Integrand size = 25, antiderivative size = 173 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=-\frac {b c x}{12 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {2 b c x}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {a+b \arcsin (c x)}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 (a+b \arcsin (c x)) \text {arctanh}\left (e^{2 i \arcsin (c x)}\right )}{d^3}+\frac {i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{2 d^3}-\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{2 d^3} \] Output:
-1/12*b*c*x/d^3/(-c^2*x^2+1)^(3/2)-2/3*b*c*x/d^3/(-c^2*x^2+1)^(1/2)+1/4*(a +b*arcsin(c*x))/d^3/(-c^2*x^2+1)^2+1/2*(a+b*arcsin(c*x))/d^3/(-c^2*x^2+1)- 2*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3+1/2*I*b*poly log(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)/d^3-1/2*I*b*polylog(2,(I*c*x+(-c^2*x^ 2+1)^(1/2))^2)/d^3
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(524\) vs. \(2(173)=346\).
Time = 0.76 (sec) , antiderivative size = 524, normalized size of antiderivative = 3.03 \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {-\frac {b \sqrt {1-c^2 x^2}}{6 (-1+c x)^2}+\frac {b c x \sqrt {1-c^2 x^2}}{12 (-1+c x)^2}+\frac {b \sqrt {1-c^2 x^2}}{6 (1+c x)^2}+\frac {b c x \sqrt {1-c^2 x^2}}{12 (1+c x)^2}+\frac {5 b \sqrt {1-c^2 x^2}}{-4+4 c x}+\frac {5 b \sqrt {1-c^2 x^2}}{4+4 c x}+\frac {a}{\left (-1+c^2 x^2\right )^2}-\frac {2 a}{-1+c^2 x^2}-4 i b \pi \arcsin (c x)+\frac {5 b \arcsin (c x)}{4-4 c x}+\frac {b \arcsin (c x)}{4 (-1+c x)^2}+\frac {b \arcsin (c x)}{4 (1+c x)^2}+\frac {5 b \arcsin (c x)}{4+4 c x}-8 b \pi \log \left (1+e^{-i \arcsin (c x)}\right )-2 b \pi \log \left (1-i e^{i \arcsin (c x)}\right )-4 b \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )+2 b \pi \log \left (1+i e^{i \arcsin (c x)}\right )-4 b \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )+4 b \arcsin (c x) \log \left (1-e^{2 i \arcsin (c x)}\right )+4 a \log (x)-2 a \log \left (1-c^2 x^2\right )+8 b \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-2 b \pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+2 b \pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+4 i b \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )+4 i b \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-2 i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )}{4 d^3} \] Input:
Integrate[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)^3),x]
Output:
(-1/6*(b*Sqrt[1 - c^2*x^2])/(-1 + c*x)^2 + (b*c*x*Sqrt[1 - c^2*x^2])/(12*( -1 + c*x)^2) + (b*Sqrt[1 - c^2*x^2])/(6*(1 + c*x)^2) + (b*c*x*Sqrt[1 - c^2 *x^2])/(12*(1 + c*x)^2) + (5*b*Sqrt[1 - c^2*x^2])/(-4 + 4*c*x) + (5*b*Sqrt [1 - c^2*x^2])/(4 + 4*c*x) + a/(-1 + c^2*x^2)^2 - (2*a)/(-1 + c^2*x^2) - ( 4*I)*b*Pi*ArcSin[c*x] + (5*b*ArcSin[c*x])/(4 - 4*c*x) + (b*ArcSin[c*x])/(4 *(-1 + c*x)^2) + (b*ArcSin[c*x])/(4*(1 + c*x)^2) + (5*b*ArcSin[c*x])/(4 + 4*c*x) - 8*b*Pi*Log[1 + E^((-I)*ArcSin[c*x])] - 2*b*Pi*Log[1 - I*E^(I*ArcS in[c*x])] - 4*b*ArcSin[c*x]*Log[1 - I*E^(I*ArcSin[c*x])] + 2*b*Pi*Log[1 + I*E^(I*ArcSin[c*x])] - 4*b*ArcSin[c*x]*Log[1 + I*E^(I*ArcSin[c*x])] + 4*b* ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 4*a*Log[x] - 2*a*Log[1 - c^2* x^2] + 8*b*Pi*Log[Cos[ArcSin[c*x]/2]] - 2*b*Pi*Log[-Cos[(Pi + 2*ArcSin[c*x ])/4]] + 2*b*Pi*Log[Sin[(Pi + 2*ArcSin[c*x])/4]] + (4*I)*b*PolyLog[2, (-I) *E^(I*ArcSin[c*x])] + (4*I)*b*PolyLog[2, I*E^(I*ArcSin[c*x])] - (2*I)*b*Po lyLog[2, E^((2*I)*ArcSin[c*x])])/(4*d^3)
