Integrand size = 14, antiderivative size = 117 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=\frac {7 \cos \left (a+b x^2\right )}{9 b^3}-\frac {x^4 \cos \left (a+b x^2\right )}{3 b}-\frac {\cos ^3\left (a+b x^2\right )}{27 b^3}+\frac {2 x^2 \sin \left (a+b x^2\right )}{3 b^2}-\frac {x^4 \cos \left (a+b x^2\right ) \sin ^2\left (a+b x^2\right )}{6 b}+\frac {x^2 \sin ^3\left (a+b x^2\right )}{9 b^2} \]
7/9*cos(b*x^2+a)/b^3-1/3*x^4*cos(b*x^2+a)/b-1/27*cos(b*x^2+a)^3/b^3+2/3*x^ 2*sin(b*x^2+a)/b^2-1/6*x^4*cos(b*x^2+a)*sin(b*x^2+a)^2/b+1/9*x^2*sin(b*x^2 +a)^3/b^2
Time = 0.20 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.64 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=\frac {-81 \left (-2+b^2 x^4\right ) \cos \left (a+b x^2\right )+\left (-2+9 b^2 x^4\right ) \cos \left (3 \left (a+b x^2\right )\right )-6 b x^2 \left (-27 \sin \left (a+b x^2\right )+\sin \left (3 \left (a+b x^2\right )\right )\right )}{216 b^3} \]
(-81*(-2 + b^2*x^4)*Cos[a + b*x^2] + (-2 + 9*b^2*x^4)*Cos[3*(a + b*x^2)] - 6*b*x^2*(-27*Sin[a + b*x^2] + Sin[3*(a + b*x^2)]))/(216*b^3)
Time = 0.64 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.11, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {3860, 3042, 3792, 3042, 3113, 2009, 3777, 3042, 3777, 25, 3042, 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.
\(\displaystyle \int x^5 \sin ^3\left (a+b x^2\right ) \, dx\) |
\(\Big \downarrow \) 3860 |
\(\displaystyle \frac {1}{2} \int x^4 \sin ^3\left (b x^2+a\right )dx^2\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \int x^4 \sin \left (b x^2+a\right )^3dx^2\) |
\(\Big \downarrow \) 3792 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 \int \sin ^3\left (b x^2+a\right )dx^2}{9 b^2}+\frac {2}{3} \int x^4 \sin \left (b x^2+a\right )dx^2+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (-\frac {2 \int \sin \left (b x^2+a\right )^3dx^2}{9 b^2}+\frac {2}{3} \int x^4 \sin \left (b x^2+a\right )dx^2+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \int \left (1-x^4\right )d\cos \left (b x^2+a\right )}{9 b^3}+\frac {2}{3} \int x^4 \sin \left (b x^2+a\right )dx^2+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \int x^4 \sin \left (b x^2+a\right )dx^2+\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {2 \int x^2 \cos \left (b x^2+a\right )dx^2}{b}-\frac {x^4 \cos \left (a+b x^2\right )}{b}\right )+\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {2 \int x^2 \sin \left (b x^2+a+\frac {\pi }{2}\right )dx^2}{b}-\frac {x^4 \cos \left (a+b x^2\right )}{b}\right )+\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {2 \left (\frac {\int -\sin \left (b x^2+a\right )dx^2}{b}+\frac {x^2 \sin \left (a+b x^2\right )}{b}\right )}{b}-\frac {x^4 \cos \left (a+b x^2\right )}{b}\right )+\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {2 \left (\frac {x^2 \sin \left (a+b x^2\right )}{b}-\frac {\int \sin \left (b x^2+a\right )dx^2}{b}\right )}{b}-\frac {x^4 \cos \left (a+b x^2\right )}{b}\right )+\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {1}{2} \left (\frac {2}{3} \left (\frac {2 \left (\frac {x^2 \sin \left (a+b x^2\right )}{b}-\frac {\int \sin \left (b x^2+a\right )dx^2}{b}\right )}{b}-\frac {x^4 \cos \left (a+b x^2\right )}{b}\right )+\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \frac {1}{2} \left (\frac {2 \left (\cos \left (a+b x^2\right )-\frac {x^6}{3}\right )}{9 b^3}+\frac {2 x^2 \sin ^3\left (a+b x^2\right )}{9 b^2}+\frac {2}{3} \left (\frac {2 \left (\frac {\cos \left (a+b x^2\right )}{b^2}+\frac {x^2 \sin \left (a+b x^2\right )}{b}\right )}{b}-\frac {x^4 \cos \left (a+b x^2\right )}{b}\right )-\frac {x^4 \sin ^2\left (a+b x^2\right ) \cos \left (a+b x^2\right )}{3 b}\right )\) |
((2*(-1/3*x^6 + Cos[a + b*x^2]))/(9*b^3) - (x^4*Cos[a + b*x^2]*Sin[a + b*x ^2]^2)/(3*b) + (2*x^2*Sin[a + b*x^2]^3)/(9*b^2) + (2*(-((x^4*Cos[a + b*x^2 ])/b) + (2*(Cos[a + b*x^2]/b^2 + (x^2*Sin[a + b*x^2])/b))/b))/3)/2
3.1.23.3.1 Defintions of rubi rules used
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[d*m*(c + d*x)^(m - 1)*((b*Sin[e + f*x])^n/(f^2*n^2)), x] + (-Sim p[b*(c + d*x)^m*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(f*n)), x] + Simp[b^ 2*((n - 1)/n) Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[d^2 *m*((m - 1)/(f^2*n^2)) Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]
Int[(x_)^(m_.)*((a_.) + (b_.)*Sin[(c_.) + (d_.)*(x_)^(n_)])^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*Sin[c + d*x])^ p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IntegerQ[Simplify[ (m + 1)/n]] && (EqQ[p, 1] || EqQ[m, n - 1] || (IntegerQ[p] && GtQ[Simplify[ (m + 1)/n], 0]))
Time = 0.20 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {3 \left (b^{2} x^{4}-2\right ) \cos \left (b \,x^{2}+a \right )}{8 b^{3}}+\frac {3 x^{2} \sin \left (b \,x^{2}+a \right )}{4 b^{2}}+\frac {\left (9 b^{2} x^{4}-2\right ) \cos \left (3 b \,x^{2}+3 a \right )}{216 b^{3}}-\frac {x^{2} \sin \left (3 b \,x^{2}+3 a \right )}{36 b^{2}}\) | \(85\) |
default | \(-\frac {3 x^{4} \cos \left (b \,x^{2}+a \right )}{8 b}+\frac {\frac {3 x^{2} \sin \left (b \,x^{2}+a \right )}{4 b}+\frac {3 \cos \left (b \,x^{2}+a \right )}{4 b^{2}}}{b}+\frac {x^{4} \cos \left (3 b \,x^{2}+3 a \right )}{24 b}-\frac {\frac {x^{2} \sin \left (3 b \,x^{2}+3 a \right )}{6 b}+\frac {\cos \left (3 b \,x^{2}+3 a \right )}{18 b^{2}}}{6 b}\) | \(113\) |
-3/8*(b^2*x^4-2)/b^3*cos(b*x^2+a)+3/4*x^2*sin(b*x^2+a)/b^2+1/216*(9*b^2*x^ 4-2)/b^3*cos(3*b*x^2+3*a)-1/36*x^2/b^2*sin(3*b*x^2+3*a)
Time = 0.28 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=\frac {{\left (9 \, b^{2} x^{4} - 2\right )} \cos \left (b x^{2} + a\right )^{3} - 3 \, {\left (9 \, b^{2} x^{4} - 14\right )} \cos \left (b x^{2} + a\right ) - 6 \, {\left (b x^{2} \cos \left (b x^{2} + a\right )^{2} - 7 \, b x^{2}\right )} \sin \left (b x^{2} + a\right )}{54 \, b^{3}} \]
1/54*((9*b^2*x^4 - 2)*cos(b*x^2 + a)^3 - 3*(9*b^2*x^4 - 14)*cos(b*x^2 + a) - 6*(b*x^2*cos(b*x^2 + a)^2 - 7*b*x^2)*sin(b*x^2 + a))/b^3
