Integrand size = 10, antiderivative size = 164 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=2 a^3 \cosh ^2(x) \coth (x) \sqrt {a \text {csch}^4(x)}-3 a^3 \cosh ^2(x) \coth ^3(x) \sqrt {a \text {csch}^4(x)}+\frac {20}{7} a^3 \cosh ^2(x) \coth ^5(x) \sqrt {a \text {csch}^4(x)}-\frac {5}{3} a^3 \cosh ^2(x) \coth ^7(x) \sqrt {a \text {csch}^4(x)}+\frac {6}{11} a^3 \cosh ^2(x) \coth ^9(x) \sqrt {a \text {csch}^4(x)}-\frac {1}{13} a^3 \cosh ^2(x) \coth ^{11}(x) \sqrt {a \text {csch}^4(x)}-a^3 \cosh (x) \sqrt {a \text {csch}^4(x)} \sinh (x) \] Output:
2*a^3*cosh(x)^2*coth(x)*(a*csch(x)^4)^(1/2)-3*a^3*cosh(x)^2*coth(x)^3*(a*c sch(x)^4)^(1/2)+20/7*a^3*cosh(x)^2*coth(x)^5*(a*csch(x)^4)^(1/2)-5/3*a^3*c osh(x)^2*coth(x)^7*(a*csch(x)^4)^(1/2)+6/11*a^3*cosh(x)^2*coth(x)^9*(a*csc h(x)^4)^(1/2)-1/13*a^3*cosh(x)^2*coth(x)^11*(a*csch(x)^4)^(1/2)-a^3*cosh(x )*(a*csch(x)^4)^(1/2)*sinh(x)
Time = 0.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.36 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=-\frac {a^3 \cosh (x) \sqrt {a \text {csch}^4(x)} \left (1024-512 \text {csch}^2(x)+384 \text {csch}^4(x)-320 \text {csch}^6(x)+280 \text {csch}^8(x)-252 \text {csch}^{10}(x)+231 \text {csch}^{12}(x)\right ) \sinh (x)}{3003} \] Input:
Integrate[(a*Csch[x]^4)^(7/2),x]
Output:
-1/3003*(a^3*Cosh[x]*Sqrt[a*Csch[x]^4]*(1024 - 512*Csch[x]^2 + 384*Csch[x] ^4 - 320*Csch[x]^6 + 280*Csch[x]^8 - 252*Csch[x]^10 + 231*Csch[x]^12)*Sinh [x])
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.51, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {3042, 4611, 25, 3042, 25, 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.
\(\displaystyle \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sec \left (\frac {\pi }{2}+i x\right )^4\right )^{7/2}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle -a^3 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int -\text {csch}^{14}(x)dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a^3 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int \text {csch}^{14}(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a^3 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int -\csc (i x)^{14}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a^3 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int \csc (i x)^{14}dx\) |
\(\Big \downarrow \) 4254 |
\(\displaystyle -i a^3 \sinh ^2(x) \sqrt {a \text {csch}^4(x)} \int \left (\coth ^{12}(x)-6 \coth ^{10}(x)+15 \coth ^8(x)-20 \coth ^6(x)+15 \coth ^4(x)-6 \coth ^2(x)+1\right )d(-i \coth (x))\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -i a^3 \sinh ^2(x) \left (-\frac {1}{13} i \coth ^{13}(x)+\frac {6}{11} i \coth ^{11}(x)-\frac {5}{3} i \coth ^9(x)+\frac {20}{7} i \coth ^7(x)-3 i \coth ^5(x)+2 i \coth ^3(x)-i \coth (x)\right ) \sqrt {a \text {csch}^4(x)}\) |
Input:
Int[(a*Csch[x]^4)^(7/2),x]
Output:
(-I)*a^3*((-I)*Coth[x] + (2*I)*Coth[x]^3 - (3*I)*Coth[x]^5 + ((20*I)/7)*Co th[x]^7 - ((5*I)/3)*Coth[x]^9 + ((6*I)/11)*Coth[x]^11 - (I/13)*Coth[x]^13) *Sqrt[a*Csch[x]^4]*Sinh[x]^2
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 = 0.80 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.44
method | result | size |
risch | \(-\frac {2048 a^{3} {\mathrm e}^{-2 x} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left (1716 \,{\mathrm e}^{12 x}-1287 \,{\mathrm e}^{10 x}+715 \,{\mathrm e}^{8 x}-286 \,{\mathrm e}^{6 x}+78 \,{\mathrm e}^{4 x}-13 \,{\mathrm e}^{2 x}+1\right )}{3003 \left ({\mathrm e}^{2 x}-1\right )^{11}}\) | \(72\) |
Input:
int((a*csch(x)^4)^(7/2),x,method=_RETURNVERBOSE)
Output:
-2048/3003*a^3*exp(-2*x)/(exp(2*x)-1)^11*(a*exp(4*x)/(exp(2*x)-1)^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. 2825 vs. \(2 (142) = 284\).
