Integrand size = 25, antiderivative size = 124 \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=-\frac {3 \sqrt {a \cosh ^2(e+f x)} \text {csch}(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^3(e+f x) \text {sech}(e+f x)}{f}-\frac {\sqrt {a \cosh ^2(e+f x)} \text {csch}^5(e+f x) \text {sech}(e+f x)}{5 f}+\frac {\sqrt {a \cosh ^2(e+f x)} \tanh (e+f x)}{f} \]
-3*csch(f*x+e)*sech(f*x+e)*(a*cosh(f*x+e)^2)^(1/2)/f-csch(f*x+e)^3*sech(f* x+e)*(a*cosh(f*x+e)^2)^(1/2)/f-1/5*csch(f*x+e)^5*sech(f*x+e)*(a*cosh(f*x+e )^2)^(1/2)/f+(a*cosh(f*x+e)^2)^(1/2)*tanh(f*x+e)/f
Time = 0.14 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.54 \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\frac {\sqrt {a \cosh ^2(e+f x)} (-182+235 \cosh (2 (e+f x))-90 \cosh (4 (e+f x))+5 \cosh (6 (e+f x))) \text {csch}^5(e+f x) \text {sech}(e+f x)}{160 f} \]
(Sqrt[a*Cosh[e + f*x]^2]*(-182 + 235*Cosh[2*(e + f*x)] - 90*Cosh[4*(e + f* x)] + 5*Cosh[6*(e + f*x)])*Csch[e + f*x]^5*Sech[e + f*x])/(160*f)
Result contains complex when optimal does not.
Time = 0.44 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.60, 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, 25, 3070, 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.
\(\displaystyle \int \coth ^6(e+f x) \sqrt {a \sinh ^2(e+f x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sqrt {a-a \sin (i e+i f x)^2}}{\tan (i e+i f x)^6}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\sqrt {a-a \sin (i e+i f x)^2}}{\tan (i e+i f x)^6}dx\) |
\(\Big \downarrow \) 3655 |
\(\displaystyle -\int -\sqrt {a \cosh ^2(e+f x)} \coth ^6(e+f x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \coth ^6(e+f x) \sqrt {a \cosh ^2(e+f x)}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \tan \left (i e+i f x+\frac {\pi }{2}\right )^6 \left (-\sqrt {a \sin \left (i e+i f x+\frac {\pi }{2}\right )^2}\right )dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \sqrt {a \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^2} \tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^6dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \text {sech}(e+f x) \left (-\sqrt {a \cosh ^2(e+f x)}\right ) \int -\cosh (e+f x) \coth ^6(e+f x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int \cosh (e+f x) \coth ^6(e+f x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int -\sin \left (i e+i f x+\frac {\pi }{2}\right ) \tan \left (i e+i f x+\frac {\pi }{2}\right )^6dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \text {sech}(e+f x) \left (-\sqrt {a \cosh ^2(e+f x)}\right ) \int \sin \left (\frac {1}{2} (2 i e+\pi )+i f x\right ) \tan \left (\frac {1}{2} (2 i e+\pi )+i f x\right )^6dx\) |
\(\Big \downarrow \) 3070 |
\(\displaystyle -\frac {i \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int -\text {csch}^6(e+f x) \left (\sinh ^2(e+f x)+1\right )^3d(-i \sinh (e+f x))}{f}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {i \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \int \left (-\text {csch}^6(e+f x)-3 \text {csch}^4(e+f x)-3 \text {csch}^2(e+f x)-1\right )d(-i \sinh (e+f x))}{f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i \text {sech}(e+f x) \sqrt {a \cosh ^2(e+f x)} \left (i \sinh (e+f x)-\frac {1}{5} i \text {csch}^5(e+f x)-i \text {csch}^3(e+f x)-3 i \text {csch}(e+f x)\right )}{f}\) |
((-I)*Sqrt[a*Cosh[e + f*x]^2]*Sech[e + f*x]*((-3*I)*Csch[e + f*x] - I*Csch [e + f*x]^3 - (I/5)*Csch[e + f*x]^5 + I*Sinh[e + f*x]))/f
3.5.36.3.1 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[sin[(e_.) + (f_.)*(x_)]^(m_.)*tan[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Simp[-f^(-1) Subst[Int[(1 - x^2)^((m + n - 1)/2)/x^n, x], x, Cos[e + f *x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n - 1)/2]
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 = 85, normalized size of antiderivative = 0.69
method | result | size |
default | \(\frac {\cosh \left (f x +e \right ) a \left (5 \sinh \left (f x +e \right )^{6}-15 \sinh \left (f x +e \right )^{4}-5 \sinh \left (f x +e \right )^{2}-1\right )}{5 \left (\cosh \left (f x +e \right )+1\right )^{2} \left (\cosh \left (f x +e \right )-1\right )^{2} \sinh \left (f x +e \right ) \sqrt {a \cosh \left (f x +e \right )^{2}}\, f}\) | \(85\) |
risch | \(-\frac {\sqrt {\left ({\mathrm e}^{2 f x +2 e}+1\right )^{2} a \,{\mathrm e}^{-2 f x -2 e}}\, \left (-5 \,{\mathrm e}^{12 f x +12 e}+90 \,{\mathrm e}^{10 f x +10 e}-235 \,{\mathrm e}^{8 f x +8 e}+364 \,{\mathrm e}^{6 f x +6 e}-235 \,{\mathrm e}^{4 f x +4 e}+90 \,{\mathrm e}^{2 f x +2 e}-5\right )}{10 \left ({\mathrm e}^{2 f x +2 e}-1\right )^{5} f \left ({\mathrm e}^{2 f x +2 e}+1\right )}\) | \(126\) |
1/5*cosh(f*x+e)*a*(5*sinh(f*x+e)^6-15*sinh(f*x+e)^4-5*sinh(f*x+e)^2-1)/(co sh(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. 1696 vs. \(2 (114) = 228\).
