Integrand size = 10, antiderivative size = 117 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=\frac {\cosh (x) \sinh (x)}{a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh (x)}{3 a^2 \sqrt {a \cosh ^4(x)}}+\frac {6 \sinh ^2(x) \tanh ^3(x)}{5 a^2 \sqrt {a \cosh ^4(x)}}-\frac {4 \sinh ^2(x) \tanh ^5(x)}{7 a^2 \sqrt {a \cosh ^4(x)}}+\frac {\sinh ^2(x) \tanh ^7(x)}{9 a^2 \sqrt {a \cosh ^4(x)}} \]
cosh(x)*sinh(x)/a^2/(a*cosh(x)^4)^(1/2)-4/3*sinh(x)^2*tanh(x)/a^2/(a*cosh( x)^4)^(1/2)+6/5*sinh(x)^2*tanh(x)^3/a^2/(a*cosh(x)^4)^(1/2)-4/7*sinh(x)^2* tanh(x)^5/a^2/(a*cosh(x)^4)^(1/2)+1/9*sinh(x)^2*tanh(x)^7/a^2/(a*cosh(x)^4 )^(1/2)
Time = 0.04 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.40 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=\frac {(128+130 \cosh (2 x)+46 \cosh (4 x)+10 \cosh (6 x)+\cosh (8 x)) \text {sech}^6(x) \tanh (x)}{315 a^2 \sqrt {a \cosh ^4(x)}} \]
((128 + 130*Cosh[2*x] + 46*Cosh[4*x] + 10*Cosh[6*x] + Cosh[8*x])*Sech[x]^6 *Tanh[x])/(315*a^2*Sqrt[a*Cosh[x]^4])
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.58, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 3686, 3042, 4254, 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.
\(\displaystyle \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sin \left (\frac {\pi }{2}+i x\right )^4\right )^{5/2}}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle \frac {\cosh ^2(x) \int \text {sech}^{10}(x)dx}{a^2 \sqrt {a \cosh ^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\cosh ^2(x) \int \csc \left (i x+\frac {\pi }{2}\right )^{10}dx}{a^2 \sqrt {a \cosh ^4(x)}}\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle \frac {i \cosh ^2(x) \int \left (\tanh ^8(x)-4 \tanh ^6(x)+6 \tanh ^4(x)-4 \tanh ^2(x)+1\right )d(-i \tanh (x))}{a^2 \sqrt {a \cosh ^4(x)}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i \cosh ^2(x) \left (-\frac {1}{9} i \tanh ^9(x)+\frac {4}{7} i \tanh ^7(x)-\frac {6}{5} i \tanh ^5(x)+\frac {4}{3} i \tanh ^3(x)-i \tanh (x)\right )}{a^2 \sqrt {a \cosh ^4(x)}}\) |
(I*Cosh[x]^2*((-I)*Tanh[x] + ((4*I)/3)*Tanh[x]^3 - ((6*I)/5)*Tanh[x]^5 + ( (4*I)/7)*Tanh[x]^7 - (I/9)*Tanh[x]^9))/(a^2*Sqrt[a*Cosh[x]^4])
3.2.39.3.1 Defintions of rubi rules used
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]])
Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp andIntegrand[(1 + x^2)^(n/2 - 1), x], x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]
Time = 0.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.51
method | result | size |
risch | \(-\frac {256 \,{\mathrm e}^{-2 x} \left (126 \,{\mathrm e}^{8 x}+84 \,{\mathrm e}^{6 x}+36 \,{\mathrm e}^{4 x}+9 \,{\mathrm e}^{2 x}+1\right )}{315 a^{2} \left (1+{\mathrm e}^{2 x}\right )^{7} \sqrt {a \left (1+{\mathrm e}^{2 x}\right )^{4} {\mathrm e}^{-4 x}}}\) | \(60\) |
default | \(\frac {4 \sqrt {8}\, \sqrt {2}\, \left (8 \cosh \left (2 x \right )^{4}+40 \cosh \left (2 x \right )^{3}+84 \cosh \left (2 x \right )^{2}+100 \cosh \left (2 x \right )+83\right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}}{315 a^{3} \left (1+\cosh \left (2 x \right )\right )^{4} \sinh \left (2 x \right ) \sqrt {\left (1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(96\) |
-256/315/a^2/(1+exp(2*x))^7*exp(-2*x)/(a*(1+exp(2*x))^4*exp(-4*x))^(1/2)*( 126*exp(8*x)+84*exp(6*x)+36*exp(4*x)+9*exp(2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 3065 vs. \(2 (99) = 198\).
