Integrand size = 10, antiderivative size = 78 \[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=\frac {5}{16} a \coth (x) \sqrt {a \sinh ^4(x)}-\frac {5}{16} a x \text {csch}^2(x) \sqrt {a \sinh ^4(x)}-\frac {5}{24} a \cosh (x) \sinh (x) \sqrt {a \sinh ^4(x)}+\frac {1}{6} a \cosh (x) \sinh ^3(x) \sqrt {a \sinh ^4(x)} \]
5/16*a*coth(x)*(a*sinh(x)^4)^(1/2)-5/16*a*x*csch(x)^2*(a*sinh(x)^4)^(1/2)- 5/24*a*cosh(x)*sinh(x)*(a*sinh(x)^4)^(1/2)+1/6*a*cosh(x)*sinh(x)^3*(a*sinh (x)^4)^(1/2)
Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 0.49 \[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=\frac {1}{192} \text {csch}^6(x) \left (a \sinh ^4(x)\right )^{3/2} (-60 x+45 \sinh (2 x)-9 \sinh (4 x)+\sinh (6 x)) \]
Time = 0.33 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.200, Rules used = {3042, 3686, 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.
\(\displaystyle \int \left (a \sinh ^4(x)\right )^{3/2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \left (a \sin (i x)^4\right )^{3/2}dx\) |
\(\Big \downarrow \) 3686 |
\(\displaystyle a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \int \sinh ^6(x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \int -\sin (i x)^6dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \int \sin (i x)^6dx\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {5}{6} \int \sinh ^4(x)dx-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \int \sin (i x)^4dx\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {5}{6} \left (\frac {3}{4} \int -\sinh ^2(x)dx+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int \sinh ^2(x)dx\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)-\frac {3}{4} \int -\sin (i x)^2dx\right )\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (-\frac {1}{6} \sinh ^5(x) \cosh (x)+\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \int \sin (i x)^2dx\right )\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )+\frac {1}{4} \sinh ^3(x) \cosh (x)\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -a \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \left (\frac {5}{6} \left (\frac {1}{4} \sinh ^3(x) \cosh (x)+\frac {3}{4} \left (\frac {x}{2}-\frac {1}{2} \sinh (x) \cosh (x)\right )\right )-\frac {1}{6} \sinh ^5(x) \cosh (x)\right )\) |
-(a*Csch[x]^2*Sqrt[a*Sinh[x]^4]*(-1/6*(Cosh[x]*Sinh[x]^5) + (5*((Cosh[x]*S inh[x]^3)/4 + (3*(x/2 - (Cosh[x]*Sinh[x])/2))/4))/6))
3.2.53.3.1 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 = 1.64 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {\sqrt {a}\, \left (-1+\cosh \left (2 x \right )\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (1+\cosh \left (2 x \right )\right )}\, \left (2 \sqrt {a}\, \sqrt {a \sinh \left (2 x \right )^{2}}\, \sinh \left (2 x \right )^{2}-9 \cosh \left (2 x \right ) \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}+24 \sqrt {a \sinh \left (2 x \right )^{2}}\, \sqrt {a}-15 \ln \left (\cosh \left (2 x \right ) \sqrt {a}+\sqrt {a \sinh \left (2 x \right )^{2}}\right ) a \right )}{96 \sinh \left (2 x \right ) \sqrt {\left (-1+\cosh \left (2 x \right )\right )^{2} a}}\) | \(125\) |
risch | \(-\frac {5 a \,{\mathrm e}^{2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}\, x}{16 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {a \,{\mathrm e}^{8 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{384 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {3 a \,{\mathrm e}^{6 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{128 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {15 a \,{\mathrm e}^{4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{128 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {15 \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}\, a}{128 \left ({\mathrm e}^{2 x}-1\right )^{2}}+\frac {3 a \,{\mathrm e}^{-2 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{128 \left ({\mathrm e}^{2 x}-1\right )^{2}}-\frac {a \,{\mathrm e}^{-4 x} \sqrt {a \left ({\mathrm e}^{2 x}-1\right )^{4} {\mathrm e}^{-4 x}}}{384 \left ({\mathrm e}^{2 x}-1\right )^{2}}\) | \(216\) |
1/96*a^(1/2)*(-1+cosh(2*x))*(a*(-1+cosh(2*x))*(1+cosh(2*x)))^(1/2)*(2*a^(1 /2)*(a*sinh(2*x)^2)^(1/2)*sinh(2*x)^2-9*cosh(2*x)*(a*sinh(2*x)^2)^(1/2)*a^ (1/2)+24*(a*sinh(2*x)^2)^(1/2)*a^(1/2)-15*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. 659 vs. \(2 (62) = 124\).
