Integrand size = 25, antiderivative size = 96 \[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {\coth (e+f x)}{f \sqrt {a \cosh ^2(e+f x)}}-\frac {2 \coth (e+f x) \text {csch}^2(e+f x)}{3 f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}^4(e+f x)}{5 f \sqrt {a \cosh ^2(e+f x)}} \]
-coth(f*x+e)/f/(a*cosh(f*x+e)^2)^(1/2)-2/3*coth(f*x+e)*csch(f*x+e)^2/f/(a* cosh(f*x+e)^2)^(1/2)-1/5*coth(f*x+e)*csch(f*x+e)^4/f/(a*cosh(f*x+e)^2)^(1/ 2)
Time = 0.07 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51 \[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {\coth (e+f x) \left (15+10 \text {csch}^2(e+f x)+3 \text {csch}^4(e+f x)\right )}{15 f \sqrt {a \cosh ^2(e+f x)}} \]
-1/15*(Coth[e + f*x]*(15 + 10*Csch[e + f*x]^2 + 3*Csch[e + f*x]^4))/(f*Sqr t[a*Cosh[e + f*x]^2])
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 25, 3655, 25, 3042, 25, 3686, 25, 3042, 3086, 210, 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.
\(\displaystyle \int \frac {\coth ^6(e+f x)}{\sqrt {a \sinh ^2(e+f x)+a}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\tan (i e+i f x)^6 \sqrt {a-a \sin (i e+i f x)^2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\sqrt {a-a \sin (i e+i f x)^2} \tan (i e+i f x)^6}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle -\int -\frac {\coth ^6(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth ^6(e+f x)}{\sqrt {a \cosh ^2(e+f x)}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (i e+i f x+\frac {\pi }{2}\right )^6}{\sqrt {a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^6}{\sqrt {a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle -\frac {\cosh (e+f x) \int -\coth ^5(e+f x) \text {csch}(e+f x)dx}{\sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\cosh (e+f x) \int \coth ^5(e+f x) \text {csch}(e+f x)dx}{\sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh (e+f x) \int \sec \left (i e+i f x-\frac {\pi }{2}\right ) \tan \left (i e+i f x-\frac {\pi }{2}\right )^5dx}{\sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {i \cosh (e+f x) \int \left (-\text {csch}^2(e+f x)-1\right )^2d(-i \text {csch}(e+f x))}{f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 210 |
\(\displaystyle -\frac {i \cosh (e+f x) \int \left (\text {csch}^4(e+f x)+2 \text {csch}^2(e+f x)+1\right )d(-i \text {csch}(e+f x))}{f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \cosh (e+f x) \left (-\frac {1}{5} i \text {csch}^5(e+f x)-\frac {2}{3} i \text {csch}^3(e+f x)-i \text {csch}(e+f x)\right )}{f \sqrt {a \cosh ^2(e+f x)}}\) |
((-I)*Cosh[e + f*x]*((-I)*Csch[e + f*x] - ((2*I)/3)*Csch[e + f*x]^3 - (I/5 )*Csch[e + f*x]^5))/(f*Sqrt[a*Cosh[e + f*x]^2])
3.5.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_.), x_Symbol] :> Simp[a/f Subst[Int[(a*x)^(m - 1)*(-1 + x^2)^((n - 1)/2 ), x], x, Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2 ] && !(IntegerQ[m/2] && LtQ[0, m, n + 1])
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]])
Time = 0.22 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.77
method | result | size |
default | \(-\frac {\cosh \left (f x +e \right ) \left (15 \cosh \left (f x +e \right )^{4}-20 \cosh \left (f x +e \right )^{2}+8\right )}{15 \left (\cosh \left (f x +e \right )+1\right )^{2} \left (\cosh \left (f x +e \right )-1\right )^{2} \sinh \left (f x +e \right ) \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(74\) |
risch | \(-\frac {2 \left ({\mathrm e}^{2 f x +2 e}+1\right ) \left (15 \,{\mathrm e}^{8 f x +8 e}-20 \,{\mathrm e}^{6 f x +6 e}+58 \,{\mathrm e}^{4 f x +4 e}-20 \,{\mathrm e}^{2 f x +2 e}+15\right )}{15 \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, f \left ({\mathrm e}^{2 f x +2 e}-1\right )^{5}}\) | \(102\) |
-1/15*cosh(f*x+e)*(15*cosh(f*x+e)^4-20*cosh(f*x+e)^2+8)/(cosh(f*x+e)+1)^2/ (cosh(f*x+e)-1)^2/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. 1399 vs. \(2 (86) = 172\).
