Integrand size = 14, antiderivative size = 79 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=-\frac {x^2 \cosh \left (a+b x^2\right )}{3 b}+\frac {\sinh \left (a+b x^2\right )}{3 b^2}+\frac {x^2 \cosh \left (a+b x^2\right ) \sinh ^2\left (a+b x^2\right )}{6 b}-\frac {\sinh ^3\left (a+b x^2\right )}{18 b^2} \]
-1/3*x^2*cosh(b*x^2+a)/b+1/3*sinh(b*x^2+a)/b^2+1/6*x^2*cosh(b*x^2+a)*sinh( b*x^2+a)^2/b-1/18*sinh(b*x^2+a)^3/b^2
Time = 0.10 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.73 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=-\frac {27 b x^2 \cosh \left (a+b x^2\right )-3 b x^2 \cosh \left (3 \left (a+b x^2\right )\right )-27 \sinh \left (a+b x^2\right )+\sinh \left (3 \left (a+b x^2\right )\right )}{72 b^2} \]
-1/72*(27*b*x^2*Cosh[a + b*x^2] - 3*b*x^2*Cosh[3*(a + b*x^2)] - 27*Sinh[a + b*x^2] + Sinh[3*(a + b*x^2)])/b^2
Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.19, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {5843, 3042, 26, 3791, 26, 3042, 26, 3777, 3042, 3117}
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.
\(\displaystyle \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 5843 |
\(\displaystyle \frac {1}{2} \int x^2 \sinh ^3\left (b x^2+a\right )dx^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int i x^2 \sin \left (i b x^2+i a\right )^3dx^2\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} i \int x^2 \sin \left (i b x^2+i a\right )^3dx^2\) |
\(\Big \downarrow \) 3791 |
\(\displaystyle \frac {1}{2} i \left (\frac {2}{3} \int i x^2 \sinh \left (b x^2+a\right )dx^2+\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} i \left (\frac {2}{3} i \int x^2 \sinh \left (b x^2+a\right )dx^2+\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} i \left (\frac {2}{3} i \int -i x^2 \sin \left (i b x^2+i a\right )dx^2+\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {1}{2} i \left (\frac {2}{3} \int x^2 \sin \left (i b x^2+i a\right )dx^2+\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{2} i \left (\frac {2}{3} \left (\frac {i x^2 \cosh \left (a+b x^2\right )}{b}-\frac {i \int \cosh \left (b x^2+a\right )dx^2}{b}\right )+\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} i \left (\frac {2}{3} \left (\frac {i x^2 \cosh \left (a+b x^2\right )}{b}-\frac {i \int \sin \left (i b x^2+i a+\frac {\pi }{2}\right )dx^2}{b}\right )+\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3117 |
\(\displaystyle \frac {1}{2} i \left (\frac {i \sinh ^3\left (a+b x^2\right )}{9 b^2}+\frac {2}{3} \left (\frac {i x^2 \cosh \left (a+b x^2\right )}{b}-\frac {i \sinh \left (a+b x^2\right )}{b^2}\right )-\frac {i x^2 \sinh ^2\left (a+b x^2\right ) \cosh \left (a+b x^2\right )}{3 b}\right )\) |
(I/2)*(((-1/3*I)*x^2*Cosh[a + b*x^2]*Sinh[a + b*x^2]^2)/b + ((I/9)*Sinh[a + b*x^2]^3)/b^2 + (2*((I*x^2*Cosh[a + b*x^2])/b - (I*Sinh[a + b*x^2])/b^2) )/3)
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[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]
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[(x_)^(m_.)*((a_.) + (b_.)*Sinh[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbo l] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sinh[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 = 0.80 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.18
method | result | size |
risch | \(\frac {\left (3 x^{2} b -1\right ) {\mathrm e}^{3 x^{2} b +3 a}}{144 b^{2}}-\frac {3 \left (x^{2} b -1\right ) {\mathrm e}^{x^{2} b +a}}{16 b^{2}}-\frac {3 \left (x^{2} b +1\right ) {\mathrm e}^{-x^{2} b -a}}{16 b^{2}}+\frac {\left (3 x^{2} b +1\right ) {\mathrm e}^{-3 x^{2} b -3 a}}{144 b^{2}}\) | \(93\) |
