Integrand size = 25, antiderivative size = 38 \[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}} \]
Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.76 \[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {\coth ^3(e+f x)}{3 f \left (a \cosh ^2(e+f x)\right )^{3/2}} \]
Time = 0.41 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 3655, 3042, 3686, 3042, 25, 3086, 15}
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.
\(\displaystyle \int \frac {\coth ^4(e+f x)}{\left (a \sinh ^2(e+f x)+a\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\tan (i e+i f x)^4 \left (a-a \sin (i e+i f x)^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle \int \frac {\coth ^4(e+f x)}{\left (a \cosh ^2(e+f x)\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\tan \left (i e+i f x+\frac {\pi }{2}\right )^4}{\left (a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2\right )^{3/2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\cosh (e+f x) \int \coth (e+f x) \text {csch}^3(e+f x)dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh (e+f x) \int -\sec \left (i e+i f x-\frac {\pi }{2}\right )^3 \tan \left (i e+i f x-\frac {\pi }{2}\right )dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\cosh (e+f x) \int \sec \left (\frac {1}{2} (2 i e-\pi )+i f x\right )^3 \tan \left (\frac {1}{2} (2 i e-\pi )+i f x\right )dx}{a \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 3086 |
\(\displaystyle \frac {i \cosh (e+f x) \int -\text {csch}^2(e+f x)d(-i \text {csch}(e+f x))}{a f \sqrt {a \cosh ^2(e+f x)}}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {\coth (e+f x) \text {csch}^2(e+f x)}{3 a f \sqrt {a \cosh ^2(e+f x)}}\) |
3.5.54.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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 = 0.20 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.92
method | result | size |
default | \(-\frac {\cosh \left (f x +e \right )}{3 \sinh \left (f x +e \right )^{3} a \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(35\) |
risch | \(-\frac {8 \left ({\mathrm e}^{2 f x +2 e}+1\right ) {\mathrm e}^{2 f x +2 e}}{3 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{3} f \sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, a}\) | \(68\) |
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 612, normalized size of antiderivative = 16.11 \[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {8 \, {\left (\cosh \left (f x + e\right )^{3} e^{\left (f x + e\right )} + 3 \, \cosh \left (f x + e\right )^{2} e^{\left (f x + e\right )} \sinh \left (f x + e\right ) + 3 \, \cosh \left (f x + e\right ) e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{2} + e^{\left (f x + e\right )} \sinh \left (f x + e\right )^{3}\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 )}}{3 \, {\left (a^{2} f \cosh \left (f x + e\right )^{6} - 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + {\left (a^{2} f e^{\left (2 \, f x + 2 \, e\right )} + a^{2} f\right )} \sinh \left (f x + e\right )^{6} + 6 \, {\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 )^{5} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{4} + 4 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} - 3 \, a^{2} f \cosh \left (f x + e\right ) + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{3} - 3 \, a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{3} - a^{2} f + 3 \, {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} - 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f + {\left (5 \, a^{2} f \cosh \left (f x + e\right )^{4} - 6 \, a^{2} f \cosh \left (f x + e\right )^{2} + a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )^{2} + {\left (a^{2} f \cosh \left (f x + e\right )^{6} - 3 \, a^{2} f \cosh \left (f x + e\right )^{4} + 3 \, a^{2} f \cosh \left (f x + e\right )^{2} - a^{2} f\right )} e^{\left (2 \, f x + 2 \, e\right )} + 6 \, {\left (a^{2} f \cosh \left (f x + e\right )^{5} - 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right ) + {\left (a^{2} f \cosh \left (f x + e\right )^{5} - 2 \, a^{2} f \cosh \left (f x + e\right )^{3} + a^{2} f \cosh \left (f x + e\right )\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} \sinh \left (f x + e\right )\right )}} \]