Time = 0.77 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.22 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=\begin {cases} - \frac {x^{4} \sin ^{2}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{2 b} - \frac {x^{4} \cos ^{3}{\left (a + b x^{2} \right )}}{3 b} + \frac {7 x^{2} \sin ^{3}{\left (a + b x^{2} \right )}}{9 b^{2}} + \frac {2 x^{2} \sin {\left (a + b x^{2} \right )} \cos ^{2}{\left (a + b x^{2} \right )}}{3 b^{2}} + \frac {7 \sin ^{2}{\left (a + b x^{2} \right )} \cos {\left (a + b x^{2} \right )}}{9 b^{3}} + \frac {20 \cos ^{3}{\left (a + b x^{2} \right )}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{6} \sin ^{3}{\left (a \right )}}{6} & \text {otherwise} \end {cases} \]
Piecewise((-x**4*sin(a + b*x**2)**2*cos(a + b*x**2)/(2*b) - x**4*cos(a + b *x**2)**3/(3*b) + 7*x**2*sin(a + b*x**2)**3/(9*b**2) + 2*x**2*sin(a + b*x* *2)*cos(a + b*x**2)**2/(3*b**2) + 7*sin(a + b*x**2)**2*cos(a + b*x**2)/(9* b**3) + 20*cos(a + b*x**2)**3/(27*b**3), Ne(b, 0)), (x**6*sin(a)**3/6, Tru e))
Time = 0.25 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.68 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=-\frac {6 \, b x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right ) - 162 \, b x^{2} \sin \left (b x^{2} + a\right ) - {\left (9 \, b^{2} x^{4} - 2\right )} \cos \left (3 \, b x^{2} + 3 \, a\right ) + 81 \, {\left (b^{2} x^{4} - 2\right )} \cos \left (b x^{2} + a\right )}{216 \, b^{3}} \]
-1/216*(6*b*x^2*sin(3*b*x^2 + 3*a) - 162*b*x^2*sin(b*x^2 + a) - (9*b^2*x^4 - 2)*cos(3*b*x^2 + 3*a) + 81*(b^2*x^4 - 2)*cos(b*x^2 + a))/b^3
Time = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=-\frac {x^{2} \sin \left (3 \, b x^{2} + 3 \, a\right )}{36 \, b^{2}} + \frac {3 \, x^{2} \sin \left (b x^{2} + a\right )}{4 \, b^{2}} + \frac {{\left (\cos \left (b x^{2} + a\right )^{3} - 3 \, \cos \left (b x^{2} + a\right )\right )} a^{2}}{6 \, b^{3}} + \frac {{\left (9 \, {\left (b x^{2} + a\right )}^{2} - 18 \, {\left (b x^{2} + a\right )} a - 2\right )} \cos \left (3 \, b x^{2} + 3 \, a\right )}{216 \, b^{3}} - \frac {3 \, {\left ({\left (b x^{2} + a\right )}^{2} - 2 \, {\left (b x^{2} + a\right )} a - 2\right )} \cos \left (b x^{2} + a\right )}{8 \, b^{3}} \]
-1/36*x^2*sin(3*b*x^2 + 3*a)/b^2 + 3/4*x^2*sin(b*x^2 + a)/b^2 + 1/6*(cos(b *x^2 + a)^3 - 3*cos(b*x^2 + a))*a^2/b^3 + 1/216*(9*(b*x^2 + a)^2 - 18*(b*x ^2 + a)*a - 2)*cos(3*b*x^2 + 3*a)/b^3 - 3/8*((b*x^2 + a)^2 - 2*(b*x^2 + a) *a - 2)*cos(b*x^2 + a)/b^3
Time = 6.40 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.80 \[ \int x^5 \sin ^3\left (a+b x^2\right ) \, dx=\frac {\frac {3\,\cos \left (b\,x^2+a\right )}{4}-\frac {\cos \left (3\,b\,x^2+3\,a\right )}{108}+b\,\left (\frac {3\,x^2\,\sin \left (b\,x^2+a\right )}{4}-\frac {x^2\,\sin \left (3\,b\,x^2+3\,a\right )}{36}\right )+b^2\,\left (\frac {x^4\,\cos \left (3\,b\,x^2+3\,a\right )}{24}-\frac {3\,x^4\,\cos \left (b\,x^2+a\right )}{8}\right )}{b^3} \]