Integrand size = 14, antiderivative size = 56 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {b c \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}-\frac {(a+b \arcsin (c x))^2}{2 x^2}+b^2 c^2 \log (x) \] Output:
-b*c*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))/x-1/2*(a+b*arcsin(c*x))^2/x^2+b^ 2*c^2*ln(x)
Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.45 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {a \left (a+2 b c x \sqrt {1-c^2 x^2}\right )+2 b \left (a+b c x \sqrt {1-c^2 x^2}\right ) \arcsin (c x)+b^2 \arcsin (c x)^2-2 b^2 c^2 x^2 \log (x)}{2 x^2} \] Input:
Integrate[(a + b*ArcSin[c*x])^2/x^3,x]
Output:
-1/2*(a*(a + 2*b*c*x*Sqrt[1 - c^2*x^2]) + 2*b*(a + b*c*x*Sqrt[1 - c^2*x^2] )*ArcSin[c*x] + b^2*ArcSin[c*x]^2 - 2*b^2*c^2*x^2*Log[x])/x^2
Time = 0.36 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.96, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {5138, 5186, 14}
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.
\(\displaystyle \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx\) |
\(\Big \downarrow \) 5138 |
\(\displaystyle b c \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{2 x^2}\) |
\(\Big \downarrow \) 5186 |
\(\displaystyle b c \left (b c \int \frac {1}{x}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}\right )-\frac {(a+b \arcsin (c x))^2}{2 x^2}\) |
\(\Big \downarrow \) 14 |
\(\displaystyle b c \left (b c \log (x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x}\right )-\frac {(a+b \arcsin (c x))^2}{2 x^2}\) |
Input:
Int[(a + b*ArcSin[c*x])^2/x^3,x]
Output:
-1/2*(a + b*ArcSin[c*x])^2/x^2 + b*c*(-((Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c *x]))/x) + b*c*Log[x])
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n /(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 *x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b *ArcSin[c*x])^n/(d*f*(m + 1))), x] - Simp[b*c*(n/(f*(m + 1)))*Simp[(d + e*x ^2)^p/(1 - c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p + 1/2)*(a + b*A rcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^ 2*d + e, 0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Time = 0.13 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.79
method | result | size |
parts | \(-\frac {a^{2}}{2 x^{2}}+b^{2} c^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+\ln \left (c x \right )\right )+2 a b \,c^{2} \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\) | \(100\) |
derivativedivides | \(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+\ln \left (c x \right )\right )+2 a b \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\) | \(101\) |
default | \(c^{2} \left (-\frac {a^{2}}{2 c^{2} x^{2}}+b^{2} \left (-\frac {\arcsin \left (c x \right )^{2}}{2 c^{2} x^{2}}-\frac {\arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}}{c x}+\ln \left (c x \right )\right )+2 a b \left (-\frac {\arcsin \left (c x \right )}{2 c^{2} x^{2}}-\frac {\sqrt {-c^{2} x^{2}+1}}{2 c x}\right )\right )\) | \(101\) |
Input:
int((a+b*arcsin(c*x))^2/x^3,x,method=_RETURNVERBOSE)
Output:
-1/2*a^2/x^2+b^2*c^2*(-1/2/c^2/x^2*arcsin(c*x)^2-arcsin(c*x)/c/x*(-c^2*x^2 +1)^(1/2)+ln(c*x))+2*a*b*c^2*(-1/2/c^2/x^2*arcsin(c*x)-1/2/c/x*(-c^2*x^2+1 )^(1/2))
Time = 0.10 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.30 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=\frac {2 \, b^{2} c^{2} x^{2} \log \left (x\right ) - b^{2} \arcsin \left (c x\right )^{2} - 2 \, a b \arcsin \left (c x\right ) - a^{2} - 2 \, {\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt {-c^{2} x^{2} + 1}}{2 \, x^{2}} \] Input:
integrate((a+b*arcsin(c*x))^2/x^3,x, algorithm="fricas")
Output:
1/2*(2*b^2*c^2*x^2*log(x) - b^2*arcsin(c*x)^2 - 2*a*b*arcsin(c*x) - a^2 - 2*(b^2*c*x*arcsin(c*x) + a*b*c*x)*sqrt(-c^2*x^2 + 1))/x^2
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \] Input:
integrate((a+b*asin(c*x))**2/x**3,x)
Output:
Integral((a + b*asin(c*x))**2/x**3, x)
Time = 0.11 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.54 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx={\left (c^{2} \log \left (x\right ) - \frac {\sqrt {-c^{2} x^{2} + 1} c \arcsin \left (c x\right )}{x}\right )} b^{2} - a b {\left (\frac {\sqrt {-c^{2} x^{2} + 1} c}{x} + \frac {\arcsin \left (c x\right )}{x^{2}}\right )} - \frac {b^{2} \arcsin \left (c x\right )^{2}}{2 \, x^{2}} - \frac {a^{2}}{2 \, x^{2}} \] Input:
integrate((a+b*arcsin(c*x))^2/x^3,x, algorithm="maxima")
Output:
(c^2*log(x) - sqrt(-c^2*x^2 + 1)*c*arcsin(c*x)/x)*b^2 - a*b*(sqrt(-c^2*x^2 + 1)*c/x + arcsin(c*x)/x^2) - 1/2*b^2*arcsin(c*x)^2/x^2 - 1/2*a^2/x^2
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (52) = 104\).