Time = 0.26 (sec) , antiderivative size = 1399, normalized size of antiderivative = 14.57 \[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
-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...
\[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\int \frac {\coth ^{6}{\left (e + f x \right )}}{\sqrt {a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1231 vs. \(2 (86) = 172\).
Time = 0.36 (sec) , antiderivative size = 1231, normalized size of antiderivative = 12.82 \[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\text {Too large to display} \]
-1/256*(120*arctan(e^(-f*x - e))/sqrt(a) + 45*log(e^(-f*x - e) + 1)/sqrt(a ) - 45*log(e^(-f*x - e) - 1)/sqrt(a) + 2*(105*sqrt(a)*e^(-f*x - e) - 530*s qrt(a)*e^(-3*f*x - 3*e) + 328*sqrt(a)*e^(-5*f*x - 5*e) - 110*sqrt(a)*e^(-7 *f*x - 7*e) + 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2*f*x - 2*e) - 10*a*e^ (-4*f*x - 4*e) + 10*a*e^(-6*f*x - 6*e) - 5*a*e^(-8*f*x - 8*e) + a*e^(-10*f *x - 10*e) - a))/f - 1/256*(120*arctan(e^(-f*x - e))/sqrt(a) - 45*log(e^(- f*x - e) + 1)/sqrt(a) + 45*log(e^(-f*x - e) - 1)/sqrt(a) + 2*(15*sqrt(a)*e ^(-f*x - e) - 110*sqrt(a)*e^(-3*f*x - 3*e) + 328*sqrt(a)*e^(-5*f*x - 5*e) - 530*sqrt(a)*e^(-7*f*x - 7*e) + 105*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2* f*x - 2*e) - 10*a*e^(-4*f*x - 4*e) + 10*a*e^(-6*f*x - 6*e) - 5*a*e^(-8*f*x - 8*e) + a*e^(-10*f*x - 10*e) - a))/f + 1/320*(60*arctan(e^(-f*x - e))/sq rt(a) + 75*log(e^(-f*x - e) + 1)/sqrt(a) - 75*log(e^(-f*x - e) - 1)/sqrt(a ) + 2*(105*sqrt(a)*e^(-f*x - e) + 130*sqrt(a)*e^(-3*f*x - 3*e) - 284*sqrt( a)*e^(-5*f*x - 5*e) + 190*sqrt(a)*e^(-7*f*x - 7*e) - 45*sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2*f*x - 2*e) - 10*a*e^(-4*f*x - 4*e) + 10*a*e^(-6*f*x - 6 *e) - 5*a*e^(-8*f*x - 8*e) + a*e^(-10*f*x - 10*e) - a))/f + 1/320*(60*arct an(e^(-f*x - e))/sqrt(a) - 75*log(e^(-f*x - e) + 1)/sqrt(a) + 75*log(e^(-f *x - e) - 1)/sqrt(a) - 2*(45*sqrt(a)*e^(-f*x - e) - 190*sqrt(a)*e^(-3*f*x - 3*e) + 284*sqrt(a)*e^(-5*f*x - 5*e) - 130*sqrt(a)*e^(-7*f*x - 7*e) - 105 *sqrt(a)*e^(-9*f*x - 9*e))/(5*a*e^(-2*f*x - 2*e) - 10*a*e^(-4*f*x - 4*e...
Exception generated. \[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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 = 1.66 (sec) , antiderivative size = 381, normalized size of antiderivative = 3.97 \[ \int \frac {\coth ^6(e+f x)}{\sqrt {a+a \sinh ^2(e+f x)}} \, dx=-\frac {4\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{a\,f\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {32\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {352\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \]
- (4*exp(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))) - (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))) - (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)))