Time = 0.20 (sec) , antiderivative size = 2825, normalized size of antiderivative = 17.23 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\text {Too large to display} \] Input:
integrate((a*csch(x)^4)^(7/2),x, algorithm="fricas")
Output:
-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...
\[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\int \left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {7}{2}}\, dx \] Input:
integrate((a*csch(x)**4)**(7/2),x)
Output:
Integral((a*csch(x)**4)**(7/2), x)
Leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (142) = 284\).
Time = 0.13 (sec) , antiderivative size = 620, normalized size of antiderivative = 3.78 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\text {Too large to display} \] Input:
integrate((a*csch(x)^4)^(7/2),x, algorithm="maxima")
Output:
-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) - 12 87*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) + 286*e^(-6*x) - 715*e^(-8*x) + 1287*e^(-10*x) - 1716*e^(-12*x) + 1716*e^(-1 4*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) + 1716*e^(-14*x) - 1287*e^(-16*x) + 715*e^(-18*x) - 286*e^(-20*x) + 78*e^(-2 2*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)/( 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) + 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 = 0.11 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.31 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=-\frac {2048 \, a^{\frac {7}{2}} {\left (1716 \, e^{\left (12 \, x\right )} - 1287 \, e^{\left (10 \, x\right )} + 715 \, e^{\left (8 \, x\right )} - 286 \, e^{\left (6 \, x\right )} + 78 \, e^{\left (4 \, x\right )} - 13 \, e^{\left (2 \, x\right )} + 1\right )}}{3003 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{13}} \] Input:
integrate((a*csch(x)^4)^(7/2),x, algorithm="giac")
Output:
-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 = 2.43 (sec) , antiderivative size = 498, normalized size of antiderivative = 3.04 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx =\text {Too large to display} \] Input:
int((a/sinh(x)^4)^(7/2),x)
Output:
- (2048*a^3*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(7*(exp(2*x) - 1)^7*(exp(2*x) - 2*exp(4*x) + e xp(6*x))) - (1536*a^3*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4*x) - 4*e xp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/((exp(2*x) - 1)^8*(exp(2*x) - 2*exp( 4*x) + exp(6*x))) - (10240*a^3*(a/(exp(-x)/2 - exp(x)/2)^4)^(1/2)*(6*exp(4 *x) - 4*exp(2*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 )*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/((exp(2*x) - 1)^1 0*(exp(2*x) - 2*exp(4*x) + exp(6*x))) - (30720*a^3*(a/(exp(-x)/2 - exp(x)/ 2)^4)^(1/2)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1))/(11*(ex p(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)*(6*exp(4*x) - 4*exp(2*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)*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + e xp(8*x) + 1))/(13*(exp(2*x) - 1)^13*(exp(2*x) - 2*exp(4*x) + exp(6*x)))
Time = 0.21 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.89 \[ \int \left (a \text {csch}^4(x)\right )^{7/2} \, dx=\frac {2048 \sqrt {a}\, a^{3} \left (-1716 e^{12 x}+1287 e^{10 x}-715 e^{8 x}+286 e^{6 x}-78 e^{4 x}+13 e^{2 x}-1\right )}{3003 e^{26 x}-39039 e^{24 x}+234234 e^{22 x}-858858 e^{20 x}+2147145 e^{18 x}-3864861 e^{16 x}+5153148 e^{14 x}-5153148 e^{12 x}+3864861 e^{10 x}-2147145 e^{8 x}+858858 e^{6 x}-234234 e^{4 x}+39039 e^{2 x}-3003} \] Input:
int((a*csch(x)^4)^(7/2),x)
Output:
(2048*sqrt(a)*a**3*( - 1716*e**(12*x) + 1287*e**(10*x) - 715*e**(8*x) + 28 6*e**(6*x) - 78*e**(4*x) + 13*e**(2*x) - 1))/(3003*(e**(26*x) - 13*e**(24* x) + 78*e**(22*x) - 286*e**(20*x) + 715*e**(18*x) - 1287*e**(16*x) + 1716* e**(14*x) - 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))