Time = 0.91 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5208, 27, 209, 208, 5208, 208, 5184, 4919, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{d^2 x \left (1-c^2 x^2\right )^2}dx}{d}-\frac {b c \int \frac {1}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^3}-\frac {b c \int \frac {1}{\left (1-c^2 x^2\right )^{5/2}}dx}{4 d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^3}-\frac {b c \left (\frac {2}{3} \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^2}dx}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 5208 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx-\frac {1}{2} b c \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}}dx+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )}dx+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 5184 |
| \(\displaystyle \frac {\int \frac {a+b \arcsin (c x)}{c x \sqrt {1-c^2 x^2}}d\arcsin (c x)+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 4919 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (a+b \arcsin (c x)) \csc (2 \arcsin (c x))d\arcsin (c x)+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 4671 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} b \int \log \left (1-e^{2 i \arcsin (c x)}\right )d\arcsin (c x)+\frac {1}{2} b \int \log \left (1+e^{2 i \arcsin (c x)}\right )d\arcsin (c x)-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {2 \left (\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1-e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\frac {1}{4} i b \int e^{-2 i \arcsin (c x)} \log \left (1+e^{2 i \arcsin (c x)}\right )de^{2 i \arcsin (c x)}-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {2 \left (-\left (\text {arctanh}\left (e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin (c x)}\right )\right )+\frac {a+b \arcsin (c x)}{2 \left (1-c^2 x^2\right )}-\frac {b c x}{2 \sqrt {1-c^2 x^2}}}{d^3}+\frac {a+b \arcsin (c x)}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c \left (\frac {2 x}{3 \sqrt {1-c^2 x^2}}+\frac {x}{3 \left (1-c^2 x^2\right )^{3/2}}\right )}{4 d^3}\) |
Input:
Int[(a + b*ArcSin[c*x])/(x*(d - c^2*d*x^2)^3),x]
Output:
-1/4*(b*c*(x/(3*(1 - c^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 - c^2*x^2])))/d^3 + (a + b*ArcSin[c*x])/(4*d^3*(1 - c^2*x^2)^2) + (-1/2*(b*c*x)/Sqrt[1 - c^2* x^2] + (a + b*ArcSin[c*x])/(2*(1 - c^2*x^2)) + 2*(-((a + b*ArcSin[c*x])*Ar cTanh[E^((2*I)*ArcSin[c*x])]) + (I/4)*b*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (I/4)*b*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/d^3
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 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[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n , x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSi n[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 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])
Time = 0.49 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.88
| method | result | size |
| parts | \(-\frac {a \left (-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\ln \left (x \right )\right )}{d^{3}}-\frac {b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(325\) |
| derivativedivides | \(-\frac {a \left (-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(327\) |
| default | \(-\frac {a \left (-\frac {1}{16 \left (c x +1\right )^{2}}-\frac {5}{16 \left (c x +1\right )}+\frac {\ln \left (c x +1\right )}{2}-\ln \left (c x \right )-\frac {1}{16 \left (c x -1\right )^{2}}+\frac {5}{16 \left (c x -1\right )}+\frac {\ln \left (c x -1\right )}{2}\right )}{d^{3}}-\frac {b \left (\frac {8 i c^{4} x^{4}-8 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}+6 c^{2} x^{2} \arcsin \left (c x \right )-16 i c^{2} x^{2}+9 c x \sqrt {-c^{2} x^{2}+1}-9 \arcsin \left (c x \right )+8 i}{12 c^{4} x^{4}-24 c^{2} x^{2}+12}+\arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )-\frac {i \operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{2}-\arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )-\arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )+i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{3}}\) | \(327\) |
