Integrand size = 10, antiderivative size = 163 \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=a^3 \cosh (x) \sqrt {a \text {sech}^4(x)} \sinh (x)-2 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh (x)+3 a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^3(x)-\frac {20}{7} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^5(x)+\frac {5}{3} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^7(x)-\frac {6}{11} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^9(x)+\frac {1}{13} a^3 \sqrt {a \text {sech}^4(x)} \sinh ^2(x) \tanh ^{11}(x) \]
a^3*cosh(x)*sinh(x)*(a*sech(x)^4)^(1/2)-2*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2 )*tanh(x)+3*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^3-20/7*a^3*sinh(x)^2 *(a*sech(x)^4)^(1/2)*tanh(x)^5+5/3*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh( x)^7-6/11*a^3*sinh(x)^2*(a*sech(x)^4)^(1/2)*tanh(x)^9+1/13*a^3*sinh(x)^2*( a*sech(x)^4)^(1/2)*tanh(x)^11
Time = 0.23 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.33 \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=\frac {\cosh (x) (2048+2380 \cosh (2 x)+1093 \cosh (4 x)+378 \cosh (6 x)+92 \cosh (8 x)+14 \cosh (10 x)+\cosh (12 x)) \left (a \text {sech}^4(x)\right )^{7/2} \sinh (x)}{6006} \]
(Cosh[x]*(2048 + 2380*Cosh[2*x] + 1093*Cosh[4*x] + 378*Cosh[6*x] + 92*Cosh [8*x] + 14*Cosh[10*x] + Cosh[12*x])*(a*Sech[x]^4)^(7/2)*Sinh[x])/6006
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.52, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3042, 4611, 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.
\(\displaystyle \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec (i x)^4\right )^{7/2}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int \text {sech}^{14}(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int \csc \left (i x+\frac {\pi }{2}\right )^{14}dx\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle i a^3 \cosh ^2(x) \sqrt {a \text {sech}^4(x)} \int \left (\tanh ^{12}(x)-6 \tanh ^{10}(x)+15 \tanh ^8(x)-20 \tanh ^6(x)+15 \tanh ^4(x)-6 \tanh ^2(x)+1\right )d(-i \tanh (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i a^3 \cosh ^2(x) \left (-\frac {1}{13} i \tanh ^{13}(x)+\frac {6}{11} i \tanh ^{11}(x)-\frac {5}{3} i \tanh ^9(x)+\frac {20}{7} i \tanh ^7(x)-3 i \tanh ^5(x)+2 i \tanh ^3(x)-i \tanh (x)\right ) \sqrt {a \text {sech}^4(x)}\) |
I*a^3*Cosh[x]^2*Sqrt[a*Sech[x]^4]*((-I)*Tanh[x] + (2*I)*Tanh[x]^3 - (3*I)* Tanh[x]^5 + ((20*I)/7)*Tanh[x]^7 - ((5*I)/3)*Tanh[x]^9 + ((6*I)/11)*Tanh[x ]^11 - (I/13)*Tanh[x]^13)
3.1.45.3.1 Defintions of rubi rules used
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]
Int[((b_.)*((c_.)*sec[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Simp[b^ IntPart[p]*((b*(c*Sec[e + f*x])^n)^FracPart[p]/(c*Sec[e + f*x])^(n*FracPart [p])) Int[(c*Sec[e + f*x])^(n*p), x], x] /; FreeQ[{b, c, e, f, n, p}, x] && !IntegerQ[p]
Time = 137.22 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {2048 a^{3} {\mathrm e}^{-2 x} \sqrt {\frac {{\mathrm e}^{4 x} a}{\left (1+{\mathrm e}^{2 x}\right )^{4}}}\, \left (1716 \,{\mathrm e}^{12 x}+1287 \,{\mathrm e}^{10 x}+715 \,{\mathrm e}^{8 x}+286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}+13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left (1+{\mathrm e}^{2 x}\right )^{11}}\) | \(72\) |
-2048/3003*a^3*exp(-2*x)/(1+exp(2*x))^11*(exp(4*x)*a/(1+exp(2*x))^4)^(1/2) *(1716*exp(12*x)+1287*exp(10*x)+715*exp(8*x)+286*exp(6*x)+78*exp(4*x)+13*e xp(2*x)+1)
Leaf count of result is larger than twice the leaf count of optimal. 2804 vs. \(2 (141) = 282\).
