Integrand size = 31, antiderivative size = 860 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arcsin (c x)}{g (f+g x)}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arcsin (c x) \log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \] Output:
-a*(-c^2*d*x^2+d)^(1/2)/g/(g*x+f)-b*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/g/(g* x+f)-a*c^3*f^2*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)/g^2/(c^2*f^2-g^2)/(-c^2*x^ 2+1)^(1/2)-1/2*b*c^3*f^2*(-c^2*d*x^2+d)^(1/2)*arcsin(c*x)^2/g^2/(c^2*f^2-g ^2)/(-c^2*x^2+1)^(1/2)+1/2*(c^2*f*x+g)^2*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin( c*x))^2/b/c/(c^2*f^2-g^2)/(g*x+f)^2/(-c^2*x^2+1)^(1/2)+1/2*(-c^2*x^2+1)^(1 /2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/b/c/(g*x+f)^2+a*c^2*f*(-c^2*d *x^2+d)^(1/2)*arctan((c^2*f*x+g)/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2))/g ^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)-I*b*c^2*f*(-c^2*d*x^2+d)^(1/2)*a rcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/ g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)+I*b*c^2*f*(-c^2*d*x^2+d)^(1/2)* arcsin(c*x)*ln(1-I*(I*c*x+(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2))) /g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2)+b*c*(-c^2*d*x^2+d)^(1/2)*ln(g* x+f)/g^2/(-c^2*x^2+1)^(1/2)-b*c^2*f*(-c^2*d*x^2+d)^(1/2)*polylog(2,I*(I*c* x+(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^2/(c^2*f^2-g^2)^(1/2) /(-c^2*x^2+1)^(1/2)+b*c^2*f*(-c^2*d*x^2+d)^(1/2)*polylog(2,I*(I*c*x+(-c^2* x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x ^2+1)^(1/2)
Time = 1.84 (sec) , antiderivative size = 600, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (c^2 f^2-g^2\right ) (a+b \arcsin (c x))^2}{g^2 (f+g x)^2}-\frac {2 c^2 f (a+b \arcsin (c x))^2}{g^2 (f+g x)}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{(f+g x)^2}+\frac {4 b c^3 f \left (-i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{g^2 \sqrt {c^2 f^2-g^2}}+\frac {2 b c^2 \left (-\frac {g \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{c f+c g x}+b \log (f+g x)+\frac {c f \left (i (a+b \arcsin (c x)) \left (\log \left (1+\frac {i e^{i \arcsin (c x)} g}{-c f+\sqrt {c^2 f^2-g^2}}\right )-\log \left (1-\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )+b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )-b \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )\right )}{\sqrt {c^2 f^2-g^2}}\right )}{g^2}\right )}{2 b c \sqrt {1-c^2 x^2}} \] Input:
Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(f + g*x)^2,x]
Output:
(Sqrt[d - c^2*d*x^2]*(((c^2*f^2 - g^2)*(a + b*ArcSin[c*x])^2)/(g^2*(f + g* x)^2) - (2*c^2*f*(a + b*ArcSin[c*x])^2)/(g^2*(f + g*x)) + ((1 - c^2*x^2)*( a + b*ArcSin[c*x])^2)/(f + g*x)^2 + (4*b*c^3*f*((-I)*(a + b*ArcSin[c*x])*( Log[1 + (I*E^(I*ArcSin[c*x])*g)/(-(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) - b*PolyLog[2, (I*E^ (I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])] + b*PolyLog[2, (I*E^(I*Arc Sin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/(g^2*Sqrt[c^2*f^2 - g^2]) + (2 *b*c^2*(-((g*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(c*f + c*g*x)) + b*Log [f + g*x] + (c*f*(I*(a + b*ArcSin[c*x])*(Log[1 + (I*E^(I*ArcSin[c*x])*g)/( -(c*f) + Sqrt[c^2*f^2 - g^2])] - Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sq rt[c^2*f^2 - g^2])]) + b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^ 2*f^2 - g^2])] - b*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])]))/Sqrt[c^2*f^2 - g^2]))/g^2))/(2*b*c*Sqrt[1 - c^2*x^2])
Time = 2.91 (sec) , antiderivative size = 632, normalized size of antiderivative = 0.73, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {5276, 5264, 27, 5254, 27, 5298, 2009}
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.
