Integrand size = 10, antiderivative size = 86 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {5 \coth (x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 x \text {csch}^2(x)}{16 a \sqrt {a \text {csch}^4(x)}}-\frac {5 \cosh (x) \sinh (x)}{24 a \sqrt {a \text {csch}^4(x)}}+\frac {\cosh (x) \sinh ^3(x)}{6 a \sqrt {a \text {csch}^4(x)}} \] Output:
5/16*coth(x)/a/(a*csch(x)^4)^(1/2)-5/16*x*csch(x)^2/a/(a*csch(x)^4)^(1/2)- 5/24*cosh(x)*sinh(x)/a/(a*csch(x)^4)^(1/2)+1/6*cosh(x)*sinh(x)^3/a/(a*csch (x)^4)^(1/2)
Time = 0.05 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.44 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {\text {csch}^6(x) (-60 x+45 \sinh (2 x)-9 \sinh (4 x)+\sinh (6 x))}{192 \left (a \text {csch}^4(x)\right )^{3/2}} \] Input:
Integrate[(a*Csch[x]^4)^(-3/2),x]
Output:
(Csch[x]^6*(-60*x + 45*Sinh[2*x] - 9*Sinh[4*x] + Sinh[6*x]))/(192*(a*Csch[ x]^4)^(3/2))
Time = 0.35 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.73, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {3042, 4611, 25, 3042, 25, 3115, 3042, 3115, 25, 3042, 25, 3115, 24}
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.
\(\displaystyle \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (a \sec \left (\frac {\pi }{2}+i x\right )^4\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4611 |
\(\displaystyle -\frac {\text {csch}^2(x) \int -\sinh ^6(x)dx}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\text {csch}^2(x) \int \sinh ^6(x)dx}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\text {csch}^2(x) \int -\sin (i x)^6dx}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\text {csch}^2(x) \int \sin (i x)^6dx}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {5}{6} \int \sinh ^4(x)dx-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \int \sin (i x)^4dx\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {5}{6} \left (\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx\right )\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx\right )\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\text {csch}^2(x) \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )}{a \sqrt {a \text {csch}^4(x)}}\) |
Input:
Int[(a*Csch[x]^4)^(-3/2),x]
Output:
-((Csch[x]^2*(-1/6*(Cosh[x]*Sinh[x]^5) + (5*((Cosh[x]*Sinh[x]^3)/4 + (3*(x /2 - (Cosh[x]*Sinh[x])/2))/4))/6))/(a*Sqrt[a*Csch[x]^4]))
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(229\) vs. \(2(70)=140\).
Time = 0.12 (sec) , antiderivative size = 230, normalized size of antiderivative = 2.67
method | result | size |
risch | \(-\frac {5 \,{\mathrm e}^{2 x} x}{16 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {{\mathrm e}^{8 x}}{384 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {3 \,{\mathrm e}^{6 x}}{128 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}+\frac {15 \,{\mathrm e}^{4 x}}{128 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {15}{128 \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}\, \left ({\mathrm e}^{2 x}-1\right )^{2} a}+\frac {3 \,{\mathrm e}^{-2 x}}{128 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}-\frac {{\mathrm e}^{-4 x}}{384 a \left ({\mathrm e}^{2 x}-1\right )^{2} \sqrt {\frac {a \,{\mathrm e}^{4 x}}{\left ({\mathrm e}^{2 x}-1\right )^{4}}}}\) | \(230\) |
Input:
int(1/(a*csch(x)^4)^(3/2),x,method=_RETURNVERBOSE)
Output:
-5/16/a*exp(2*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)*x+1/384/ a*exp(8*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-3/128/a*exp(6* x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)+15/128/a*exp(4*x)/(exp (2*x)-1)^2/(a*exp(4*x)/(exp(2*x)-1)^4)^(1/2)-15/128/(a*exp(4*x)/(exp(2*x)- 1)^4)^(1/2)/(exp(2*x)-1)^2/a+3/128/a*exp(-2*x)/(exp(2*x)-1)^2/(a*exp(4*x)/ (exp(2*x)-1)^4)^(1/2)-1/384/a*exp(-4*x)/(exp(2*x)-1)^2/(a*exp(4*x)/(exp(2* x)-1)^4)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1141 vs. \(2 (70) = 140\).
