Integrand size = 27, antiderivative size = 220 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}+\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}}-\frac {i b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d \sqrt {d-c^2 d x^2}} \]
(a+b*arcsin(c*x))/d/(-c^2*d*x^2+d)^(1/2)-2*(a+b*arcsin(c*x))*arctanh(I*c*x +(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-b*arctanh(c *x)*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)+I*b*polylog(2,-I*c*x-(-c^2*x ^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)-I*b*polylog(2,I*c*x +(-c^2*x^2+1)^(1/2))*(-c^2*x^2+1)^(1/2)/d/(-c^2*d*x^2+d)^(1/2)
Time = 0.74 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.36 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-\frac {a \sqrt {d-c^2 d x^2}}{-1+c^2 x^2}+a \sqrt {d} \log (x)-a \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {b d \left (\arcsin (c x)+\sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )-\sqrt {1-c^2 x^2} \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-\sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}}{d^2} \]
(-((a*Sqrt[d - c^2*d*x^2])/(-1 + c^2*x^2)) + a*Sqrt[d]*Log[x] - a*Sqrt[d]* Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]] + (b*d*(ArcSin[c*x] + Sqrt[1 - c^2*x^ 2]*ArcSin[c*x]*Log[1 - E^(I*ArcSin[c*x])] - Sqrt[1 - c^2*x^2]*ArcSin[c*x]* Log[1 + E^(I*ArcSin[c*x])] + Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] - Si n[ArcSin[c*x]/2]] - Sqrt[1 - c^2*x^2]*Log[Cos[ArcSin[c*x]/2] + Sin[ArcSin[ c*x]/2]] + I*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^(I*ArcSin[c*x])] - I*Sqrt[1 - c^2*x^2]*PolyLog[2, E^(I*ArcSin[c*x])]))/Sqrt[d - c^2*d*x^2])/d^2
Time = 0.64 (sec) , antiderivative size = 158, 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 = {5208, 219, 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.
\(\displaystyle \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 5208 |
\(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}-\frac {b c \sqrt {1-c^2 x^2} \int \frac {1}{1-c^2 x^2}dx}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \sqrt {d-c^2 d x^2}}dx}{d}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 5218 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 4671 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {\sqrt {1-c^2 x^2} \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{d \sqrt {d-c^2 d x^2}}+\frac {a+b \arcsin (c x)}{d \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \text {arctanh}(c x)}{d \sqrt {d-c^2 d x^2}}\) |
(a + b*ArcSin[c*x])/(d*Sqrt[d - c^2*d*x^2]) - (b*Sqrt[1 - c^2*x^2]*ArcTanh [c*x])/(d*Sqrt[d - c^2*d*x^2]) + (Sqrt[1 - c^2*x^2]*(-2*(a + b*ArcSin[c*x] )*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*Po lyLog[2, E^(I*ArcSin[c*x])]))/(d*Sqrt[d - c^2*d*x^2])
3.2.25.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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_)*((d_) + (e_. )*(x_)^2)^(p_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1)) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp[b*c *(n/(2*f*(p + 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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 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 = 0.16 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.02
method | result | size |
default | \(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(224\) |
parts | \(\frac {a}{d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {a \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {-c^{2} d \,x^{2}+d}}{x}\right )}{d^{\frac {3}{2}}}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} \left (c^{2} x^{2}-1\right )}+\frac {\sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2} \left (c^{2} x^{2}-1\right )}\right )\) | \(224\) |
a/d/(-c^2*d*x^2+d)^(1/2)-a/d^(3/2)*ln((2*d+2*d^(1/2)*(-c^2*d*x^2+d)^(1/2)) /x)+b*(-(-d*(c^2*x^2-1))^(1/2)/d^2/(c^2*x^2-1)*arcsin(c*x)+(-c^2*x^2+1)^(1 /2)*(-d*(c^2*x^2-1))^(1/2)*(arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))-2*I *arctan(I*c*x+(-c^2*x^2+1)^(1/2))-I*dilog(I*c*x+(-c^2*x^2+1)^(1/2))-I*dilo g(1+I*c*x+(-c^2*x^2+1)^(1/2)))/d^2/(c^2*x^2-1))
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{x \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
-a*(log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/d^(3/2) - 1/(s qrt(-c^2*d*x^2 + d)*d)) - b*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/((c^2*d*x^3 - d*x)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]