Integrand size = 25, antiderivative size = 77 \[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}}-\frac {\coth (e+f x) \text {csch}^4(e+f x)}{5 a f \sqrt {a \cosh ^2(e+f x)}} \]
-1/3*coth(f*x+e)*csch(f*x+e)^2/a/f/(a*cosh(f*x+e)^2)^(1/2)-1/5*coth(f*x+e) *csch(f*x+e)^4/a/f/(a*cosh(f*x+e)^2)^(1/2)
Time = 0.08 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.53 \[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\coth ^3(e+f x) \left (5+3 \text {csch}^2(e+f x)\right )}{15 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Result contains complex when optimal does not.
Time = 0.45 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.77, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 25, 3655, 25, 3042, 25, 3686, 25, 3042, 3086, 25, 244, 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.
\(\displaystyle \int \frac {\coth ^6(e+f x)}{\left (a \sinh ^2(e+f x)+a\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {1}{\tan (i e+i f x)^6 \left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {1}{\left (a-a \sin (i e+i f x)^2\right )^{3/2} \tan (i e+i f x)^6}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle -\int -\frac {\coth ^6(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\coth ^6(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\tan \left (i e+i f x+\frac {\pi }{2}\right )^6}{\left (a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^6}{\left (a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle -\frac {\cosh (e+f x) \int -\coth ^3(e+f x) \text {csch}^3(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\cosh (e+f x) \int \coth ^3(e+f x) \text {csch}^3(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh (e+f x) \int \sec \left (i e+i f x-\frac {\pi }{2}\right )^3 \tan \left (i e+i f x-\frac {\pi }{2}\right )^3dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle -\frac {i \cosh (e+f x) \int \text {csch}^2(e+f x) \left (\text {csch}^2(e+f x)+1\right )d(-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {i \cosh (e+f x) \int -\text {csch}^2(e+f x) \left (\text {csch}^2(e+f x)+1\right )d(-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {i \cosh (e+f x) \int \left (-\text {csch}^4(e+f x)-\text {csch}^2(e+f x)\right )d(-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \cosh (e+f x) \left (-\frac {1}{5} i \text {csch}^5(e+f x)-\frac {1}{3} i \text {csch}^3(e+f x)\right )}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
((-I)*Cosh[e + f*x]*((-1/3*I)*Csch[e + f*x]^3 - (I/5)*Csch[e + f*x]^5))/(a *f*Sqrt[a*Cosh[e + f*x]^2])
3.5.55.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, 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 = 0.19 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.61
method | result | size |
default | \(-\frac {\cosh \left (f x +e \right ) \left (5 \sinh \left (f x +e \right )^{2}+3\right )}{15 a \sinh \left (f x +e \right )^{5} \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(47\) |
risch | \(-\frac {8 \left (5 \,{\mathrm e}^{4 f x +4 e}+2 \,{\mathrm e}^{2 f x +2 e}+5\right ) \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{2 f x +2 e}}{15 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{5} f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\) | \(92\) |
Leaf count of result is larger than twice the leaf count of optimal. 1410 vs. \(2 (69) = 138\).
