Integrand size = 10, antiderivative size = 132 \[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=\frac {63}{256} a^2 \coth (x) \sqrt {a \sinh ^4(x)}-\frac {63}{256} a^2 x \text {csch}^2(x) \sqrt {a \sinh ^4(x)}-\frac {21}{128} a^2 \cosh (x) \sinh (x) \sqrt {a \sinh ^4(x)}+\frac {21}{160} a^2 \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^4(x)}-\frac {9}{80} a^2 \cosh (x) \sinh ^5(x) \sqrt {a \sinh ^4(x)}+\frac {1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt {a \sinh ^4(x)} \]
63/256*a^2*coth(x)*(a*sinh(x)^4)^(1/2)-63/256*a^2*x*csch(x)^2*(a*sinh(x)^4 )^(1/2)-21/128*a^2*cosh(x)*sinh(x)*(a*sinh(x)^4)^(1/2)+21/160*a^2*cosh(x)* sinh(x)^3*(a*sinh(x)^4)^(1/2)-9/80*a^2*cosh(x)*sinh(x)^5*(a*sinh(x)^4)^(1/ 2)+1/10*a^2*cosh(x)*sinh(x)^7*(a*sinh(x)^4)^(1/2)
Time = 0.15 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.40 \[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=\frac {a \text {csch}^6(x) \left (a \sinh ^4(x)\right )^{3/2} (-2520 x+2100 \sinh (2 x)-600 \sinh (4 x)+150 \sinh (6 x)-25 \sinh (8 x)+2 \sinh (10 x))}{10240} \]
(a*Csch[x]^6*(a*Sinh[x]^4)^(3/2)*(-2520*x + 2100*Sinh[2*x] - 600*Sinh[4*x] + 150*Sinh[6*x] - 25*Sinh[8*x] + 2*Sinh[10*x]))/10240
Time = 0.48 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.70, number of steps used = 18, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.800, Rules used = {3042, 3686, 3042, 25, 3115, 3042, 3115, 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.
| \(\displaystyle \int \left (a \sinh ^4(x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a \sin (i x)^4\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3686 |
| \(\displaystyle a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \int \sinh ^{10}(x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \int -\sin (i x)^{10}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \int \sin (i x)^{10}dx\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \int \sinh ^8(x)dx-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \int \sin (i x)^8dx\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \int -\sinh ^6(x)dx+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)-\frac {7}{8} \int \sinh ^6(x)dx\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)-\frac {7}{8} \int -\sin (i x)^6dx\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \int \sin (i x)^6dx\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \int \sinh ^4(x)dx-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \int \sin (i x)^4dx\right )\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{10} \sinh ^9(x) \cosh (x)+\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {7}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )+\frac {1}{8} \sinh ^7(x) \cosh (x)\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle -a^2 \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {9}{10} \left (\frac {1}{8} \sinh ^7(x) \cosh (x)+\frac {7}{8} \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\right )-\frac {1}{10} \sinh ^9(x) \cosh (x)\right )\) |
-(a^2*Csch[x]^2*Sqrt[a*Sinh[x]^4]*(-1/10*(Cosh[x]*Sinh[x]^9) + (9*((Cosh[x ]*Sinh[x]^7)/8 + (7*(-1/6*(Cosh[x]*Sinh[x]^5) + (5*((Cosh[x]*Sinh[x]^3)/4 + (3*(x/2 - (Cosh[x]*Sinh[x])/2))/4))/6))/8))/10))
3.2.52.3.1 Defintions of rubi rules used
