Integrand size = 27, antiderivative size = 147 \[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d x^3}-\frac {2 c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d x}+\frac {2 b c^3 \sqrt {1-c^2 x^2} \log (x)}{3 \sqrt {d-c^2 d x^2}} \]
-1/6*b*c*(-c^2*x^2+1)^(1/2)/x^2/(-c^2*d*x^2+d)^(1/2)+2/3*b*c^3*ln(x)*(-c^2 *x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1 /2)/d/x^3-2/3*c^2*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/d/x
Time = 0.20 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.92 \[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=-\frac {\sqrt {d-c^2 d x^2} \left (b c x+6 b c^3 x^3+2 a \sqrt {1-c^2 x^2}+4 a c^2 x^2 \sqrt {1-c^2 x^2}+2 b \sqrt {1-c^2 x^2} \left (1+2 c^2 x^2\right ) \arcsin (c x)-4 b c^3 x^3 \log (x)\right )}{6 d x^3 \sqrt {1-c^2 x^2}} \]
-1/6*(Sqrt[d - c^2*d*x^2]*(b*c*x + 6*b*c^3*x^3 + 2*a*Sqrt[1 - c^2*x^2] + 4 *a*c^2*x^2*Sqrt[1 - c^2*x^2] + 2*b*Sqrt[1 - c^2*x^2]*(1 + 2*c^2*x^2)*ArcSi n[c*x] - 4*b*c^3*x^3*Log[x]))/(d*x^3*Sqrt[1 - c^2*x^2])
Time = 0.43 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.99, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {5204, 15, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 5204 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {d-c^2 d x^2}}dx+\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{x^3}dx}{3 \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d x^3}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {2}{3} c^2 \int \frac {a+b \arcsin (c x)}{x^2 \sqrt {d-c^2 d x^2}}dx-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d x^3}-\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5186 |
| \(\displaystyle \frac {2}{3} c^2 \left (\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{x}dx}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{d x}\right )-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d x^3}-\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 14 |
| \(\displaystyle \frac {2}{3} c^2 \left (\frac {b c \sqrt {1-c^2 x^2} \log (x)}{\sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{d x}\right )-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{3 d x^3}-\frac {b c \sqrt {1-c^2 x^2}}{6 x^2 \sqrt {d-c^2 d x^2}}\) |
-1/6*(b*c*Sqrt[1 - c^2*x^2])/(x^2*Sqrt[d - c^2*d*x^2]) - (Sqrt[d - c^2*d*x ^2]*(a + b*ArcSin[c*x]))/(3*d*x^3) + (2*c^2*(-((Sqrt[d - c^2*d*x^2]*(a + b *ArcSin[c*x]))/(d*x)) + (b*c*Sqrt[1 - c^2*x^2]*Log[x])/Sqrt[d - c^2*d*x^2] ))/3
3.2.18.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && 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]
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[c^2*((m + 2*p + 3)/(f^2*(m + 1)) ) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], 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*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]
Result contains complex when optimal does not.
Time = 0.17 (sec) , antiderivative size = 850, normalized size of antiderivative = 5.78
| method | result | size |
| default | \(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arcsin \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arcsin \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}}{2 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d x}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{3}}-\frac {2 b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}\) | \(850\) |
| parts | \(a \left (-\frac {\sqrt {-c^{2} d \,x^{2}+d}}{3 d \,x^{3}}-\frac {2 c^{2} \sqrt {-c^{2} d \,x^{2}+d}}{3 d x}\right )-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \left (-c^{2} x^{2}+1\right ) c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {4 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{5}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{5} c^{8}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} \arcsin \left (c x \right ) c^{6}}{\left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \,c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x^{3} c^{6}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {2 i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c^{3}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \arcsin \left (c x \right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}-\frac {i b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, x \left (-c^{2} x^{2}+1\right ) c^{4}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, c^{3} \sqrt {-c^{2} x^{2}+1}}{2 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d}+\frac {4 b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) c^{2}}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d x}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, c}{6 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{2}}+\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{3 \left (3 c^{4} x^{4}-2 c^{2} x^{2}-1\right ) d \,x^{3}}-\frac {2 b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \ln \left (\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}-1\right ) c^{3}}{3 d \left (c^{2} x^{2}-1\right )}\) | \(850\) |