Time = 0.30 (sec) , antiderivative size = 659, normalized size of antiderivative = 8.45 \[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=\text {Too large to display} \]
1/384*(12*a*cosh(x)*e^(2*x)*sinh(x)^11 + a*e^(2*x)*sinh(x)^12 + 3*(22*a*co sh(x)^2 - 3*a)*e^(2*x)*sinh(x)^10 + 10*(22*a*cosh(x)^3 - 9*a*cosh(x))*e^(2 *x)*sinh(x)^9 + 45*(11*a*cosh(x)^4 - 9*a*cosh(x)^2 + a)*e^(2*x)*sinh(x)^8 + 72*(11*a*cosh(x)^5 - 15*a*cosh(x)^3 + 5*a*cosh(x))*e^(2*x)*sinh(x)^7 + 6 *(154*a*cosh(x)^6 - 315*a*cosh(x)^4 + 210*a*cosh(x)^2 - 20*a*x)*e^(2*x)*si nh(x)^6 + 36*(22*a*cosh(x)^7 - 63*a*cosh(x)^5 + 70*a*cosh(x)^3 - 20*a*x*co sh(x))*e^(2*x)*sinh(x)^5 + 45*(11*a*cosh(x)^8 - 42*a*cosh(x)^6 + 70*a*cosh (x)^4 - 40*a*x*cosh(x)^2 - a)*e^(2*x)*sinh(x)^4 + 20*(11*a*cosh(x)^9 - 54* a*cosh(x)^7 + 126*a*cosh(x)^5 - 120*a*x*cosh(x)^3 - 9*a*cosh(x))*e^(2*x)*s inh(x)^3 + 3*(22*a*cosh(x)^10 - 135*a*cosh(x)^8 + 420*a*cosh(x)^6 - 600*a* x*cosh(x)^4 - 90*a*cosh(x)^2 + 3*a)*e^(2*x)*sinh(x)^2 + 6*(2*a*cosh(x)^11 - 15*a*cosh(x)^9 + 60*a*cosh(x)^7 - 120*a*x*cosh(x)^5 - 30*a*cosh(x)^3 + 3 *a*cosh(x))*e^(2*x)*sinh(x) + (a*cosh(x)^12 - 9*a*cosh(x)^10 + 45*a*cosh(x )^8 - 120*a*x*cosh(x)^6 - 45*a*cosh(x)^4 + 9*a*cosh(x)^2 - a)*e^(2*x))*sqr t(a*e^(8*x) - 4*a*e^(6*x) + 6*a*e^(4*x) - 4*a*e^(2*x) + a)*e^(-2*x)/(cosh( x)^6*e^(4*x) - 2*cosh(x)^6*e^(2*x) + (e^(4*x) - 2*e^(2*x) + 1)*sinh(x)^6 + cosh(x)^6 + 6*(cosh(x)*e^(4*x) - 2*cosh(x)*e^(2*x) + cosh(x))*sinh(x)^5 + 15*(cosh(x)^2*e^(4*x) - 2*cosh(x)^2*e^(2*x) + cosh(x)^2)*sinh(x)^4 + 20*( cosh(x)^3*e^(4*x) - 2*cosh(x)^3*e^(2*x) + cosh(x)^3)*sinh(x)^3 + 15*(cosh( x)^4*e^(4*x) - 2*cosh(x)^4*e^(2*x) + cosh(x)^4)*sinh(x)^2 + 6*(cosh(x)^...
\[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=\int \left (a \sinh ^{4}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.81 \[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=-\frac {5}{16} \, a^{\frac {3}{2}} x - \frac {1}{384} \, {\left (9 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )} - 45 \, a^{\frac {3}{2}} e^{\left (-4 \, x\right )} + 45 \, a^{\frac {3}{2}} e^{\left (-8 \, x\right )} - 9 \, a^{\frac {3}{2}} e^{\left (-10 \, x\right )} + a^{\frac {3}{2}} e^{\left (-12 \, x\right )} - a^{\frac {3}{2}}\right )} e^{\left (6 \, x\right )} \]
-5/16*a^(3/2)*x - 1/384*(9*a^(3/2)*e^(-2*x) - 45*a^(3/2)*e^(-4*x) + 45*a^( 3/2)*e^(-8*x) - 9*a^(3/2)*e^(-10*x) + a^(3/2)*e^(-12*x) - a^(3/2))*e^(6*x)
Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.64 \[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=\frac {1}{384} \, {\left ({\left (110 \, e^{\left (6 \, x\right )} - 45 \, e^{\left (4 \, x\right )} + 9 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-6 \, x\right )} - 120 \, x + e^{\left (6 \, x\right )} - 9 \, e^{\left (4 \, x\right )} + 45 \, e^{\left (2 \, x\right )}\right )} a^{\frac {3}{2}} \]
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. \[ \int \left (a \sinh ^4(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {sinh}\left (x\right )}^4\right )}^{3/2} \,d x \]