Integrand size = 14, antiderivative size = 66 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=-\frac {b c \sqrt {1-c^2 x^4}}{24 x^6}-\frac {b c^3 \sqrt {1-c^2 x^4}}{12 x^2}-\frac {a+b \arcsin \left (c x^2\right )}{8 x^8} \] Output:
-1/24*b*c*(-c^2*x^4+1)^(1/2)/x^6-1/12*b*c^3*(-c^2*x^4+1)^(1/2)/x^2-1/8*(a+ b*arcsin(c*x^2))/x^8
Time = 0.06 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.91 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=-\frac {a}{8 x^8}+\frac {1}{2} b \left (-\frac {c \sqrt {1-c^2 x^4} \left (1+2 c^2 x^4\right )}{12 x^6}-\frac {\arcsin \left (c x^2\right )}{4 x^8}\right ) \] Input:
Integrate[(a + b*ArcSin[c*x^2])/x^9,x]
Output:
-1/8*a/x^8 + (b*(-1/12*(c*Sqrt[1 - c^2*x^4]*(1 + 2*c^2*x^4))/x^6 - ArcSin[ c*x^2]/(4*x^8)))/2
Time = 0.24 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.06, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5341, 27, 803, 796}
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.
\(\displaystyle \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx\) |
\(\Big \downarrow \) 5341 |
\(\displaystyle \frac {1}{8} b \int \frac {2 c}{x^7 \sqrt {1-c^2 x^4}}dx-\frac {a+b \arcsin \left (c x^2\right )}{8 x^8}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} b c \int \frac {1}{x^7 \sqrt {1-c^2 x^4}}dx-\frac {a+b \arcsin \left (c x^2\right )}{8 x^8}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle \frac {1}{4} b c \left (\frac {2}{3} c^2 \int \frac {1}{x^3 \sqrt {1-c^2 x^4}}dx-\frac {\sqrt {1-c^2 x^4}}{6 x^6}\right )-\frac {a+b \arcsin \left (c x^2\right )}{8 x^8}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle \frac {1}{4} b c \left (-\frac {\sqrt {1-c^2 x^4}}{6 x^6}-\frac {c^2 \sqrt {1-c^2 x^4}}{3 x^2}\right )-\frac {a+b \arcsin \left (c x^2\right )}{8 x^8}\) |
Input:
Int[(a + b*ArcSin[c*x^2])/x^9,x]
Output:
(b*c*(-1/6*Sqrt[1 - c^2*x^4]/x^6 - (c^2*Sqrt[1 - c^2*x^4])/(3*x^2)))/4 - ( a + b*ArcSin[c*x^2])/(8*x^8)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(c + d*x)^(m + 1)*((a + b*ArcSin[u])/(d*(m + 1))), x] - Simp[b/(d*(m + 1) ) Int[SimplifyIntegrand[(c + d*x)^(m + 1)*(D[u, x]/Sqrt[1 - u^2]), x], x] , x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] && !FunctionOfQ[(c + d*x)^(m + 1), u, x] && !FunctionOfExponentialQ[u, x]
Time = 0.07 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.97
method | result | size |
default | \(-\frac {a}{8 x^{8}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{8 x^{8}}+\frac {c \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (2 c^{2} x^{4}+1\right )}{24 x^{6} \sqrt {-c^{2} x^{4}+1}}\right )\) | \(64\) |
parts | \(-\frac {a}{8 x^{8}}+b \left (-\frac {\arcsin \left (c \,x^{2}\right )}{8 x^{8}}+\frac {c \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (2 c^{2} x^{4}+1\right )}{24 x^{6} \sqrt {-c^{2} x^{4}+1}}\right )\) | \(64\) |
orering | \(\frac {\left (\frac {3}{8} c^{4} x^{9}-\frac {3}{16} c^{2} x^{5}-\frac {5}{16} x \right ) \left (a +b \arcsin \left (c \,x^{2}\right )\right )}{x^{9}}+\frac {\left (2 c^{2} x^{4}+1\right ) x^{2} \left (c \,x^{2}-1\right ) \left (c \,x^{2}+1\right ) \left (\frac {2 b c}{x^{8} \sqrt {-c^{2} x^{4}+1}}-\frac {9 \left (a +b \arcsin \left (c \,x^{2}\right )\right )}{x^{10}}\right )}{48}\) | \(100\) |
Input:
int((a+b*arcsin(c*x^2))/x^9,x,method=_RETURNVERBOSE)
Output:
-1/8*a/x^8+b*(-1/8/x^8*arcsin(c*x^2)+1/24*c*(c*x^2-1)*(c*x^2+1)*(2*c^2*x^4 +1)/x^6/(-c^2*x^4+1)^(1/2))
Time = 0.11 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.82 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=\frac {3 \, a x^{8} - 3 \, b \arcsin \left (c x^{2}\right ) - {\left (2 \, b c^{3} x^{6} + b c x^{2}\right )} \sqrt {-c^{2} x^{4} + 1} - 3 \, a}{24 \, x^{8}} \] Input:
integrate((a+b*arcsin(c*x^2))/x^9,x, algorithm="fricas")
Output:
