Integrand size = 14, antiderivative size = 79 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {\cosh \left (a+b x^2\right )}{3 b^2}-\frac {\cosh ^3\left (a+b x^2\right )}{18 b^2}+\frac {x^2 \sinh \left (a+b x^2\right )}{3 b}+\frac {x^2 \cosh ^2\left (a+b x^2\right ) \sinh \left (a+b x^2\right )}{6 b} \]
-1/3*cosh(b*x^2+a)/b^2-1/18*cosh(b*x^2+a)^3/b^2+1/3*x^2*sinh(b*x^2+a)/b+1/ 6*x^2*cosh(b*x^2+a)^2*sinh(b*x^2+a)/b
Time = 0.13 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.70 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {27 \cosh \left (a+b x^2\right )+\cosh \left (3 \left (a+b x^2\right )\right )-3 b x^2 \left (9 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )\right )}{72 b^2} \]
-1/72*(27*Cosh[a + b*x^2] + Cosh[3*(a + b*x^2)] - 3*b*x^2*(9*Sinh[a + b*x^ 2] + Sinh[3*(a + b*x^2)]))/b^2
Time = 0.40 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.05, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {5844, 3042, 3791, 3042, 3777, 26, 3042, 26, 3118}
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.
\(\displaystyle \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 5844 |
\(\displaystyle \frac {1}{2} \int x^2 \cosh ^3\left (b x^2+a\right )dx^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int x^2 \sin \left (i b x^2+i a+\frac {\pi }{2}\right )^3dx^2\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \int x^2 \cosh \left (b x^2+a\right )dx^2-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \int x^2 \sin \left (i b x^2+i a+\frac {\pi }{2}\right )dx^2-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {x^2 \sinh \left (a+b x^2\right )}{b}-\frac {i \int -i \sinh \left (b x^2+a\right )dx^2}{b}\right )-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {x^2 \sinh \left (a+b x^2\right )}{b}-\frac {\int \sinh \left (b x^2+a\right )dx^2}{b}\right )-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {x^2 \sinh \left (a+b x^2\right )}{b}-\frac {\int -i \sin \left (i b x^2+i a\right )dx^2}{b}\right )-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {x^2 \sinh \left (a+b x^2\right )}{b}+\frac {i \int \sin \left (i b x^2+i a\right )dx^2}{b}\right )-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {1}{2} \left (-\frac {\cosh ^3\left (a+b x^2\right )}{9 b^2}+\frac {2}{3} \left (\frac {x^2 \sinh \left (a+b x^2\right )}{b}-\frac {\cosh \left (a+b x^2\right )}{b^2}\right )+\frac {x^2 \sinh \left (a+b x^2\right ) \cosh ^2\left (a+b x^2\right )}{3 b}\right )\) |
(-1/9*Cosh[a + b*x^2]^3/b^2 + (x^2*Cosh[a + b*x^2]^2*Sinh[a + b*x^2])/(3*b ) + (2*(-(Cosh[a + b*x^2]/b^2) + (x^2*Sinh[a + b*x^2])/b))/3)/2
3.1.15.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[d*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Simp[b*(c + d*x)*Cos[e + f*x ]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^2*((n - 1)/n) Int[(c + d* x)*(b*Sin[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]
Int[((a_.) + Cosh[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Cosh[c + d*x] )^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplif y[(m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplif y[(m + 1)/n], 0]))
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {\left (3 b \,x^{2}-1\right ) {\mathrm e}^{3 b \,x^{2}+3 a}}{144 b^{2}}+\frac {3 \left (b \,x^{2}-1\right ) {\mathrm e}^{b \,x^{2}+a}}{16 b^{2}}-\frac {3 \left (b \,x^{2}+1\right ) {\mathrm e}^{-b \,x^{2}-a}}{16 b^{2}}-\frac {\left (3 b \,x^{2}+1\right ) {\mathrm e}^{-3 b \,x^{2}-3 a}}{144 b^{2}}\) | \(93\) |
parallelrisch | \(\frac {-9 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{5} x^{2} b +6 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{3} x^{2} b +9 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{4}-9 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right ) x^{2} b -12 \tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )^{2}+7}{9 b^{2} {\left (1+\tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )\right )}^{3} {\left (\tanh \left (\frac {b \,x^{2}}{2}+\frac {a}{2}\right )-1\right )}^{3}}\) | \(123\) |
