \text {sech}^3(e+f x)dx\right )}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\cosh (e+f x) \left (\frac {\tanh (e+f x) \text {sech}^3(e+f x)}{4 f}-\frac {1}{4} \int \csc \left (i e+i f x+\frac {\pi }{2}\right )^3dx\right )}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 4255 |
| \(\displaystyle -\frac {\cosh (e+f x) \left (\frac {1}{4} \left (-\frac {1}{2} \int \text {sech}(e+f x)dx-\frac {\tanh (e+f x) \text {sech}(e+f x)}{2 f}\right )+\frac {\tanh (e+f x) \text {sech}^3(e+f x)}{4 f}\right )}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\cosh (e+f x) \left (\frac {\tanh (e+f x) \text {sech}^3(e+f x)}{4 f}+\frac {1}{4} \left (-\frac {\tanh (e+f x) \text {sech}(e+f x)}{2 f}-\frac {1}{2} \int \csc \left (i e+i f x+\frac {\pi }{2}\right )dx\right )\right )}{a \sqrt {a \cosh ^2(e+f x)}}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\cosh (e+f x) \left (\frac {1}{4} \left (-\frac {\arctan (\sinh (e+f x))}{2 f}-\frac {\tanh (e+f x) \text {sech}(e+f x)}{2 f}\right )+\frac {\tanh (e+f x) \text {sech}^3(e+f x)}{4 f}\right )}{a \sqrt {a \cosh ^2(e+f x)}}\) |
-((Cosh[e + f*x]*((Sech[e + f*x]^3*Tanh[e + f*x])/(4*f) + (-1/2*ArcTan[Sin h[e + f*x]]/f - (Sech[e + f*x]*Tanh[e + f*x])/(2*f))/4))/(a*Sqrt[a*Cosh[e + f*x]^2]))
3.5.52.3.1 Defintions of rubi rules used
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
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[(u_.)*((b_.)*sin[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Sin[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Sin[e + f*x]^ n)^FracPart[p]/(Sin[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Si n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Csc[c + d*x])^(n - 1)/(d*(n - 1))), x] + Simp[b^2*((n - 2)/(n - 1)) Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(\frac {\arctan \left (\sinh \left (f x +e \right )\right ) \cosh \left (f x +e \right )^{4}+\cosh \left (f x +e \right )^{2} \sinh \left (f x +e \right )-2 \sinh \left (f x +e \right )}{8 a \cosh \left (f x +e \right )^{3} \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(69\) |
| risch | \(\frac {{\mathrm e}^{6 f x +6 e}-7 \,{\mathrm e}^{4 f x +4 e}+7 \,{\mathrm e}^{2 f x +2 e}-1}{4 a \left ({\mathrm e}^{2 f x +2 e}+1\right )^{3} \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f}+\frac {i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{-f x -e}}{8 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}-\frac {i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{-f x -e}}{8 f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\) | \(218\) |
1/8/a*(arctan(sinh(f*x+e))*cosh(f*x+e)^4+cosh(f*x+e)^2*sinh(f*x+e)-2*sinh( f*x+e))/cosh(f*x+e)^3/(a*cosh(f*x+e)^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 1423 vs. \(2 (94) = 188\).
Time = 0.28 (sec) , antiderivative size = 1423, normalized size of antiderivative = 13.42 \[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
1/4*(7*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^6 + e^(f*x + e)*sinh(f*x + e)^7 + 7*(3*cosh(f*x + e)^2 - 1)*e^(f*x + e)*sinh(f*x + e)^5 + 35*(cosh(f* x + e)^3 - cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 7*(5*cosh(f*x + e) ^4 - 10*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^3 + 7*(3*cosh(f*x + e)^5 - 10*cosh(f*x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^2 + (7*cosh(f*x + e)^6 - 35*cosh(f*x + e)^4 + 21*cosh(f*x + e)^2 - 1)*e^(f*x + e)*sinh(f*x + e) + (8*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^7 + e^(f* x + e)*sinh(f*x + e)^8 + 4*(7*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^6 + 8*(7*cosh(f*x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^5 + 2*(35*cosh(f*x + e)^4 + 30*cosh(f*x + e)^2 + 3)*e^(f*x + e)*sinh(f*x + e)^4 + 8*(7*cosh(f*x + e)^5 + 10*cosh(f*x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^3 + 4*(7*cosh(f*x + e)^6 + 15*cosh(f*x + e)^4 + 9*cosh (f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^2 + 8*(cosh(f*x + e)^7 + 3*cosh (f*x + e)^5 + 3*cosh(f*x + e)^3 + cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) + (cosh(f*x + e)^8 + 4*cosh(f*x + e)^6 + 6*cosh(f*x + e)^4 + 4*cosh(f*x + e)^2 + 1)*e^(f*x + e))*arctan(cosh(f*x + e) + sinh(f*x + e)) + (cosh(f*x + e)^7 - 7*cosh(f*x + e)^5 + 7*cosh(f*x + e)^3 - cosh(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^2*f*c osh(f*x + e)^8 + 4*a^2*f*cosh(f*x + e)^6 + (a^2*f*e^(2*f*x + 2*e) + a^2*f) *sinh(f*x + e)^8 + 8*(a^2*f*cosh(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(...