1/24*(3*a*x^8 - 3*b*arcsin(c*x^2) - (2*b*c^3*x^6 + b*c*x^2)*sqrt(-c^2*x^4 + 1) - 3*a)/x^8
Time = 1.85 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.70 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=- \frac {a}{8 x^{8}} + \frac {b c \left (\begin {cases} - \frac {i c^{2} \sqrt {c^{2} x^{4} - 1}}{3 x^{2}} - \frac {i \sqrt {c^{2} x^{4} - 1}}{6 x^{6}} & \text {for}\: \left |{c^{2} x^{4}}\right | > 1 \\- \frac {c^{2} \sqrt {- c^{2} x^{4} + 1}}{3 x^{2}} - \frac {\sqrt {- c^{2} x^{4} + 1}}{6 x^{6}} & \text {otherwise} \end {cases}\right )}{4} - \frac {b \operatorname {asin}{\left (c x^{2} \right )}}{8 x^{8}} \] Input:
integrate((a+b*asin(c*x**2))/x**9,x)
Output:
-a/(8*x**8) + b*c*Piecewise((-I*c**2*sqrt(c**2*x**4 - 1)/(3*x**2) - I*sqrt (c**2*x**4 - 1)/(6*x**6), Abs(c**2*x**4) > 1), (-c**2*sqrt(-c**2*x**4 + 1) /(3*x**2) - sqrt(-c**2*x**4 + 1)/(6*x**6), True))/4 - b*asin(c*x**2)/(8*x* *8)
Time = 0.10 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=-\frac {1}{24} \, {\left (c {\left (\frac {3 \, \sqrt {-c^{2} x^{4} + 1} c^{2}}{x^{2}} + \frac {{\left (-c^{2} x^{4} + 1\right )}^{\frac {3}{2}}}{x^{6}}\right )} + \frac {3 \, \arcsin \left (c x^{2}\right )}{x^{8}}\right )} b - \frac {a}{8 \, x^{8}} \] Input:
integrate((a+b*arcsin(c*x^2))/x^9,x, algorithm="maxima")
Output:
-1/24*(c*(3*sqrt(-c^2*x^4 + 1)*c^2/x^2 + (-c^2*x^4 + 1)^(3/2)/x^6) + 3*arc sin(c*x^2)/x^8)*b - 1/8*a/x^8
Leaf count of result is larger than twice the leaf count of optimal. 342 vs. \(2 (56) = 112\).
Time = 0.14 (sec) , antiderivative size = 342, normalized size of antiderivative = 5.18 \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=-\frac {\frac {3 \, b c^{9} x^{8} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} + \frac {3 \, a c^{9} x^{8}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}} - \frac {2 \, b c^{8} x^{6}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac {12 \, b c^{7} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac {12 \, a c^{7} x^{4}}{{\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac {18 \, b c^{6} x^{2}}{\sqrt {-c^{2} x^{4} + 1} + 1} + 18 \, b c^{5} \arcsin \left (c x^{2}\right ) + 18 \, a c^{5} + \frac {18 \, b c^{4} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac {12 \, b c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac {12 \, a c^{3} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac {2 \, b c^{2} {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}} + \frac {3 \, b c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4} \arcsin \left (c x^{2}\right )}{x^{8}} + \frac {3 \, a c {\left (\sqrt {-c^{2} x^{4} + 1} + 1\right )}^{4}}{x^{8}}}{384 \, c} \] Input:
integrate((a+b*arcsin(c*x^2))/x^9,x, algorithm="giac")
Output:
-1/384*(3*b*c^9*x^8*arcsin(c*x^2)/(sqrt(-c^2*x^4 + 1) + 1)^4 + 3*a*c^9*x^8 /(sqrt(-c^2*x^4 + 1) + 1)^4 - 2*b*c^8*x^6/(sqrt(-c^2*x^4 + 1) + 1)^3 + 12* b*c^7*x^4*arcsin(c*x^2)/(sqrt(-c^2*x^4 + 1) + 1)^2 + 12*a*c^7*x^4/(sqrt(-c ^2*x^4 + 1) + 1)^2 - 18*b*c^6*x^2/(sqrt(-c^2*x^4 + 1) + 1) + 18*b*c^5*arcs in(c*x^2) + 18*a*c^5 + 18*b*c^4*(sqrt(-c^2*x^4 + 1) + 1)/x^2 + 12*b*c^3*(s qrt(-c^2*x^4 + 1) + 1)^2*arcsin(c*x^2)/x^4 + 12*a*c^3*(sqrt(-c^2*x^4 + 1) + 1)^2/x^4 + 2*b*c^2*(sqrt(-c^2*x^4 + 1) + 1)^3/x^6 + 3*b*c*(sqrt(-c^2*x^4 + 1) + 1)^4*arcsin(c*x^2)/x^8 + 3*a*c*(sqrt(-c^2*x^4 + 1) + 1)^4/x^8)/c
Timed out. \[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x^2\right )}{x^9} \,d x \] Input:
int((a + b*asin(c*x^2))/x^9,x)
Output:
int((a + b*asin(c*x^2))/x^9, x)
\[ \int \frac {a+b \arcsin \left (c x^2\right )}{x^9} \, dx=\frac {8 \left (\int \frac {\mathit {asin} \left (c \,x^{2}\right )}{x^{9}}d x \right ) b \,x^{8}-a}{8 x^{8}} \] Input:
int((a+b*asin(c*x^2))/x^9,x)
Output:
(8*int(asin(c*x**2)/x**9,x)*b*x**8 - a)/(8*x**8)