a*(-1/3/d/x^3*(-c^2*d*x^2+d)^(1/2)-2/3*c^2/d/x*(-c^2*d*x^2+d)^(1/2))-2/3*I *b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*(-c^2*x^2+1)*c^6+4 /3*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c^2*x^2-1)*arcsin(c*x) *c^3-2*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^2*arcsin(c*x )*(-c^2*x^2+1)^(1/2)*c^5-2/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x ^2-1)/d*x^5*c^8-2*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*a rcsin(c*x)*c^6+1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x* c^4+1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x^3*c^6-2/3*I *b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*arcsin(c*x)*(-c^2*x^2+ 1)^(1/2)*c^3+1/3*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*arcs in(c*x)*c^4-1/3*I*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*x*(-c ^2*x^2+1)*c^4+1/2*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d*c^3*( -c^2*x^2+1)^(1/2)+4/3*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x *arcsin(c*x)*c^2+1/6*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1)/d/x^ 2*(-c^2*x^2+1)^(1/2)*c+1/3*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-2*c^2*x^2-1 )/d/x^3*arcsin(c*x)-2/3*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2 *x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c^3
Time = 0.30 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.95 \[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\left [\frac {2 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {d} \log \left (\frac {c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} - \sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{4} - 1\right )} \sqrt {d} - d}{c^{2} x^{4} - x^{2}}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} - 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} + {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \arcsin \left (c x\right ) - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}, \frac {4 \, {\left (b c^{5} x^{5} - b c^{3} x^{3}\right )} \sqrt {-d} \arctan \left (\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {-c^{2} x^{2} + 1} {\left (x^{2} + 1\right )} \sqrt {-d}}{c^{2} d x^{4} - {\left (c^{2} + 1\right )} d x^{2} + d}\right ) - \sqrt {-c^{2} d x^{2} + d} {\left (b c x^{3} - b c x\right )} \sqrt {-c^{2} x^{2} + 1} - 2 \, {\left (2 \, a c^{4} x^{4} - a c^{2} x^{2} + {\left (2 \, b c^{4} x^{4} - b c^{2} x^{2} - b\right )} \arcsin \left (c x\right ) - a\right )} \sqrt {-c^{2} d x^{2} + d}}{6 \, {\left (c^{2} d x^{5} - d x^{3}\right )}}\right ] \]
[1/6*(2*(b*c^5*x^5 - b*c^3*x^3)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 - sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 - x^2)) - sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x)*sqrt(-c^2*x^2 + 1) - 2*( 2*a*c^4*x^4 - a*c^2*x^2 + (2*b*c^4*x^4 - b*c^2*x^2 - b)*arcsin(c*x) - a)*s qrt(-c^2*d*x^2 + d))/(c^2*d*x^5 - d*x^3), 1/6*(4*(b*c^5*x^5 - b*c^3*x^3)*s qrt(-d)*arctan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^2 + 1)*sqrt(-d)/ (c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) - sqrt(-c^2*d*x^2 + d)*(b*c*x^3 - b*c*x )*sqrt(-c^2*x^2 + 1) - 2*(2*a*c^4*x^4 - a*c^2*x^2 + (2*b*c^4*x^4 - b*c^2*x ^2 - b)*arcsin(c*x) - a)*sqrt(-c^2*d*x^2 + d))/(c^2*d*x^5 - d*x^3)]
\[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x^{4} \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\frac {1}{6} \, {\left (\frac {4 \, c^{2} \log \left (x\right )}{\sqrt {d}} - \frac {1}{\sqrt {d} x^{2}}\right )} b c - \frac {1}{3} \, b {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \arcsin \left (c x\right ) - \frac {1}{3} \, a {\left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} c^{2}}{d x} + \frac {\sqrt {-c^{2} d x^{2} + d}}{d x^{3}}\right )} \]
1/6*(4*c^2*log(x)/sqrt(d) - 1/(sqrt(d)*x^2))*b*c - 1/3*b*(2*sqrt(-c^2*d*x^ 2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))*arcsin(c*x) - 1/3*a*(2*sq rt(-c^2*d*x^2 + d)*c^2/(d*x) + sqrt(-c^2*d*x^2 + d)/(d*x^3))
Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x^4 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x^4\,\sqrt {d-c^2\,d\,x^2}} \,d x \]