Time = 0.29 (sec) , antiderivative size = 1410, normalized size of antiderivative = 18.31 \[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-8/15*(35*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^6 + 5*e^(f*x + e)*sinh(f *x + e)^7 + (105*cosh(f*x + e)^2 + 2)*e^(f*x + e)*sinh(f*x + e)^5 + 5*(35* cosh(f*x + e)^3 + 2*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^4 + 5*(35*cos h(f*x + e)^4 + 4*cosh(f*x + e)^2 + 1)*e^(f*x + e)*sinh(f*x + e)^3 + 5*(21* cosh(f*x + e)^5 + 4*cosh(f*x + e)^3 + 3*cosh(f*x + e))*e^(f*x + e)*sinh(f* x + e)^2 + 5*(7*cosh(f*x + e)^6 + 2*cosh(f*x + e)^4 + 3*cosh(f*x + e)^2)*e ^(f*x + e)*sinh(f*x + e) + (5*cosh(f*x + e)^7 + 2*cosh(f*x + e)^5 + 5*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)^10 - 5*a^2*f*cosh(f*x + e)^8 + (a^2*f* e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^10 + 10*(a^2*f*cosh(f*x + e)*e^(2*f *x + 2*e) + a^2*f*cosh(f*x + e))*sinh(f*x + e)^9 + 10*a^2*f*cosh(f*x + e)^ 6 + 5*(9*a^2*f*cosh(f*x + e)^2 - a^2*f + (9*a^2*f*cosh(f*x + e)^2 - a^2*f) *e^(2*f*x + 2*e))*sinh(f*x + e)^8 + 40*(3*a^2*f*cosh(f*x + e)^3 - a^2*f*co sh(f*x + e) + (3*a^2*f*cosh(f*x + e)^3 - a^2*f*cosh(f*x + e))*e^(2*f*x + 2 *e))*sinh(f*x + e)^7 - 10*a^2*f*cosh(f*x + e)^4 + 10*(21*a^2*f*cosh(f*x + e)^4 - 14*a^2*f*cosh(f*x + e)^2 + a^2*f + (21*a^2*f*cosh(f*x + e)^4 - 14*a ^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2*e))*sinh(f*x + e)^6 + 4*(63*a^2 *f*cosh(f*x + e)^5 - 70*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e) + ( 63*a^2*f*cosh(f*x + e)^5 - 70*a^2*f*cosh(f*x + e)^3 + 15*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e)^5 + 5*a^2*f*cosh(f*x + e)^2 + 10*(21...
\[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\coth ^{6}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 1531 vs. \(2 (69) = 138\).
Time = 0.39 (sec) , antiderivative size = 1531, normalized size of antiderivative = 19.88 \[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
-3/256*(2*(105*e^(-f*x - e) - 300*e^(-3*f*x - 3*e) + 81*e^(-5*f*x - 5*e) - 248*e^(-7*f*x - 7*e) + 51*e^(-9*f*x - 9*e) + 100*e^(-11*f*x - 11*e) - 45* e^(-13*f*x - 13*e))/(3*a^(3/2)*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-1 0*f*x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2)) + 60*arctan(e^(-f*x - e))/a^(3/2) + 75*log(e^(-f*x - e) + 1)/a^ (3/2) - 75*log(e^(-f*x - e) - 1)/a^(3/2))/f + 1/48*((105*e^(-f*x - e) - 35 0*e^(-3*f*x - 3*e) + 231*e^(-5*f*x - 5*e) + 412*e^(-7*f*x - 7*e) + 231*e^( -9*f*x - 9*e) - 350*e^(-11*f*x - 11*e) + 105*e^(-13*f*x - 13*e))/(3*a^(3/2 )*e^(-2*f*x - 2*e) - a^(3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^( -12*f*x - 12*e) + a^(3/2)*e^(-14*f*x - 14*e) - a^(3/2)) + 105*arctan(e^(-f *x - e))/a^(3/2))/f + 3/256*(2*(45*e^(-f*x - e) - 100*e^(-3*f*x - 3*e) - 5 1*e^(-5*f*x - 5*e) + 248*e^(-7*f*x - 7*e) - 81*e^(-9*f*x - 9*e) + 300*e^(- 11*f*x - 11*e) - 105*e^(-13*f*x - 13*e))/(3*a^(3/2)*e^(-2*f*x - 2*e) - a^( 3/2)*e^(-4*f*x - 4*e) - 5*a^(3/2)*e^(-6*f*x - 6*e) + 5*a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - 3*a^(3/2)*e^(-12*f*x - 12*e) + a^(3/2 )*e^(-14*f*x - 14*e) - a^(3/2)) - 60*arctan(e^(-f*x - e))/a^(3/2) + 75*log (e^(-f*x - e) + 1)/a^(3/2) - 75*log(e^(-f*x - e) - 1)/a^(3/2))/f - 3/320*( 4*(45*e^(-f*x - e) - 135*e^(-3*f*x - 3*e) + 54*e^(-5*f*x - 5*e) + 198*e...
Exception generated. \[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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 = 0.18 (sec) , antiderivative size = 305, normalized size of antiderivative = 3.96 \[ \int \frac {\coth ^6(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {272\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{15\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \]
- (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))) - ( 272*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))) - (1 28*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/( 5*a^2*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 ^2*f*(exp(2*e + 2*f*x) - 1)^5*(exp(e + f*x) + exp(3*e + 3*f*x)))