Integrand size = 25, antiderivative size = 64 \[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\arctan (\sinh (e+f x)) \cosh (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x)}{a f \sqrt {a \cosh ^2(e+f x)}} \]
-arctan(sinh(f*x+e))*cosh(f*x+e)/a/f/(a*cosh(f*x+e)^2)^(1/2)-coth(f*x+e)/a /f/(a*cosh(f*x+e)^2)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.06 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.72 \[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\coth (e+f x) \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\sinh ^2(e+f x)\right )}{a f \sqrt {a \cosh ^2(e+f x)}} \]
-((Coth[e + f*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Sinh[e + f*x]^2])/(a*f*S qrt[a*Cosh[e + f*x]^2]))
Result contains complex when optimal does not.
Time = 0.43 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {3042, 25, 3655, 25, 3042, 25, 3686, 25, 3042, 25, 3101, 25, 262, 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.
\(\displaystyle \int \frac {\coth ^2(e+f x)}{\left (a \sinh ^2(e+f x)+a\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\tan (i e+i f x)^2 \left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\left (a-a \sin (i e+i f x)^2\right )^{3/2} \tan (i e+i f x)^2}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle -\int -\frac {\coth ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth ^2(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (i e+i f x+\frac {\pi }{2}\right )^2}{\left (a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2}{\left (a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle -\frac {\cosh (e+f x) \int -\text {csch}^2(e+f x) \text {sech}(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\cosh (e+f x) \int \text {csch}^2(e+f x) \text {sech}(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh (e+f x) \int -\csc (i e+i f x)^2 \sec (i e+i f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cosh (e+f x) \int \csc (i e+i f x)^2 \sec (i e+i f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3101 |
\(\displaystyle -\frac {i \cosh (e+f x) \int \frac {\text {csch}^2(e+f x)}{\text {csch}^2(e+f x)+1}d(-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {i \cosh (e+f x) \int -\frac {\text {csch}^2(e+f x)}{\text {csch}^2(e+f x)+1}d(-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {i \cosh (e+f x) \left (-\int \frac {1}{\text {csch}^2(e+f x)+1}d(-i \text {csch}(e+f x))-i \text {csch}(e+f x)\right )}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {i \cosh (e+f x) (i \arctan (\text {csch}(e+f x))-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
((-I)*Cosh[e + f*x]*(I*ArcTan[Csch[e + f*x]] - I*Csch[e + f*x]))/(a*f*Sqrt [a*Cosh[e + f*x]^2])
3.5.53.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_S ymbol] :> Simp[-(f*a^n)^(-1) Subst[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n + 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, 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]])
Leaf count of result is larger than twice the leaf count of optimal. \(134\) vs. \(2(60)=120\).
Time = 0.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 2.11
method | result | size |
default | \(-\frac {\cosh \left (f x +e \right ) \sqrt {a \sinh \left (f x +e \right )^{2}}\, \left (-\ln \left (\frac {2 \sqrt {-a}\, \sqrt {a \sinh \left (f x +e \right )^{2}}-2 a}{\cosh \left (f x +e \right )}\right ) a \sinh \left (f x +e \right )^{2}+\sqrt {-a}\, \sqrt {a \sinh \left (f x +e \right )^{2}}\right )}{a^{2} \sqrt {-a}\, \left (\cosh \left (f x +e \right )-1\right ) \left (\cosh \left (f x +e \right )+1\right ) \sinh \left (f x +e \right ) \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(135\) |
risch | \(\frac {i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) {\mathrm e}^{3 f x +3 e}-i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) {\mathrm e}^{3 f x +3 e}-i \ln \left ({\mathrm e}^{f x}-i {\mathrm e}^{-e}\right ) {\mathrm e}^{-f x -e}+i \ln \left ({\mathrm e}^{f x}+i {\mathrm e}^{-e}\right ) {\mathrm e}^{-f x -e}-2 \,{\mathrm e}^{2 f x +2 e}-2}{a \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f \left ({\mathrm e}^{2 f x +2 e}-1\right )}\) | \(160\) |
-1/a^2*cosh(f*x+e)*(a*sinh(f*x+e)^2)^(1/2)*(-ln(2/cosh(f*x+e)*((-a)^(1/2)* (a*sinh(f*x+e)^2)^(1/2)-a))*a*sinh(f*x+e)^2+(-a)^(1/2)*(a*sinh(f*x+e)^2)^( 1/2))/(-a)^(1/2)/(cosh(f*x+e)-1)/(cosh(f*x+e)+1)/sinh(f*x+e)/(a*cosh(f*x+e )^2)^(1/2)/f
Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (60) = 120\).
Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 3.97 \[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {2 \, {\left ({\left (2 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + {\left (\cosh \left (f x + e\right )^{2} - 1\right )} e^{\left (f x + e\right )}\right )} \arctan \left (\cosh \left (f x + e\right ) + \sinh \left (f x + e\right )\right ) + \cosh \left (f x + e\right ) e^{\left (f x + e\right )} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )\right )} \sqrt {a e^{\left (4 \, f x + 4 \, e\right )} + 2 \, a e^{\left (2 \, f x + 2 \, e\right )} + a} e^{\left (-f x - e\right )}}{a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 2 \, {\left (a^{2} f \cosh \left (f x + e\right ) e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f \cosh \left (f x + e\right )\right )} \sinh \left (f x + e\right )} \]
-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))*arctan(cosh(f*x + e) + sinh(f*x + e)) + cosh(f*x + e)*e^(f*x + e) + e^(f*x + e)*sinh(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)^2 - a^2*f + (a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^2 + (a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e) + 2*(a^2*f*cosh(f*x + e)*e^(2*f*x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e))
\[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\coth ^{2}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 321 vs. \(2 (60) = 120\).
Time = 0.30 (sec) , antiderivative size = 321, normalized size of antiderivative = 5.02 \[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} + 2 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} + 3 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}} - \frac {3 \, \arctan \left (e^{\left (-f x - e\right )}\right )}{a^{\frac {3}{2}}}}{2 \, f} - \frac {5 \, \sqrt {a} e^{\left (-f x - e\right )} + 6 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 3 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{4 \, {\left (a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}\right )} f} + \frac {3 \, \sqrt {a} e^{\left (-f x - e\right )} - 6 \, \sqrt {a} e^{\left (-3 \, f x - 3 \, e\right )} - 5 \, \sqrt {a} e^{\left (-5 \, f x - 5 \, e\right )}}{4 \, {\left (a^{2} e^{\left (-2 \, f x - 2 \, e\right )} - a^{2} e^{\left (-4 \, f x - 4 \, e\right )} - a^{2} e^{\left (-6 \, f x - 6 \, e\right )} + a^{2}\right )} f} + \frac {\arctan \left (e^{\left (-f x - e\right )}\right )}{2 \, a^{\frac {3}{2}} f} \]
-1/2*((3*sqrt(a)*e^(-f*x - e) + 2*sqrt(a)*e^(-3*f*x - 3*e) + 3*sqrt(a)*e^( -5*f*x - 5*e))/(a^2*e^(-2*f*x - 2*e) - a^2*e^(-4*f*x - 4*e) - a^2*e^(-6*f* x - 6*e) + a^2) - 3*arctan(e^(-f*x - e))/a^(3/2))/f - 1/4*(5*sqrt(a)*e^(-f *x - e) + 6*sqrt(a)*e^(-3*f*x - 3*e) - 3*sqrt(a)*e^(-5*f*x - 5*e))/((a^2*e ^(-2*f*x - 2*e) - a^2*e^(-4*f*x - 4*e) - a^2*e^(-6*f*x - 6*e) + a^2)*f) + 1/4*(3*sqrt(a)*e^(-f*x - e) - 6*sqrt(a)*e^(-3*f*x - 3*e) - 5*sqrt(a)*e^(-5 *f*x - 5*e))/((a^2*e^(-2*f*x - 2*e) - a^2*e^(-4*f*x - 4*e) - a^2*e^(-6*f*x - 6*e) + a^2)*f) + 1/2*arctan(e^(-f*x - e))/(a^(3/2)*f)
Exception generated. \[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\coth ^2(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {{\mathrm {coth}\left (e+f\,x\right )}^2}{{\left (a\,{\mathrm {sinh}\left (e+f\,x\right )}^2+a\right )}^{3/2}} \,d x \]