Time = 0.35 (sec) , antiderivative size = 3065, normalized size of antiderivative = 26.20 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=\text {Too large to display} \]
-256/315*(1008*cosh(x)*e^(2*x)*sinh(x)^7 + 126*e^(2*x)*sinh(x)^8 + 84*(42* cosh(x)^2 + 1)*e^(2*x)*sinh(x)^6 + 504*(14*cosh(x)^3 + cosh(x))*e^(2*x)*si nh(x)^5 + 36*(245*cosh(x)^4 + 35*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^4 + 48*(14 7*cosh(x)^5 + 35*cosh(x)^3 + 3*cosh(x))*e^(2*x)*sinh(x)^3 + 9*(392*cosh(x) ^6 + 140*cosh(x)^4 + 24*cosh(x)^2 + 1)*e^(2*x)*sinh(x)^2 + 18*(56*cosh(x)^ 7 + 28*cosh(x)^5 + 8*cosh(x)^3 + cosh(x))*e^(2*x)*sinh(x) + (126*cosh(x)^8 + 84*cosh(x)^6 + 36*cosh(x)^4 + 9*cosh(x)^2 + 1)*e^(2*x))*sqrt(a*e^(8*x) + 4*a*e^(6*x) + 6*a*e^(4*x) + 4*a*e^(2*x) + a)*e^(-2*x)/(a^3*cosh(x)^18 + 9*a^3*cosh(x)^16 + (a^3*e^(4*x) + 2*a^3*e^(2*x) + a^3)*sinh(x)^18 + 18*(a^ 3*cosh(x)*e^(4*x) + 2*a^3*cosh(x)*e^(2*x) + a^3*cosh(x))*sinh(x)^17 + 36*a ^3*cosh(x)^14 + 9*(17*a^3*cosh(x)^2 + a^3 + (17*a^3*cosh(x)^2 + a^3)*e^(4* x) + 2*(17*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^16 + 48*(17*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (17*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(17*a^3* cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^15 + 84*a^3*cosh(x)^12 + 36*(8 5*a^3*cosh(x)^4 + 30*a^3*cosh(x)^2 + a^3 + (85*a^3*cosh(x)^4 + 30*a^3*cosh (x)^2 + a^3)*e^(4*x) + 2*(85*a^3*cosh(x)^4 + 30*a^3*cosh(x)^2 + a^3)*e^(2* x))*sinh(x)^14 + 504*(17*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + a^3*cosh(x) + (17*a^3*cosh(x)^5 + 10*a^3*cosh(x)^3 + a^3*cosh(x))*e^(4*x) + 2*(17*a^3*co sh(x)^5 + 10*a^3*cosh(x)^3 + a^3*cosh(x))*e^(2*x))*sinh(x)^13 + 126*a^3*co sh(x)^10 + 84*(221*a^3*cosh(x)^6 + 195*a^3*cosh(x)^4 + 39*a^3*cosh(x)^2...
Timed out. \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (99) = 198\).