1/144*(3*b*x^2-1)/b^2*exp(3*b*x^2+3*a)+3/16*(b*x^2-1)/b^2*exp(b*x^2+a)-3/1 6*(b*x^2+1)/b^2*exp(-b*x^2-a)-1/144*(3*b*x^2+1)/b^2*exp(-3*b*x^2-3*a)
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.22 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 \, b x^{2} \sinh \left (b x^{2} + a\right )^{3} - \cosh \left (b x^{2} + a\right )^{3} - 3 \, \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} + 9 \, {\left (b x^{2} \cosh \left (b x^{2} + a\right )^{2} + 3 \, b x^{2}\right )} \sinh \left (b x^{2} + a\right ) - 27 \, \cosh \left (b x^{2} + a\right )}{72 \, b^{2}} \]
1/72*(3*b*x^2*sinh(b*x^2 + a)^3 - cosh(b*x^2 + a)^3 - 3*cosh(b*x^2 + a)*si nh(b*x^2 + a)^2 + 9*(b*x^2*cosh(b*x^2 + a)^2 + 3*b*x^2)*sinh(b*x^2 + a) - 27*cosh(b*x^2 + a))/b^2
Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=\begin {cases} - \frac {x^{2} \sinh ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {x^{2} \sinh {\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{2 b} + \frac {\sinh ^{2}{\left (a + b x^{2} \right )} \cosh {\left (a + b x^{2} \right )}}{3 b^{2}} - \frac {7 \cosh ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \cosh ^{3}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((-x**2*sinh(a + b*x**2)**3/(3*b) + x**2*sinh(a + b*x**2)*cosh(a + b*x**2)**2/(2*b) + sinh(a + b*x**2)**2*cosh(a + b*x**2)/(3*b**2) - 7*cos h(a + b*x**2)**3/(18*b**2), Ne(b, 0)), (x**4*cosh(a)**3/4, True))
Time = 0.20 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.27 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {{\left (3 \, b x^{2} e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x^{2}\right )}}{144 \, b^{2}} + \frac {3 \, {\left (b x^{2} e^{a} - e^{a}\right )} e^{\left (b x^{2}\right )}}{16 \, b^{2}} - \frac {3 \, {\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{16 \, b^{2}} - \frac {{\left (3 \, b x^{2} + 1\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} \]
1/144*(3*b*x^2*e^(3*a) - e^(3*a))*e^(3*b*x^2)/b^2 + 3/16*(b*x^2*e^a - e^a) *e^(b*x^2)/b^2 - 3/16*(b*x^2 + 1)*e^(-b*x^2 - a)/b^2 - 1/144*(3*b*x^2 + 1) *e^(-3*b*x^2 - 3*a)/b^2
Leaf count of result is larger than twice the leaf count of optimal. 192 vs. \(2 (71) = 142\).
Time = 0.27 (sec) , antiderivative size = 192, normalized size of antiderivative = 2.43 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 \, {\left (b x^{2} + a\right )} e^{\left (3 \, b x^{2} + 3 \, a\right )} + 27 \, {\left (b x^{2} + a\right )} e^{\left (b x^{2} + a\right )} - 27 \, {\left (b x^{2} + a\right )} e^{\left (-b x^{2} - a\right )} - 3 \, {\left (b x^{2} + a\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )} - e^{\left (3 \, b x^{2} + 3 \, a\right )} - 27 \, e^{\left (b x^{2} + a\right )} - 27 \, e^{\left (-b x^{2} - a\right )} - e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} - \frac {a e^{\left (3 \, b x^{2} + 3 \, a\right )} + 9 \, a e^{\left (b x^{2} + a\right )} - {\left (9 \, a e^{\left (2 \, b x^{2} + 2 \, a\right )} + a\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b^{2}} \]
1/144*(3*(b*x^2 + a)*e^(3*b*x^2 + 3*a) + 27*(b*x^2 + a)*e^(b*x^2 + a) - 27 *(b*x^2 + a)*e^(-b*x^2 - a) - 3*(b*x^2 + a)*e^(-3*b*x^2 - 3*a) - e^(3*b*x^ 2 + 3*a) - 27*e^(b*x^2 + a) - 27*e^(-b*x^2 - a) - e^(-3*b*x^2 - 3*a))/b^2 - 1/48*(a*e^(3*b*x^2 + 3*a) + 9*a*e^(b*x^2 + a) - (9*a*e^(2*b*x^2 + 2*a) + a)*e^(-3*b*x^2 - 3*a))/b^2
Time = 1.62 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int x^3 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {\frac {x^2\,\mathrm {sinh}\left (b\,x^2+a\right )}{3}+\frac {x^2\,{\mathrm {cosh}\left (b\,x^2+a\right )}^2\,\mathrm {sinh}\left (b\,x^2+a\right )}{6}}{b}-\frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^3}{18\,b^2}-\frac {\mathrm {cosh}\left (b\,x^2+a\right )}{3\,b^2} \]