Time = 0.10 (sec) , antiderivative size = 1141, normalized size of antiderivative = 13.27 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(1/(a*csch(x)^4)^(3/2),x, algorithm="fricas")
Output:
1/384*((e^(4*x) - 2*e^(2*x) + 1)*sinh(x)^12 + cosh(x)^12 + 12*(cosh(x)*e^( 4*x) - 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^11 + 3*(22*cosh(x)^2 + (22*cos h(x)^2 - 3)*e^(4*x) - 2*(22*cosh(x)^2 - 3)*e^(2*x) - 3)*sinh(x)^10 - 9*cos h(x)^10 + 10*(22*cosh(x)^3 + (22*cosh(x)^3 - 9*cosh(x))*e^(4*x) - 2*(22*co sh(x)^3 - 9*cosh(x))*e^(2*x) - 9*cosh(x))*sinh(x)^9 + 45*(11*cosh(x)^4 - 9 *cosh(x)^2 + (11*cosh(x)^4 - 9*cosh(x)^2 + 1)*e^(4*x) - 2*(11*cosh(x)^4 - 9*cosh(x)^2 + 1)*e^(2*x) + 1)*sinh(x)^8 + 45*cosh(x)^8 + 72*(11*cosh(x)^5 - 15*cosh(x)^3 + (11*cosh(x)^5 - 15*cosh(x)^3 + 5*cosh(x))*e^(4*x) - 2*(11 *cosh(x)^5 - 15*cosh(x)^3 + 5*cosh(x))*e^(2*x) + 5*cosh(x))*sinh(x)^7 - 12 0*x*cosh(x)^6 + 6*(154*cosh(x)^6 - 315*cosh(x)^4 + 210*cosh(x)^2 + (154*co sh(x)^6 - 315*cosh(x)^4 + 210*cosh(x)^2 - 20*x)*e^(4*x) - 2*(154*cosh(x)^6 - 315*cosh(x)^4 + 210*cosh(x)^2 - 20*x)*e^(2*x) - 20*x)*sinh(x)^6 + 36*(2 2*cosh(x)^7 - 63*cosh(x)^5 + 70*cosh(x)^3 - 20*x*cosh(x) + (22*cosh(x)^7 - 63*cosh(x)^5 + 70*cosh(x)^3 - 20*x*cosh(x))*e^(4*x) - 2*(22*cosh(x)^7 - 6 3*cosh(x)^5 + 70*cosh(x)^3 - 20*x*cosh(x))*e^(2*x))*sinh(x)^5 + 45*(11*cos h(x)^8 - 42*cosh(x)^6 + 70*cosh(x)^4 - 40*x*cosh(x)^2 + (11*cosh(x)^8 - 42 *cosh(x)^6 + 70*cosh(x)^4 - 40*x*cosh(x)^2 - 1)*e^(4*x) - 2*(11*cosh(x)^8 - 42*cosh(x)^6 + 70*cosh(x)^4 - 40*x*cosh(x)^2 - 1)*e^(2*x) - 1)*sinh(x)^4 - 45*cosh(x)^4 + 20*(11*cosh(x)^9 - 54*cosh(x)^7 + 126*cosh(x)^5 - 120*x* cosh(x)^3 + (11*cosh(x)^9 - 54*cosh(x)^7 + 126*cosh(x)^5 - 120*x*cosh(x...
\[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{4}{\left (x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a*csch(x)**4)**(3/2),x)
Output:
Integral((a*csch(x)**4)**(-3/2), x)
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.53 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=-\frac {{\left (9 \, e^{\left (-2 \, x\right )} - 45 \, e^{\left (-4 \, x\right )} + 45 \, e^{\left (-8 \, x\right )} - 9 \, e^{\left (-10 \, x\right )} + e^{\left (-12 \, x\right )} - 1\right )} e^{\left (6 \, x\right )}}{384 \, a^{\frac {3}{2}}} - \frac {5 \, x}{16 \, a^{\frac {3}{2}}} \] Input:
integrate(1/(a*csch(x)^4)^(3/2),x, algorithm="maxima")
Output:
-1/384*(9*e^(-2*x) - 45*e^(-4*x) + 45*e^(-8*x) - 9*e^(-10*x) + e^(-12*x) - 1)*e^(6*x)/a^(3/2) - 5/16*x/a^(3/2)
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {{\left (110 \, e^{\left (6 \, x\right )} - 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-6 \, x\right )} - 120 \, x + e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} + 45 \, e^{\left (2 \, x\right )}}{384 \, a^{\frac {3}{2}}} \] Input:
integrate(1/(a*csch(x)^4)^(3/2),x, algorithm="giac")
Output:
1/384*((110*e^(6*x) - 45*e^(4*x) + 9*e^(2*x) - 1)*e^(-6*x) - 120*x + e^(6* x) - 9*e^(4*x) + 45*e^(2*x))/a^(3/2)
Timed out. \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^4}\right )}^{3/2}} \,d x \] Input:
int(1/(a/sinh(x)^4)^(3/2),x)
Output:
int(1/(a/sinh(x)^4)^(3/2), x)
Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \frac {1}{\left (a \text {csch}^4(x)\right )^{3/2}} \, dx=\frac {\sqrt {a}\, \left (e^{12 x}-9 e^{10 x}+45 e^{8 x}-120 e^{6 x} x -45 e^{4 x}+9 e^{2 x}-1\right )}{384 e^{6 x} a^{2}} \] Input:
int(1/(a*csch(x)^4)^(3/2),x)
Output:
(sqrt(a)*(e**(12*x) - 9*e**(10*x) + 45*e**(8*x) - 120*e**(6*x)*x - 45*e**( 4*x) + 9*e**(2*x) - 1))/(384*e**(6*x)*a**2)