Time = 0.27 (sec) , antiderivative size = 457, normalized size of antiderivative = 3.91 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=\frac {256 \, e^{\left (-2 \, x\right )}}{35 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {1024 \, e^{\left (-4 \, x\right )}}{35 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {1024 \, e^{\left (-6 \, x\right )}}{15 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {512 \, e^{\left (-8 \, x\right )}}{5 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} + \frac {256}{315 \, {\left (9 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 126 \, a^{\frac {5}{2}} e^{\left (-10 \, x\right )} + 84 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} + 36 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 9 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} + a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + a^{\frac {5}{2}}\right )}} \]
256/35*e^(-2*x)/(9*a^(5/2)*e^(-2*x) + 36*a^(5/2)*e^(-4*x) + 84*a^(5/2)*e^( -6*x) + 126*a^(5/2)*e^(-8*x) + 126*a^(5/2)*e^(-10*x) + 84*a^(5/2)*e^(-12*x ) + 36*a^(5/2)*e^(-14*x) + 9*a^(5/2)*e^(-16*x) + a^(5/2)*e^(-18*x) + a^(5/ 2)) + 1024/35*e^(-4*x)/(9*a^(5/2)*e^(-2*x) + 36*a^(5/2)*e^(-4*x) + 84*a^(5 /2)*e^(-6*x) + 126*a^(5/2)*e^(-8*x) + 126*a^(5/2)*e^(-10*x) + 84*a^(5/2)*e ^(-12*x) + 36*a^(5/2)*e^(-14*x) + 9*a^(5/2)*e^(-16*x) + a^(5/2)*e^(-18*x) + a^(5/2)) + 1024/15*e^(-6*x)/(9*a^(5/2)*e^(-2*x) + 36*a^(5/2)*e^(-4*x) + 84*a^(5/2)*e^(-6*x) + 126*a^(5/2)*e^(-8*x) + 126*a^(5/2)*e^(-10*x) + 84*a^ (5/2)*e^(-12*x) + 36*a^(5/2)*e^(-14*x) + 9*a^(5/2)*e^(-16*x) + a^(5/2)*e^( -18*x) + a^(5/2)) + 512/5*e^(-8*x)/(9*a^(5/2)*e^(-2*x) + 36*a^(5/2)*e^(-4* x) + 84*a^(5/2)*e^(-6*x) + 126*a^(5/2)*e^(-8*x) + 126*a^(5/2)*e^(-10*x) + 84*a^(5/2)*e^(-12*x) + 36*a^(5/2)*e^(-14*x) + 9*a^(5/2)*e^(-16*x) + a^(5/2 )*e^(-18*x) + a^(5/2)) + 256/315/(9*a^(5/2)*e^(-2*x) + 36*a^(5/2)*e^(-4*x) + 84*a^(5/2)*e^(-6*x) + 126*a^(5/2)*e^(-8*x) + 126*a^(5/2)*e^(-10*x) + 84 *a^(5/2)*e^(-12*x) + 36*a^(5/2)*e^(-14*x) + 9*a^(5/2)*e^(-16*x) + a^(5/2)* e^(-18*x) + a^(5/2))
Time = 0.27 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=-\frac {256 \, {\left (126 \, e^{\left (8 \, x\right )} + 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \, a^{\frac {5}{2}} {\left (e^{\left (2 \, x\right )} + 1\right )}^{9}} \]
Time = 1.71 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.19 \[ \int \frac {1}{\left (a \cosh ^4(x)\right )^{5/2}} \, dx=\frac {4096\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{3\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^6\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{5\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^5\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {12288\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{7\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^8\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,{\mathrm {e}}^{4\,x}\,\sqrt {a\,{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}{9\,a^3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]
(4096*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(3*a^3*(exp(2*x) + 1)^6 *(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (2048*exp(4*x)*(a*(exp(-x)/2 + exp( x)/2)^4)^(1/2))/(5*a^3*(exp(2*x) + 1)^5*(exp(2*x) + 2*exp(4*x) + exp(6*x)) ) - (12288*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(7*a^3*(exp(2*x) + 1)^7*(exp(2*x) + 2*exp(4*x) + exp(6*x))) + (1024*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(a^3*(exp(2*x) + 1)^8*(exp(2*x) + 2*exp(4*x) + exp(6* x))) - (2048*exp(4*x)*(a*(exp(-x)/2 + exp(x)/2)^4)^(1/2))/(9*a^3*(exp(2*x) + 1)^9*(exp(2*x) + 2*exp(4*x) + exp(6*x)))