Integrand size = 10, antiderivative size = 74 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=\frac {\coth (x)}{7 \left (a \text {csch}^2(x)\right )^{7/2}}-\frac {6 \coth (x)}{35 a \left (a \text {csch}^2(x)\right )^{5/2}}+\frac {8 \coth (x)}{35 a^2 \left (a \text {csch}^2(x)\right )^{3/2}}-\frac {16 \coth (x)}{35 a^3 \sqrt {a \text {csch}^2(x)}} \]
1/7*coth(x)/(a*csch(x)^2)^(7/2)-6/35*coth(x)/a/(a*csch(x)^2)^(5/2)+8/35*co th(x)/a^2/(a*csch(x)^2)^(3/2)-16/35*coth(x)/a^3/(a*csch(x)^2)^(1/2)
Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.57 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=\frac {(-1225 \cosh (x)+245 \cosh (3 x)-49 \cosh (5 x)+5 \cosh (7 x)) \sqrt {a \text {csch}^2(x)} \sinh (x)}{2240 a^4} \]
((-1225*Cosh[x] + 245*Cosh[3*x] - 49*Cosh[5*x] + 5*Cosh[7*x])*Sqrt[a*Csch[ x]^2]*Sinh[x])/(2240*a^4)
Time = 0.25 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.51, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3042, 4610, 209, 209, 209, 208}
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.
\(\displaystyle \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (-a \sec \left (\frac {\pi }{2}+i x\right )^2\right )^{7/2}}dx\) |
\(\Big \downarrow \) 4610 |
\(\displaystyle -a \int \frac {1}{\left (a \coth ^2(x)-a\right )^{9/2}}d\coth (x)\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -a \left (-\frac {6 \int \frac {1}{\left (a \coth ^2(x)-a\right )^{7/2}}d\coth (x)}{7 a}-\frac {\coth (x)}{7 a \left (a \coth ^2(x)-a\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -a \left (-\frac {6 \left (-\frac {4 \int \frac {1}{\left (a \coth ^2(x)-a\right )^{5/2}}d\coth (x)}{5 a}-\frac {\coth (x)}{5 a \left (a \coth ^2(x)-a\right )^{5/2}}\right )}{7 a}-\frac {\coth (x)}{7 a \left (a \coth ^2(x)-a\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 209 |
\(\displaystyle -a \left (-\frac {6 \left (-\frac {4 \left (-\frac {2 \int \frac {1}{\left (a \coth ^2(x)-a\right )^{3/2}}d\coth (x)}{3 a}-\frac {\coth (x)}{3 a \left (a \coth ^2(x)-a\right )^{3/2}}\right )}{5 a}-\frac {\coth (x)}{5 a \left (a \coth ^2(x)-a\right )^{5/2}}\right )}{7 a}-\frac {\coth (x)}{7 a \left (a \coth ^2(x)-a\right )^{7/2}}\right )\) |
\(\Big \downarrow \) 208 |
\(\displaystyle -a \left (-\frac {6 \left (-\frac {4 \left (\frac {2 \coth (x)}{3 a^2 \sqrt {a \coth ^2(x)-a}}-\frac {\coth (x)}{3 a \left (a \coth ^2(x)-a\right )^{3/2}}\right )}{5 a}-\frac {\coth (x)}{5 a \left (a \coth ^2(x)-a\right )^{5/2}}\right )}{7 a}-\frac {\coth (x)}{7 a \left (a \coth ^2(x)-a\right )^{7/2}}\right )\) |
-(a*(-1/7*Coth[x]/(a*(-a + a*Coth[x]^2)^(7/2)) - (6*(-1/5*Coth[x]/(a*(-a + a*Coth[x]^2)^(5/2)) - (4*(-1/3*Coth[x]/(a*(-a + a*Coth[x]^2)^(3/2)) + (2* Coth[x])/(3*a^2*Sqrt[-a + a*Coth[x]^2])))/(5*a)))/(7*a)))
3.1.35.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFac tors[Tan[e + f*x], x]}, Simp[b*(ff/f) Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] && !IntegerQ[p]
Leaf count of result is larger than twice the leaf count of optimal. \(261\) vs. \(2(58)=116\).
Time = 0.15 (sec) , antiderivative size = 262, normalized size of antiderivative = 3.54
method | result | size |
risch | \(\frac {{\mathrm e}^{8 x}}{896 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {7 \,{\mathrm e}^{6 x}}{640 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {7 \,{\mathrm e}^{4 x}}{128 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {35 \,{\mathrm e}^{2 x}}{128 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {35}{128 \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left ({\mathrm e}^{2 x}-1\right ) a^{3}}+\frac {7 \,{\mathrm e}^{-2 x}}{128 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}-\frac {7 \,{\mathrm e}^{-4 x}}{640 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}+\frac {{\mathrm e}^{-6 x}}{896 a^{3} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}}\) | \(262\) |
1/896/a^3*exp(8*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-7/640/a^ 3*exp(6*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)+7/128/a^3*exp(4* x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-35/128/a^3*exp(2*x)/(exp (2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^(1/2)-35/128/(a*exp(2*x)/(exp(2*x)-1) ^2)^(1/2)/(exp(2*x)-1)/a^3+7/128/a^3*exp(-2*x)/(exp(2*x)-1)/(a*exp(2*x)/(e xp(2*x)-1)^2)^(1/2)-7/640/a^3*exp(-4*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x) -1)^2)^(1/2)+1/896/a^3*exp(-6*x)/(exp(2*x)-1)/(a*exp(2*x)/(exp(2*x)-1)^2)^ (1/2)
Leaf count of result is larger than twice the leaf count of optimal. 984 vs. \(2 (58) = 116\).