Time = 0.37 (sec) , antiderivative size = 2804, normalized size of antiderivative = 17.20 \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=\text {Too large to display} \]
-2048/3003*(1716*a^3*cosh(x)^12 + 1287*a^3*cosh(x)^10 + 1716*(a^3*e^(4*x) + 2*a^3*e^(2*x) + a^3)*sinh(x)^12 + 20592*(a^3*cosh(x)*e^(4*x) + 2*a^3*cos h(x)*e^(2*x) + a^3*cosh(x))*sinh(x)^11 + 715*a^3*cosh(x)^8 + 1287*(88*a^3* cosh(x)^2 + a^3 + (88*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(88*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^10 + 4290*(88*a^3*cosh(x)^3 + 3*a^3*cosh(x) + (88*a ^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(88*a^3*cosh(x)^3 + 3*a^3*cosh(x ))*e^(2*x))*sinh(x)^9 + 286*a^3*cosh(x)^6 + 715*(1188*a^3*cosh(x)^4 + 81*a ^3*cosh(x)^2 + a^3 + (1188*a^3*cosh(x)^4 + 81*a^3*cosh(x)^2 + a^3)*e^(4*x) + 2*(1188*a^3*cosh(x)^4 + 81*a^3*cosh(x)^2 + a^3)*e^(2*x))*sinh(x)^8 + 11 44*(1188*a^3*cosh(x)^5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x) + (1188*a^3*cos h(x)^5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x))*e^(4*x) + 2*(1188*a^3*cosh(x)^ 5 + 135*a^3*cosh(x)^3 + 5*a^3*cosh(x))*e^(2*x))*sinh(x)^7 + 78*a^3*cosh(x) ^4 + 286*(5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3 + (5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3)*e^(4*x ) + 2*(5544*a^3*cosh(x)^6 + 945*a^3*cosh(x)^4 + 70*a^3*cosh(x)^2 + a^3)*e^ (2*x))*sinh(x)^6 + 572*(2376*a^3*cosh(x)^7 + 567*a^3*cosh(x)^5 + 70*a^3*co sh(x)^3 + 3*a^3*cosh(x) + (2376*a^3*cosh(x)^7 + 567*a^3*cosh(x)^5 + 70*a^3 *cosh(x)^3 + 3*a^3*cosh(x))*e^(4*x) + 2*(2376*a^3*cosh(x)^7 + 567*a^3*cosh (x)^5 + 70*a^3*cosh(x)^3 + 3*a^3*cosh(x))*e^(2*x))*sinh(x)^5 + 13*a^3*cosh (x)^2 + 26*(32670*a^3*cosh(x)^8 + 10395*a^3*cosh(x)^6 + 1925*a^3*cosh(x...
Timed out. \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (141) = 282\).
Time = 0.29 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.80 \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=\text {Too large to display} \]
2048/231*a^(7/2)*e^(-2*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715* e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16* x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26* x) + 1) + 4096/77*a^(7/2)*e^(-4*x)/(13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6* x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 128 7*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 4096/21*a^(7/2)*e^(-6*x)/(13*e^(-2*x) + 78*e^(-4*x) + 2 86*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14 *x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e ^(-24*x) + e^(-26*x) + 1) + 10240/21*a^(7/2)*e^(-8*x)/(13*e^(-2*x) + 78*e^ (-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1 716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22 *x) + 13*e^(-24*x) + e^(-26*x) + 1) + 6144/7*a^(7/2)*e^(-10*x)/(13*e^(-2*x ) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(- 12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^(-20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 8192/7*a^(7/2)*e^(-12*x)/(1 3*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*x) + 1716*e^(-12*x) + 1716*e^(-14*x) + 1287*e^(-16*x) + 715*e^(-18*x) + 286*e^( -20*x) + 78*e^(-22*x) + 13*e^(-24*x) + e^(-26*x) + 1) + 2048/3003*a^(7/2)/ (13*e^(-2*x) + 78*e^(-4*x) + 286*e^(-6*x) + 715*e^(-8*x) + 1287*e^(-10*...
Time = 0.28 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.31 \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=-\frac {2048 \, a^{\frac {7}{2}} {\left (1716 \, e^{\left (12 \, x\right )} + 1287 \, e^{\left (10 \, x\right )} + 715 \, e^{\left (8 \, x\right )} + 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} + 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} + 1\right )}^{13}} \]
-2048/3003*a^(7/2)*(1716*e^(12*x) + 1287*e^(10*x) + 715*e^(8*x) + 286*e^(6 *x) + 78*e^(4*x) + 13*e^(2*x) + 1)/(e^(2*x) + 1)^13
Time = 2.04 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.06 \[ \int \left (a \text {sech}^4(x)\right )^{7/2} \, dx=\frac {1536\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\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\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{7\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^7\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {10240\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{3\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^9\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {4096\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{10}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {30720\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{11\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{11}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}+\frac {1024\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{{\left ({\mathrm {e}}^{2\,x}+1\right )}^{12}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )}-\frac {2048\,a^3\,\sqrt {\frac {a}{{\left (\frac {{\mathrm {e}}^{-x}}{2}+\frac {{\mathrm {e}}^x}{2}\right )}^4}}\,\left (4\,{\mathrm {e}}^{2\,x}+6\,{\mathrm {e}}^{4\,x}+4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}{13\,{\left ({\mathrm {e}}^{2\,x}+1\right )}^{13}\,\left ({\mathrm {e}}^{2\,x}+2\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}\right )} \]
(1536*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4* exp(6*x) + exp(8*x) + 1))/((exp(2*x) + 1)^8*(exp(2*x) + 2*exp(4*x) + exp(6 *x))) - (2048*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4 *x) + 4*exp(6*x) + exp(8*x) + 1))/(7*(exp(2*x) + 1)^7*(exp(2*x) + 2*exp(4* x) + exp(6*x))) - (10240*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x ) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(3*(exp(2*x) + 1)^9*(exp(2*x) + 2*exp(4*x) + exp(6*x))) + (4096*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)* (4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/((exp(2*x) + 1)^10* (exp(2*x) + 2*exp(4*x) + exp(6*x))) - (30720*a^3*(a/(exp(-x)/2 + exp(x)/2) ^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1))/(11*(exp( 2*x) + 1)^11*(exp(2*x) + 2*exp(4*x) + exp(6*x))) + (1024*a^3*(a/(exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp(8*x) + 1 ))/((exp(2*x) + 1)^12*(exp(2*x) + 2*exp(4*x) + exp(6*x))) - (2048*a^3*(a/( exp(-x)/2 + exp(x)/2)^4)^(1/2)*(4*exp(2*x) + 6*exp(4*x) + 4*exp(6*x) + exp (8*x) + 1))/(13*(exp(2*x) + 1)^13*(exp(2*x) + 2*exp(4*x) + exp(6*x)))