\[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\tanh ^{2}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 369 vs. \(2 (94) = 188\).
Time = 0.30 (sec) , antiderivative size = 369, normalized size of antiderivative = 3.48 \[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {3 \, e^{\left (-f x - e\right )} + 11 \, e^{\left (-3 \, f x - 3 \, e\right )} - 11 \, e^{\left (-5 \, f x - 5 \, e\right )} - 3 \, e^{\left (-7 \, f x - 7 \, e\right )}}{4 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}}} - \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}}}{8 \, f} + \frac {15 \, e^{\left (-f x - e\right )} + 55 \, e^{\left (-3 \, f x - 3 \, e\right )} + 73 \, e^{\left (-5 \, f x - 5 \, e\right )} - 15 \, e^{\left (-7 \, f x - 7 \, e\right )}}{48 \, {\left (4 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}}\right )} f} + \frac {15 \, e^{\left (-f x - e\right )} - 73 \, e^{\left (-3 \, f x - 3 \, e\right )} - 55 \, e^{\left (-5 \, f x - 5 \, e\right )} - 15 \, e^{\left (-7 \, f x - 7 \, e\right )}}{48 \, {\left (4 \, a^{\frac {3}{2}} e^{\left (-2 \, f x - 2 \, e\right )} + 6 \, a^{\frac {3}{2}} e^{\left (-4 \, f x - 4 \, e\right )} + 4 \, a^{\frac {3}{2}} e^{\left (-6 \, f x - 6 \, e\right )} + a^{\frac {3}{2}} e^{\left (-8 \, f x - 8 \, e\right )} + a^{\frac {3}{2}}\right )} f} - \frac {5 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{8 \, a^{\frac {3}{2}} f} \]
-1/8*((3*e^(-f*x - e) + 11*e^(-3*f*x - 3*e) - 11*e^(-5*f*x - 5*e) - 3*e^(- 7*f*x - 7*e))/(4*a^(3/2)*e^(-2*f*x - 2*e) + 6*a^(3/2)*e^(-4*f*x - 4*e) + 4 *a^(3/2)*e^(-6*f*x - 6*e) + a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)) - 3*arctan (e^(-f*x - e))/a^(3/2))/f + 1/48*(15*e^(-f*x - e) + 55*e^(-3*f*x - 3*e) + 73*e^(-5*f*x - 5*e) - 15*e^(-7*f*x - 7*e))/((4*a^(3/2)*e^(-2*f*x - 2*e) + 6*a^(3/2)*e^(-4*f*x - 4*e) + 4*a^(3/2)*e^(-6*f*x - 6*e) + a^(3/2)*e^(-8*f* x - 8*e) + a^(3/2))*f) + 1/48*(15*e^(-f*x - e) - 73*e^(-3*f*x - 3*e) - 55* e^(-5*f*x - 5*e) - 15*e^(-7*f*x - 7*e))/((4*a^(3/2)*e^(-2*f*x - 2*e) + 6*a ^(3/2)*e^(-4*f*x - 4*e) + 4*a^(3/2)*e^(-6*f*x - 6*e) + a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2))*f) - 5/8*arctan(e^(-f*x - e))/(a^(3/2)*f)
Exception generated. \[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {\tanh ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {tanh}\left (e+f\,x\right )}^2}{{\left (a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]