parallelrisch | \(\frac {-27 \cosh \left (x^{2} b +a \right ) b \,x^{2}+3 x^{2} b \cosh \left (3 x^{2} b +3 a \right )-24 x^{2} b +27 \sinh \left (x^{2} b +a \right )-24 \ln \left (1-\tanh \left (\frac {x^{2} b}{2}+\frac {a}{2}\right )\right )+24 \ln \left (\tanh \left (\frac {x^{2} b}{2}+\frac {a}{2}\right )+1\right )-\sinh \left (3 x^{2} b +3 a \right )}{72 b^{2}}\) | \(101\) |
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 = 94, normalized size of antiderivative = 1.19 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=\frac {3 \, b x^{2} \cosh \left (b x^{2} + a\right )^{3} + 9 \, b x^{2} \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right )^{2} - 27 \, b x^{2} \cosh \left (b x^{2} + a\right ) - \sinh \left (b x^{2} + a\right )^{3} - 3 \, {\left (\cosh \left (b x^{2} + a\right )^{2} - 9\right )} \sinh \left (b x^{2} + a\right )}{72 \, b^{2}} \]
1/72*(3*b*x^2*cosh(b*x^2 + a)^3 + 9*b*x^2*cosh(b*x^2 + a)*sinh(b*x^2 + a)^ 2 - 27*b*x^2*cosh(b*x^2 + a) - sinh(b*x^2 + a)^3 - 3*(cosh(b*x^2 + a)^2 - 9)*sinh(b*x^2 + a))/b^2
Time = 0.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.16 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=\begin {cases} \frac {x^{2} \sinh ^{2}{\left (a + b x^{2} \right )} \cosh {\left (a + b x^{2} \right )}}{2 b} - \frac {x^{2} \cosh ^{3}{\left (a + b x^{2} \right )}}{3 b} - \frac {7 \sinh ^{3}{\left (a + b x^{2} \right )}}{18 b^{2}} + \frac {\sinh {\left (a + b x^{2} \right )} \cosh ^{2}{\left (a + b x^{2} \right )}}{3 b^{2}} & \text {for}\: b \neq 0 \\\frac {x^{4} \sinh ^{3}{\left (a \right )}}{4} & \text {otherwise} \end {cases} \]
Piecewise((x**2*sinh(a + b*x**2)**2*cosh(a + b*x**2)/(2*b) - x**2*cosh(a + b*x**2)**3/(3*b) - 7*sinh(a + b*x**2)**3/(18*b**2) + sinh(a + b*x**2)*cos h(a + b*x**2)**2/(3*b**2), Ne(b, 0)), (x**4*sinh(a)**3/4, True))
Time = 0.23 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.27 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=\frac {{\left (3 \, b x^{2} e^{\left (3 \, a\right )} - e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x^{2}\right )}}{144 \, b^{2}} - \frac {3 \, {\left (b x^{2} e^{a} - e^{a}\right )} e^{\left (b x^{2}\right )}}{16 \, b^{2}} - \frac {3 \, {\left (b x^{2} + 1\right )} e^{\left (-b x^{2} - a\right )}}{16 \, b^{2}} + \frac {{\left (3 \, b x^{2} + 1\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} \]
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 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=\frac {3 \, {\left (b x^{2} + a\right )} e^{\left (3 \, b x^{2} + 3 \, a\right )} - 27 \, {\left (b x^{2} + a\right )} e^{\left (b x^{2} + a\right )} - 27 \, {\left (b x^{2} + a\right )} e^{\left (-b x^{2} - a\right )} + 3 \, {\left (b x^{2} + a\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )} - e^{\left (3 \, b x^{2} + 3 \, a\right )} + 27 \, e^{\left (b x^{2} + a\right )} - 27 \, e^{\left (-b x^{2} - a\right )} + e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{144 \, b^{2}} - \frac {a e^{\left (3 \, b x^{2} + 3 \, a\right )} - 9 \, a e^{\left (b x^{2} + a\right )} - {\left (9 \, a e^{\left (2 \, b x^{2} + 2 \, a\right )} - a\right )} e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b^{2}} \]
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 = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int x^3 \sinh ^3\left (a+b x^2\right ) \, dx=\frac {\frac {x^2\,{\mathrm {cosh}\left (b\,x^2+a\right )}^3}{6}-\frac {x^2\,\mathrm {cosh}\left (b\,x^2+a\right )}{2}}{b}+\frac {7\,\mathrm {sinh}\left (b\,x^2+a\right )}{18\,b^2}-\frac {{\mathrm {cosh}\left (b\,x^2+a\right )}^2\,\mathrm {sinh}\left (b\,x^2+a\right )}{18\,b^2} \]