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[(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]])
Time = 8.39 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.30
| method | result | size |
| default | \(\frac {a^{\frac {3}{2}} \left (-1+\cosh \left (2 x \right )\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \left (8 \sqrt {a}\, \sqrt {a \sinh \left (2 x \right )^{2}}\, \sinh \left (2 x \right )^{4}-50 \sqrt {a}\, \sqrt {a \sinh \left (2 x \right )^{2}}\, \cosh \left (2 x \right ) \sinh \left (2 x \right )^{2}+160 \sqrt {a}\, \sqrt {a \sinh \left (2 x \right )^{2}}\, \sinh \left (2 x \right )^{2}-325 \cosh \left (2 x \right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+640 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}-315 \ln \left (\cosh \left (2 x \right ) \sqrt {a}+\sqrt {a \sinh \left (2 x \right )^{2}}\right ) a \right )}{2560 \sinh \left (2 x \right ) \sqrt {\left (-1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(171\) |
| risch | \(-\frac {63 a^{2} {\mathrm e}^{2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}\, x}{256 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {a^{2} {\mathrm e}^{12 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{10240 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {5 a^{2} {\mathrm e}^{10 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{4096 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {15 a^{2} {\mathrm e}^{8 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{2048 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {15 a^{2} {\mathrm e}^{6 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{512 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {105 a^{2} {\mathrm e}^{4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{1024 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {105 \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}\, a^{2}}{1024 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {15 a^{2} {\mathrm e}^{-2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{512 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {15 a^{2} {\mathrm e}^{-4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{2048 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {5 a^{2} {\mathrm e}^{-6 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{4096 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {a^{2} {\mathrm e}^{-8 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{10240 \left ({\mathrm e}^{2 x}-1\right )^{2}}\) | \(362\) |
1/2560*a^(3/2)*(-1+cosh(2*x))*(a*(-1+cosh(2*x))*(1+cosh(2*x)))^(1/2)*(8*a^ (1/2)*(a*sinh(2*x)^2)^(1/2)*sinh(2*x)^4-50*a^(1/2)*(a*sinh(2*x)^2)^(1/2)*c osh(2*x)*sinh(2*x)^2+160*a^(1/2)*(a*sinh(2*x)^2)^(1/2)*sinh(2*x)^2-325*cos h(2*x)*(a*sinh(2*x)^2)^(1/2)*a^(1/2)+640*(a*sinh(2*x)^2)^(1/2)*a^(1/2)-315 *ln(cosh(2*x)*a^(1/2)+(a*sinh(2*x)^2)^(1/2))*a)/sinh(2*x)/((-1+cosh(2*x))^ 2*a)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 1597 vs. \(2 (108) = 216\).
Time = 0.31 (sec) , antiderivative size = 1597, normalized size of antiderivative = 12.10 \[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=\text {Too large to display} \]
