Integrand size = 27, antiderivative size = 225 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}+\frac {c^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{\sqrt {1-c^2 x^2}}-\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}}+\frac {i b c^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{2 \sqrt {1-c^2 x^2}} \]
-1/2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^2-1/2*b*c*(-c^2*d*x^2+d)^(1/ 2)/x/(-c^2*x^2+1)^(1/2)+c^2*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1)^( 1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/2*I*b*c^2*polylog(2,-I*c*x -(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+1/2*I*b*c^2*p olylog(2,I*c*x+(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)
Time = 1.68 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=\frac {1}{8} \left (-\frac {4 a \sqrt {d-c^2 d x^2}}{x^2}-4 a c^2 \sqrt {d} \log (x)+4 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b c^2 d \sqrt {1-c^2 x^2} \left (-2 \cot \left (\frac {1}{2} \arcsin (c x)\right )-\arcsin (c x) \csc ^2\left (\frac {1}{2} \arcsin (c x)\right )-4 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+4 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-4 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+4 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )+\arcsin (c x) \sec ^2\left (\frac {1}{2} \arcsin (c x)\right )-2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{\sqrt {d-c^2 d x^2}}\right ) \]
((-4*a*Sqrt[d - c^2*d*x^2])/x^2 - 4*a*c^2*Sqrt[d]*Log[x] + 4*a*c^2*Sqrt[d] *Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*c^2*d*Sqrt[1 - c^2*x^2]*(-2*Cot [ArcSin[c*x]/2] - ArcSin[c*x]*Csc[ArcSin[c*x]/2]^2 - 4*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] + 4*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (4*I)*Pol yLog[2, -E^(I*ArcSin[c*x])] + (4*I)*PolyLog[2, E^(I*ArcSin[c*x])] + ArcSin [c*x]*Sec[ArcSin[c*x]/2]^2 - 2*Tan[ArcSin[c*x]/2]))/Sqrt[d - c^2*d*x^2])/8
Time = 0.61 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.72, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {5196, 15, 5218, 3042, 4671, 2715, 2838}
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.
\(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx\) |
\(\Big \downarrow \) 5196 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \int \frac {1}{x^2}dx}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {c^2 \sqrt {d-c^2 d x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c \sqrt {d-c^2 d x^2}}{2 x \sqrt {1-c^2 x^2}}\) |
-1/2*(b*c*Sqrt[d - c^2*d*x^2])/(x*Sqrt[1 - c^2*x^2]) - (Sqrt[d - c^2*d*x^2 ]*(a + b*ArcSin[c*x]))/(2*x^2) - (c^2*Sqrt[d - c^2*d*x^2]*(-2*(a + b*ArcSi n[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/(2*Sqrt[1 - c^2*x^2])
3.1.66.3.1 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[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[- 2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*x))]/f), x] + (-Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x )^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG tQ[m, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcS in[c*x])^n/(f*(m + 1))), x] + (-Simp[b*c*(n/(f*(m + 1)))*Simp[Sqrt[d + e*x^ 2]/Sqrt[1 - c^2*x^2]] Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x] + Simp[(c^2/(f^2*(m + 1)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]] Int [(f*x)^(m + 2)*((a + b*ArcSin[c*x])^n/Sqrt[1 - c^2*x^2]), x], x]) /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[m, -1]
Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[d + e* x^2]] Subst[Int[(a + b*x)^n*Sin[x]^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a , b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Time = 0.17 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.28
method | result | size |
default | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )\) | \(289\) |
parts | \(a \left (-\frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{2 d \,x^{2}}-\frac {c^{2} \left (\sqrt {-c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )\right )}{2}\right )+b \left (-\frac {\left (c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right ) \sqrt {-d \left (c^{2} x^{2}-1\right )}}{2 x^{2} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )-\operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )+\operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )\right ) c^{2}}{2 c^{2} x^{2}-2}\right )\) | \(289\) |
a*(-1/2/d/x^2*(-c^2*d*x^2+d)^(3/2)-1/2*c^2*((-c^2*d*x^2+d)^(1/2)-d^(1/2)*l n((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2))/x)))+b*(-1/2*(c^2*x^2*arcsin(c*x)-c *x*(-c^2*x^2+1)^(1/2)-arcsin(c*x))*(-d*(c^2*x^2-1))^(1/2)/x^2/(c^2*x^2-1)+ I*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arcsin(c*x)*ln(1+I*c*x+(-c^ 2*x^2+1)^(1/2))-I*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-polylog(2,I*c *x+(-c^2*x^2+1)^(1/2))+polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2)))*c^2/(2*c^2*x^ 2-2))
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{x^{3}}\, dx \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
b*sqrt(d)*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1 )*sqrt(-c*x + 1))/x^3, x) + 1/2*(c^2*sqrt(d)*log(2*sqrt(-c^2*d*x^2 + d)*sq rt(d)/abs(x) + 2*d/abs(x)) - sqrt(-c^2*d*x^2 + d)*c^2 - (-c^2*d*x^2 + d)^( 3/2)/(d*x^2))*a
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{x^3} \,d x \]