Time = 0.28 (sec) , antiderivative size = 1696, normalized size of antiderivative = 13.68 \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\text {Too large to display} \]
1/10*(60*cosh(f*x + e)*e^(f*x + e)*sinh(f*x + e)^11 + 5*e^(f*x + e)*sinh(f *x + e)^12 + 30*(11*cosh(f*x + e)^2 - 3)*e^(f*x + e)*sinh(f*x + e)^10 + 10 0*(11*cosh(f*x + e)^3 - 9*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^9 + 5*( 495*cosh(f*x + e)^4 - 810*cosh(f*x + e)^2 + 47)*e^(f*x + e)*sinh(f*x + e)^ 8 + 40*(99*cosh(f*x + e)^5 - 270*cosh(f*x + e)^3 + 47*cosh(f*x + e))*e^(f* x + e)*sinh(f*x + e)^7 + 28*(165*cosh(f*x + e)^6 - 675*cosh(f*x + e)^4 + 2 35*cosh(f*x + e)^2 - 13)*e^(f*x + e)*sinh(f*x + e)^6 + 8*(495*cosh(f*x + e )^7 - 2835*cosh(f*x + e)^5 + 1645*cosh(f*x + e)^3 - 273*cosh(f*x + e))*e^( f*x + e)*sinh(f*x + e)^5 + 5*(495*cosh(f*x + e)^8 - 3780*cosh(f*x + e)^6 + 3290*cosh(f*x + e)^4 - 1092*cosh(f*x + e)^2 + 47)*e^(f*x + e)*sinh(f*x + e)^4 + 20*(55*cosh(f*x + e)^9 - 540*cosh(f*x + e)^7 + 658*cosh(f*x + e)^5 - 364*cosh(f*x + e)^3 + 47*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e)^3 + 10 *(33*cosh(f*x + e)^10 - 405*cosh(f*x + e)^8 + 658*cosh(f*x + e)^6 - 546*co sh(f*x + e)^4 + 141*cosh(f*x + e)^2 - 9)*e^(f*x + e)*sinh(f*x + e)^2 + 4*( 15*cosh(f*x + e)^11 - 225*cosh(f*x + e)^9 + 470*cosh(f*x + e)^7 - 546*cosh (f*x + e)^5 + 235*cosh(f*x + e)^3 - 45*cosh(f*x + e))*e^(f*x + e)*sinh(f*x + e) + (5*cosh(f*x + e)^12 - 90*cosh(f*x + e)^10 + 235*cosh(f*x + e)^8 - 364*cosh(f*x + e)^6 + 235*cosh(f*x + e)^4 - 90*cosh(f*x + e)^2 + 5)*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)/(f*c osh(f*x + e)^11 + (f*e^(2*f*x + 2*e) + f)*sinh(f*x + e)^11 + 11*(f*cosh...
Timed out. \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 1050 vs. \(2 (114) = 228\).