1/20480*(40*a^2*cosh(x)*e^(2*x)*sinh(x)^19 + 2*a^2*e^(2*x)*sinh(x)^20 + 5* (76*a^2*cosh(x)^2 - 5*a^2)*e^(2*x)*sinh(x)^18 + 30*(76*a^2*cosh(x)^3 - 15* a^2*cosh(x))*e^(2*x)*sinh(x)^17 + 15*(646*a^2*cosh(x)^4 - 255*a^2*cosh(x)^ 2 + 10*a^2)*e^(2*x)*sinh(x)^16 + 48*(646*a^2*cosh(x)^5 - 425*a^2*cosh(x)^3 + 50*a^2*cosh(x))*e^(2*x)*sinh(x)^15 + 60*(1292*a^2*cosh(x)^6 - 1275*a^2* cosh(x)^4 + 300*a^2*cosh(x)^2 - 10*a^2)*e^(2*x)*sinh(x)^14 + 120*(1292*a^2 *cosh(x)^7 - 1785*a^2*cosh(x)^5 + 700*a^2*cosh(x)^3 - 70*a^2*cosh(x))*e^(2 *x)*sinh(x)^13 + 60*(4199*a^2*cosh(x)^8 - 7735*a^2*cosh(x)^6 + 4550*a^2*co sh(x)^4 - 910*a^2*cosh(x)^2 + 35*a^2)*e^(2*x)*sinh(x)^12 + 80*(4199*a^2*co sh(x)^9 - 9945*a^2*cosh(x)^7 + 8190*a^2*cosh(x)^5 - 2730*a^2*cosh(x)^3 + 3 15*a^2*cosh(x))*e^(2*x)*sinh(x)^11 + 2*(184756*a^2*cosh(x)^10 - 546975*a^2 *cosh(x)^8 + 600600*a^2*cosh(x)^6 - 300300*a^2*cosh(x)^4 + 69300*a^2*cosh( x)^2 - 2520*a^2*x)*e^(2*x)*sinh(x)^10 + 20*(16796*a^2*cosh(x)^11 - 60775*a ^2*cosh(x)^9 + 85800*a^2*cosh(x)^7 - 60060*a^2*cosh(x)^5 + 23100*a^2*cosh( x)^3 - 2520*a^2*x*cosh(x))*e^(2*x)*sinh(x)^9 + 30*(8398*a^2*cosh(x)^12 - 3 6465*a^2*cosh(x)^10 + 64350*a^2*cosh(x)^8 - 60060*a^2*cosh(x)^6 + 34650*a^ 2*cosh(x)^4 - 7560*a^2*x*cosh(x)^2 - 70*a^2)*e^(2*x)*sinh(x)^8 + 240*(646* a^2*cosh(x)^13 - 3315*a^2*cosh(x)^11 + 7150*a^2*cosh(x)^9 - 8580*a^2*cosh( x)^7 + 6930*a^2*cosh(x)^5 - 2520*a^2*x*cosh(x)^3 - 70*a^2*cosh(x))*e^(2*x) *sinh(x)^7 + 60*(1292*a^2*cosh(x)^14 - 7735*a^2*cosh(x)^12 + 20020*a^2*...
\[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=\int \left (a \sinh ^{4}{\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=-\frac {63}{256} \, a^{\frac {5}{2}} x - \frac {1}{20480} \, {\left (25 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 150 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} - 2100 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 2100 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 150 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + 2 \, a^{\frac {5}{2}} e^{\left (-20 \, x\right )} - 2 \, a^{\frac {5}{2}}\right )} e^{\left (10 \, x\right )} \]
-63/256*a^(5/2)*x - 1/20480*(25*a^(5/2)*e^(-2*x) - 150*a^(5/2)*e^(-4*x) + 600*a^(5/2)*e^(-6*x) - 2100*a^(5/2)*e^(-8*x) + 2100*a^(5/2)*e^(-12*x) - 60 0*a^(5/2)*e^(-14*x) + 150*a^(5/2)*e^(-16*x) - 25*a^(5/2)*e^(-18*x) + 2*a^( 5/2)*e^(-20*x) - 2*a^(5/2))*e^(10*x)
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.86 \[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=-\frac {1}{20480} \, {\left (5040 \, a^{2} x - 2 \, a^{2} e^{\left (10 \, x\right )} + 25 \, a^{2} e^{\left (8 \, x\right )} - 150 \, a^{2} e^{\left (6 \, x\right )} + 600 \, a^{2} e^{\left (4 \, x\right )} - 2100 \, a^{2} e^{\left (2 \, x\right )} - {\left (5754 \, a^{2} e^{\left (10 \, x\right )} - 2100 \, a^{2} e^{\left (8 \, x\right )} + 600 \, a^{2} e^{\left (6 \, x\right )} - 150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} - 2 \, a^{2}\right )} e^{\left (-10 \, x\right )}\right )} \sqrt {a} \]
-1/20480*(5040*a^2*x - 2*a^2*e^(10*x) + 25*a^2*e^(8*x) - 150*a^2*e^(6*x) + 600*a^2*e^(4*x) - 2100*a^2*e^(2*x) - (5754*a^2*e^(10*x) - 2100*a^2*e^(8*x ) + 600*a^2*e^(6*x) - 150*a^2*e^(4*x) + 25*a^2*e^(2*x) - 2*a^2)*e^(-10*x)) *sqrt(a)
Timed out. \[ \int \left (a \sinh ^4(x)\right )^{5/2} \, dx=\int {\left (a\,{\mathrm {sinh}\left (x\right )}^4\right )}^{5/2} \,d x \]