Time = 0.19 (sec) , antiderivative size = 377, normalized size of antiderivative = 6.73 \[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=-\frac {b^{2} c^{4} x^{2} \arcsin \left (c x\right )^{2}}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {a b c^{4} x^{2} \arcsin \left (c x\right )}{4 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {a^{2} c^{4} x^{2}}{8 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {b^{2} c^{3} x \arcsin \left (c x\right )}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {1}{4} \, b^{2} c^{2} \arcsin \left (c x\right )^{2} + \frac {a b c^{3} x}{2 \, {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {1}{2} \, a b c^{2} \arcsin \left (c x\right ) + 2 \, b^{2} c^{2} \log \left (2\right ) + b^{2} c^{2} \log \left ({\left | c \right |} {\left | x \right |}\right ) - b^{2} c^{2} \log \left (2 \, \sqrt {-c^{2} x^{2} + 1} + 2\right ) + b^{2} c^{2} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - \frac {1}{4} \, a^{2} c^{2} - \frac {b^{2} c {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )} \arcsin \left (c x\right )}{2 \, x} - \frac {b^{2} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2} \arcsin \left (c x\right )^{2}}{8 \, x^{2}} - \frac {a b c {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}}{2 \, x} - \frac {a b {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2} \arcsin \left (c x\right )}{4 \, x^{2}} - \frac {a^{2} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}}{8 \, x^{2}} \] Input:
integrate((a+b*arcsin(c*x))^2/x^3,x, algorithm="giac")
Output:
-1/8*b^2*c^4*x^2*arcsin(c*x)^2/(sqrt(-c^2*x^2 + 1) + 1)^2 - 1/4*a*b*c^4*x^ 2*arcsin(c*x)/(sqrt(-c^2*x^2 + 1) + 1)^2 - 1/8*a^2*c^4*x^2/(sqrt(-c^2*x^2 + 1) + 1)^2 + 1/2*b^2*c^3*x*arcsin(c*x)/(sqrt(-c^2*x^2 + 1) + 1) - 1/4*b^2 *c^2*arcsin(c*x)^2 + 1/2*a*b*c^3*x/(sqrt(-c^2*x^2 + 1) + 1) - 1/2*a*b*c^2* arcsin(c*x) + 2*b^2*c^2*log(2) + b^2*c^2*log(abs(c)*abs(x)) - b^2*c^2*log( 2*sqrt(-c^2*x^2 + 1) + 2) + b^2*c^2*log(sqrt(-c^2*x^2 + 1) + 1) - 1/4*a^2* c^2 - 1/2*b^2*c*(sqrt(-c^2*x^2 + 1) + 1)*arcsin(c*x)/x - 1/8*b^2*(sqrt(-c^ 2*x^2 + 1) + 1)^2*arcsin(c*x)^2/x^2 - 1/2*a*b*c*(sqrt(-c^2*x^2 + 1) + 1)/x - 1/4*a*b*(sqrt(-c^2*x^2 + 1) + 1)^2*arcsin(c*x)/x^2 - 1/8*a^2*(sqrt(-c^2 *x^2 + 1) + 1)^2/x^2
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^3} \,d x \] Input:
int((a + b*asin(c*x))^2/x^3,x)
Output:
int((a + b*asin(c*x))^2/x^3, x)
\[ \int \frac {(a+b \arcsin (c x))^2}{x^3} \, dx=\frac {-2 \mathit {asin} \left (c x \right ) a b -2 \sqrt {-c^{2} x^{2}+1}\, a b c x +2 \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}-a^{2}}{2 x^{2}} \] Input:
int((a+b*asin(c*x))^2/x^3,x)
Output:
( - 2*asin(c*x)*a*b - 2*sqrt( - c**2*x**2 + 1)*a*b*c*x + 2*int(asin(c*x)** 2/x**3,x)*b**2*x**2 - a**2)/(2*x**2)