Time = 0.28 (sec) , antiderivative size = 984, normalized size of antiderivative = 13.30 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=\text {Too large to display} \]
1/4480*(5*(e^(2*x) - 1)*sinh(x)^14 - 5*cosh(x)^14 + 70*(cosh(x)*e^(2*x) - cosh(x))*sinh(x)^13 - 7*(65*cosh(x)^2 - (65*cosh(x)^2 - 7)*e^(2*x) - 7)*si nh(x)^12 + 49*cosh(x)^12 - 28*(65*cosh(x)^3 - (65*cosh(x)^3 - 21*cosh(x))* e^(2*x) - 21*cosh(x))*sinh(x)^11 - 7*(715*cosh(x)^4 - 462*cosh(x)^2 - (715 *cosh(x)^4 - 462*cosh(x)^2 + 35)*e^(2*x) + 35)*sinh(x)^10 - 245*cosh(x)^10 - 70*(143*cosh(x)^5 - 154*cosh(x)^3 - (143*cosh(x)^5 - 154*cosh(x)^3 + 35 *cosh(x))*e^(2*x) + 35*cosh(x))*sinh(x)^9 - 35*(429*cosh(x)^6 - 693*cosh(x )^4 + 315*cosh(x)^2 - (429*cosh(x)^6 - 693*cosh(x)^4 + 315*cosh(x)^2 - 35) *e^(2*x) - 35)*sinh(x)^8 + 1225*cosh(x)^8 - 8*(2145*cosh(x)^7 - 4851*cosh( x)^5 + 3675*cosh(x)^3 - (2145*cosh(x)^7 - 4851*cosh(x)^5 + 3675*cosh(x)^3 - 1225*cosh(x))*e^(2*x) - 1225*cosh(x))*sinh(x)^7 - 7*(2145*cosh(x)^8 - 64 68*cosh(x)^6 + 7350*cosh(x)^4 - 4900*cosh(x)^2 - (2145*cosh(x)^8 - 6468*co sh(x)^6 + 7350*cosh(x)^4 - 4900*cosh(x)^2 - 175)*e^(2*x) - 175)*sinh(x)^6 + 1225*cosh(x)^6 - 14*(715*cosh(x)^9 - 2772*cosh(x)^7 + 4410*cosh(x)^5 - 4 900*cosh(x)^3 - (715*cosh(x)^9 - 2772*cosh(x)^7 + 4410*cosh(x)^5 - 4900*co sh(x)^3 - 525*cosh(x))*e^(2*x) - 525*cosh(x))*sinh(x)^5 - 35*(143*cosh(x)^ 10 - 693*cosh(x)^8 + 1470*cosh(x)^6 - 2450*cosh(x)^4 - 525*cosh(x)^2 - (14 3*cosh(x)^10 - 693*cosh(x)^8 + 1470*cosh(x)^6 - 2450*cosh(x)^4 - 525*cosh( x)^2 + 7)*e^(2*x) + 7)*sinh(x)^4 - 245*cosh(x)^4 - 140*(13*cosh(x)^11 - 77 *cosh(x)^9 + 210*cosh(x)^7 - 490*cosh(x)^5 - 175*cosh(x)^3 - (13*cosh(x...
\[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=\int \frac {1}{\left (a \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {7}{2}}}\, dx \]
Time = 0.27 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=-\frac {e^{\left (7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (5 \, x\right )}}{640 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (3 \, x\right )}}{128 \, a^{\frac {7}{2}}} + \frac {35 \, e^{\left (-x\right )}}{128 \, a^{\frac {7}{2}}} - \frac {7 \, e^{\left (-3 \, x\right )}}{128 \, a^{\frac {7}{2}}} + \frac {7 \, e^{\left (-5 \, x\right )}}{640 \, a^{\frac {7}{2}}} - \frac {e^{\left (-7 \, x\right )}}{896 \, a^{\frac {7}{2}}} + \frac {35 \, e^{x}}{128 \, a^{\frac {7}{2}}} \]
-1/896*e^(7*x)/a^(7/2) + 7/640*e^(5*x)/a^(7/2) - 7/128*e^(3*x)/a^(7/2) + 3 5/128*e^(-x)/a^(7/2) - 7/128*e^(-3*x)/a^(7/2) + 7/640*e^(-5*x)/a^(7/2) - 1 /896*e^(-7*x)/a^(7/2) + 35/128*e^x/a^(7/2)
Time = 0.27 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=-\frac {{\left (1225 \, e^{\left (6 \, x\right )} - 245 \, e^{\left (4 \, x\right )} + 49 \, e^{\left (2 \, x\right )} - 5\right )} e^{\left (-7 \, x\right )} - 5 \, e^{\left (7 \, x\right )} + 49 \, e^{\left (5 \, x\right )} - 245 \, e^{\left (3 \, x\right )} + 1225 \, e^{x}}{4480 \, a^{\frac {7}{2}} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right )} \]
-1/4480*((1225*e^(6*x) - 245*e^(4*x) + 49*e^(2*x) - 5)*e^(-7*x) - 5*e^(7*x ) + 49*e^(5*x) - 245*e^(3*x) + 1225*e^x)/(a^(7/2)*sgn(e^(3*x) - e^x))
Timed out. \[ \int \frac {1}{\left (a \text {csch}^2(x)\right )^{7/2}} \, dx=\int \frac {1}{{\left (\frac {a}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{7/2}} \,d x \]