Input:
int((a+b*arcsin(c*x))/x/(-c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
-a/d^3*(-1/16/(c*x-1)^2+5/16/(c*x-1)+1/2*ln(c*x-1)-1/16/(c*x+1)^2-5/16/(c* x+1)+1/2*ln(c*x+1)-ln(x))-b/d^3*(1/12*(8*I*c^4*x^4-8*c^3*x^3*(-c^2*x^2+1)^ (1/2)+6*c^2*x^2*arcsin(c*x)-16*I*c^2*x^2+9*c*x*(-c^2*x^2+1)^(1/2)-9*arcsin (c*x)+8*I)/(c^4*x^4-2*c^2*x^2+1)+arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2 ))^2)-1/2*I*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)-arcsin(c*x)*ln(1+I*c* x+(-c^2*x^2+1)^(1/2))+I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))-arcsin(c*x)*l n(1-I*c*x-(-c^2*x^2+1)^(1/2))+I*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2)))
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/x/(-c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral(-(b*arcsin(c*x) + a)/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
Timed out. \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((a+b*asin(c*x))/x/(-c**2*d*x**2+d)**3,x)
Output:
Timed out
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int { -\frac {b \arcsin \left (c x\right ) + a}{{\left (c^{2} d x^{2} - d\right )}^{3} x} \,d x } \] Input:
integrate((a+b*arcsin(c*x))/x/(-c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/4*a*((2*c^2*x^2 - 3)/(c^4*d^3*x^4 - 2*c^2*d^3*x^2 + d^3) + 2*log(c*x + 1)/d^3 + 2*log(c*x - 1)/d^3 - 4*log(x)/d^3) - b*integrate(arctan2(c*x, sqr t(c*x + 1)*sqrt(-c*x + 1))/(c^6*d^3*x^7 - 3*c^4*d^3*x^5 + 3*c^2*d^3*x^3 - d^3*x), x)
Exception generated. \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arcsin(c*x))/x/(-c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
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. \[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{x\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \] Input:
int((a + b*asin(c*x))/(x*(d - c^2*d*x^2)^3),x)
Output:
int((a + b*asin(c*x))/(x*(d - c^2*d*x^2)^3), x)
\[ \int \frac {a+b \arcsin (c x)}{x \left (d-c^2 d x^2\right )^3} \, dx=\frac {-4 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{7}-3 c^{4} x^{5}+3 c^{2} x^{3}-x}d x \right ) b \,c^{4} x^{4}+8 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{7}-3 c^{4} x^{5}+3 c^{2} x^{3}-x}d x \right ) b \,c^{2} x^{2}-4 \left (\int \frac {\mathit {asin} \left (c x \right )}{c^{6} x^{7}-3 c^{4} x^{5}+3 c^{2} x^{3}-x}d x \right ) b -2 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{4} x^{4}+4 \,\mathrm {log}\left (c^{2} x -c \right ) a \,c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x -c \right ) a -2 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{4} x^{4}+4 \,\mathrm {log}\left (c^{2} x +c \right ) a \,c^{2} x^{2}-2 \,\mathrm {log}\left (c^{2} x +c \right ) a +4 \,\mathrm {log}\left (x \right ) a \,c^{4} x^{4}-8 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}+4 \,\mathrm {log}\left (x \right ) a -a \,c^{4} x^{4}+2 a}{4 d^{3} \left (c^{4} x^{4}-2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asin(c*x))/x/(-c^2*d*x^2+d)^3,x)
Output:
( - 4*int(asin(c*x)/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x),x)*b*c**4* x**4 + 8*int(asin(c*x)/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x),x)*b*c* *2*x**2 - 4*int(asin(c*x)/(c**6*x**7 - 3*c**4*x**5 + 3*c**2*x**3 - x),x)*b - 2*log(c**2*x - c)*a*c**4*x**4 + 4*log(c**2*x - c)*a*c**2*x**2 - 2*log(c **2*x - c)*a - 2*log(c**2*x + c)*a*c**4*x**4 + 4*log(c**2*x + c)*a*c**2*x* *2 - 2*log(c**2*x + c)*a + 4*log(x)*a*c**4*x**4 - 8*log(x)*a*c**2*x**2 + 4 *log(x)*a - a*c**4*x**4 + 2*a)/(4*d**3*(c**4*x**4 - 2*c**2*x**2 + 1))