-8/3*(cosh(f*x + e)^3*e^(f*x + e) + 3*cosh(f*x + e)^2*e^(f*x + e)*sinh(f*x + e) + 3*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^2 + e^(f*x + e)*sinh(f*x + e)^3)*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)^6 - 3*a^2*f*cosh(f*x + e)^4 + (a^2*f*e^(2*f*x + 2*e) + a^2*f)*sinh(f*x + e)^6 + 6*(a^2*f*cosh(f*x + e)*e^(2*f*x + 2*e) + a^2*f*co sh(f*x + e))*sinh(f*x + e)^5 + 3*a^2*f*cosh(f*x + e)^2 + 3*(5*a^2*f*cosh(f *x + e)^2 - a^2*f + (5*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e))*sin h(f*x + e)^4 + 4*(5*a^2*f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e) + (5*a^2 *f*cosh(f*x + e)^3 - 3*a^2*f*cosh(f*x + e))*e^(2*f*x + 2*e))*sinh(f*x + e) ^3 - a^2*f + 3*(5*a^2*f*cosh(f*x + e)^4 - 6*a^2*f*cosh(f*x + e)^2 + a^2*f + (5*a^2*f*cosh(f*x + e)^4 - 6*a^2*f*cosh(f*x + e)^2 + a^2*f)*e^(2*f*x + 2 *e))*sinh(f*x + e)^2 + (a^2*f*cosh(f*x + e)^6 - 3*a^2*f*cosh(f*x + e)^4 + 3*a^2*f*cosh(f*x + e)^2 - a^2*f)*e^(2*f*x + 2*e) + 6*(a^2*f*cosh(f*x + e)^ 5 - 2*a^2*f*cosh(f*x + e)^3 + a^2*f*cosh(f*x + e) + (a^2*f*cosh(f*x + e)^5 - 2*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))
\[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\int \frac {\coth ^{4}{\left (e + f x \right )}}{\left (a \left (\sinh ^{2}{\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (34) = 68\).
Time = 0.33 (sec) , antiderivative size = 823, normalized size of antiderivative = 21.66 \[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Too large to display} \]
1/12*((21*e^(-f*x - e) - 16*e^(-3*f*x - 3*e) + 34*e^(-5*f*x - 5*e) + 8*e^( -7*f*x - 7*e) - 15*e^(-9*f*x - 9*e))/(a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2) *e^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - a^(3/2)) + 3*arctan(e^(-f*x - e))/a^(3/2) + 9*log(e^(-f*x - e) + 1)/a^(3/2) - 9*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/ 12*((15*e^(-f*x - e) - 8*e^(-3*f*x - 3*e) - 34*e^(-5*f*x - 5*e) + 16*e^(-7 *f*x - 7*e) - 21*e^(-9*f*x - 9*e))/(a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2)*e ^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^(3/2)*e^(-10*f*x - 10*e) - a^(3/2)) - 3*arctan(e^(-f*x - e))/a^(3/2) + 9 *log(e^(-f*x - e) + 1)/a^(3/2) - 9*log(e^(-f*x - e) - 1)/a^(3/2))/f - 1/8* ((15*e^(-f*x - e) - 20*e^(-3*f*x - 3*e) - 22*e^(-5*f*x - 5*e) - 20*e^(-7*f *x - 7*e) + 15*e^(-9*f*x - 9*e))/(a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2)*e^( -4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e) + a^ (3/2)*e^(-10*f*x - 10*e) - a^(3/2)) + 15*arctan(e^(-f*x - e))/a^(3/2))/f + 1/48*(45*e^(-f*x - e) - 52*e^(-3*f*x - 3*e) - 74*e^(-5*f*x - 5*e) + 92*e^ (-7*f*x - 7*e) + 21*e^(-9*f*x - 9*e))/((a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/ 2)*e^(-4*f*x - 4*e) - 2*a^(3/2)*e^(-6*f*x - 6*e) - a^(3/2)*e^(-8*f*x - 8*e ) + a^(3/2)*e^(-10*f*x - 10*e) - a^(3/2))*f) + 1/48*(21*e^(-f*x - e) + 92* e^(-3*f*x - 3*e) - 74*e^(-5*f*x - 5*e) - 52*e^(-7*f*x - 7*e) + 45*e^(-9*f* x - 9*e))/((a^(3/2)*e^(-2*f*x - 2*e) + 2*a^(3/2)*e^(-4*f*x - 4*e) - 2*a...
Exception generated. \[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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 = 1.74 (sec) , antiderivative size = 71, normalized size of antiderivative = 1.87 \[ \int \frac {\coth ^4(e+f x)}{\left (a+a \sinh ^2(e+f x)\right )^{3/2}} \, dx=-\frac {16\,{\mathrm {e}}^{4\,e+4\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{3\,a^2\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}+1\right )} \]