Time = 0.31 (sec) , antiderivative size = 1050, normalized size of antiderivative = 8.47 \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=\text {Too large to display} \]
-1/320*(105*sqrt(a)*log(e^(-f*x - e) + 1) - 105*sqrt(a)*log(e^(-f*x - e) - 1) - 2*(375*sqrt(a)*e^(-f*x - e) - 790*sqrt(a)*e^(-3*f*x - 3*e) + 896*sqr t(a)*e^(-5*f*x - 5*e) - 490*sqrt(a)*e^(-7*f*x - 7*e) + 105*sqrt(a)*e^(-9*f *x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f + 1/320*(105*sqrt(a)*lo g(e^(-f*x - e) + 1) - 105*sqrt(a)*log(e^(-f*x - e) - 1) + 2*(105*sqrt(a)*e ^(-f*x - e) - 490*sqrt(a)*e^(-3*f*x - 3*e) + 896*sqrt(a)*e^(-5*f*x - 5*e) - 790*sqrt(a)*e^(-7*f*x - 7*e) + 375*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f* x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f + 1/256*(15*sqrt(a)*log(e^(-f*x - e) + 1) - 1 5*sqrt(a)*log(e^(-f*x - e) - 1) + 2*(15*sqrt(a)*e^(-f*x - e) + 250*sqrt(a) *e^(-3*f*x - 3*e) - 128*sqrt(a)*e^(-5*f*x - 5*e) + 70*sqrt(a)*e^(-7*f*x - 7*e) - 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4 *e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/ f - 1/256*(15*sqrt(a)*log(e^(-f*x - e) + 1) - 15*sqrt(a)*log(e^(-f*x - e) - 1) + 2*(15*sqrt(a)*e^(-f*x - e) - 70*sqrt(a)*e^(-3*f*x - 3*e) + 128*sqrt (a)*e^(-5*f*x - 5*e) - 250*sqrt(a)*e^(-7*f*x - 7*e) - 15*sqrt(a)*e^(-9*f*x - 9*e))/(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) - 5*e^(-8*f*x - 8*e) + e^(-10*f*x - 10*e) - 1))/f + 2*sqrt(a)*e^(-5*f*x - 5 *e)/(f*(5*e^(-2*f*x - 2*e) - 10*e^(-4*f*x - 4*e) + 10*e^(-6*f*x - 6*e) ...
Time = 0.31 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77 \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=-\frac {\sqrt {a} {\left (\frac {4 \, {\left (15 \, {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{4} + 20 \, {\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{2} + 16\right )}}{{\left (e^{\left (f x + e\right )} - e^{\left (-f x - e\right )}\right )}^{5}} - 5 \, e^{\left (f x + e\right )} + 5 \, e^{\left (-f x - e\right )}\right )}}{10 \, f} \]
-1/10*sqrt(a)*(4*(15*(e^(f*x + e) - e^(-f*x - e))^4 + 20*(e^(f*x + e) - e^ (-f*x - e))^2 + 16)/(e^(f*x + e) - e^(-f*x - e))^5 - 5*e^(f*x + e) + 5*e^( -f*x - e))/f
Time = 1.62 (sec) , antiderivative size = 427, normalized size of antiderivative = 3.44 \[ \int \coth ^6(e+f x) \sqrt {a+a \sinh ^2(e+f x)} \, dx=-\frac {\left (\frac {1}{f}-\frac {{\mathrm {e}}^{2\,e+2\,f\,x}}{f}\right )\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{{\mathrm {e}}^{2\,e+2\,f\,x}+1}-\frac {12\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{f\,\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {16\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^2\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {144\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^3\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {128\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^4\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )}-\frac {64\,{\mathrm {e}}^{3\,e+3\,f\,x}\,\sqrt {a+a\,{\left (\frac {{\mathrm {e}}^{e+f\,x}}{2}-\frac {{\mathrm {e}}^{-e-f\,x}}{2}\right )}^2}}{5\,f\,{\left ({\mathrm {e}}^{2\,e+2\,f\,x}-1\right )}^5\,\left ({\mathrm {e}}^{e+f\,x}+{\mathrm {e}}^{3\,e+3\,f\,x}\right )} \]
- ((1/f - exp(2*e + 2*f*x)/f)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2 )^(1/2))/(exp(2*e + 2*f*x) + 1) - (12*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x )/2 - exp(- e - f*x)/2)^2)^(1/2))/(f*(exp(2*e + 2*f*x) - 1)*(exp(e + f*x) + exp(3*e + 3*f*x))) - (16*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x)/2)^2)^(1/2))/(f*(exp(2*e + 2*f*x) - 1)^2*(exp(e + f*x) + exp(3*e + 3*f*x))) - (144*exp(3*e + 3*f*x)*(a + a*(exp(e + f*x)/2 - exp(- e - f*x )/2)^2)^(1/2))/(5*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*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*f*(exp(2*e + 2*f*x) - 1)^5*(exp(e + f*x) + exp(3*e + 3*f*x)))