\(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx\) |
\(\Big \downarrow \) 5276 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{(f+g x)^2}dx}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5264 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}-\frac {\int -\frac {2 \left (f x c^2+g\right ) (a+b \arcsin (c x))^2}{(f+g x)^3}dx}{2 b c}\right )}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\int \frac {\left (f x c^2+g\right ) (a+b \arcsin (c x))^2}{(f+g x)^3}dx}{b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5254 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\frac {\left (c^2 f x+g\right )^2 (a+b \arcsin (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}-2 b c \int \frac {\left (f x c^2+g\right )^2 (a+b \arcsin (c x))}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}dx}{b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\frac {\left (c^2 f x+g\right )^2 (a+b \arcsin (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}-\frac {b c \int \frac {\left (f x c^2+g\right )^2 (a+b \arcsin (c x))}{(f+g x)^2 \sqrt {1-c^2 x^2}}dx}{c^2 f^2-g^2}}{b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 5298 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\frac {\left (c^2 f x+g\right )^2 (a+b \arcsin (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}-\frac {b c \int \left (\frac {b \arcsin (c x) \left (f x c^2+g\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {a \left (f x c^2+g\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}\right )dx}{c^2 f^2-g^2}}{b c}+\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 b c (f+g x)^2}+\frac {\frac {\left (c^2 f x+g\right )^2 (a+b \arcsin (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}-\frac {b c \left (\frac {a c^3 f^2 \arcsin (c x)}{g^2}-\frac {a c^2 f \sqrt {c^2 f^2-g^2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {a \sqrt {1-c^2 x^2} (c f-g) (c f+g)}{g (f+g x)}+\frac {b c^3 f^2 \arcsin (c x)^2}{2 g^2}+\frac {b c^2 f \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {b c^2 f \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {i b c^2 f \arcsin (c x) \sqrt {c^2 f^2-g^2} \log \left (1-\frac {i g e^{i \arcsin (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {i b c^2 f \arcsin (c x) \sqrt {c^2 f^2-g^2} \log \left (1-\frac {i g e^{i \arcsin (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2}+\frac {b \sqrt {1-c^2 x^2} \arcsin (c x) (c f-g) (c f+g)}{g (f+g x)}+b c \left (1-\frac {c^2 f^2}{g^2}\right ) \log (f+g x)\right )}{c^2 f^2-g^2}}{b c}\right )}{\sqrt {1-c^2 x^2}}\) |
Input:
Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(f + g*x)^2,x]
Output:
(Sqrt[d - c^2*d*x^2]*(((1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*b*c*(f + g* x)^2) + (((g + c^2*f*x)^2*(a + b*ArcSin[c*x])^2)/(2*(c^2*f^2 - g^2)*(f + g *x)^2) - (b*c*((a*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2])/(g*(f + g*x)) + ( a*c^3*f^2*ArcSin[c*x])/g^2 + (b*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*ArcS in[c*x])/(g*(f + g*x)) + (b*c^3*f^2*ArcSin[c*x]^2)/(2*g^2) - (a*c^2*f*Sqrt [c^2*f^2 - g^2]*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2 ])])/g^2 + (I*b*c^2*f*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcS in[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 - (I*b*c^2*f*Sqrt[c^2*f^2 - g^2]*ArcSin[c*x]*Log[1 - (I*E^(I*ArcSin[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2 ])])/g^2 + b*c*(1 - (c^2*f^2)/g^2)*Log[f + g*x] + (b*c^2*f*Sqrt[c^2*f^2 - g^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^2 - (b*c^2*f*Sqrt[c^2*f^2 - g^2]*PolyLog[2, (I*E^(I*ArcSin[c*x])*g)/(c*f + S qrt[c^2*f^2 - g^2])])/g^2))/(c^2*f^2 - g^2))/(b*c)))/Sqrt[1 - c^2*x^2]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(p_.), x_Symbol] :> With[{u = IntHide[(f + g*x)^p*(d + e*x)^ m, x]}, Simp[(a + b*ArcSin[c*x])^n u, x] - Simp[b*c*n Int[SimplifyInteg rand[u*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x], x]] /; Free Q[{a, b, c, d, e, f, g}, x] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && Lt Q[m + p + 1, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_)*Sqrt[ (d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f + g*x)^m*(d + e*x^2)*((a + b*Arc Sin[c*x])^(n + 1)/(b*c*Sqrt[d]*(n + 1))), x] - Simp[1/(b*c*Sqrt[d]*(n + 1)) Int[(d*g*m + 2*e*f*x + e*g*(m + 2)*x^2)*(f + g*x)^(m - 1)*(a + b*ArcSin[ c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && ILtQ[m, 0] && GtQ[d, 0] && IGtQ[n, 0]
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[Simp[(d + e*x^2)^p/(1 - c^2*x^2)^ p] Int[(f + g*x)^m*(1 - c^2*x^2)^p*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ [{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && Intege rQ[p - 1/2] && !GtQ[d, 0]
Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p _), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x]) ^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] && IGt Q[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]
Time = 1.20 (sec) , antiderivative size = 1352, normalized size of antiderivative = 1.57
method | result | size |
default | \(\text {Expression too large to display}\) | \(1352\) |
parts | \(\text {Expression too large to display}\) | \(1352\) |
Input:
int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(g*x+f)^2,x,method=_RETURNVERBO SE)
Output:
a/g^2*(1/d/(c^2*f^2-g^2)*g^2/(x+1/g*f)*(-(x+1/g*f)^2*c^2*d+2*c^2*d*f/g*(x+ 1/g*f)-d*(c^2*f^2-g^2)/g^2)^(3/2)-c^2*f*g/(c^2*f^2-g^2)*((-(x+1/g*f)^2*c^2 *d+2*c^2*d*f/g*(x+1/g*f)-d*(c^2*f^2-g^2)/g^2)^(1/2)+c^2*d*f/g/(c^2*d)^(1/2 )*arctan((c^2*d)^(1/2)*x/(-(x+1/g*f)^2*c^2*d+2*c^2*d*f/g*(x+1/g*f)-d*(c^2* f^2-g^2)/g^2)^(1/2))+d*(c^2*f^2-g^2)/g^2/(-d*(c^2*f^2-g^2)/g^2)^(1/2)*ln(( -2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+1/g*f)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2 )*(-(x+1/g*f)^2*c^2*d+2*c^2*d*f/g*(x+1/g*f)-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x +1/g*f)))+2*c^2/(c^2*f^2-g^2)*g^2*(-1/4*(-2*(x+1/g*f)*c^2*d+2*c^2*d*f/g)/c ^2/d*(-(x+1/g*f)^2*c^2*d+2*c^2*d*f/g*(x+1/g*f)-d*(c^2*f^2-g^2)/g^2)^(1/2)- 1/8*(4*c^2*d^2*(c^2*f^2-g^2)/g^2-4*c^4*d^2*f^2/g^2)/c^2/d/(c^2*d)^(1/2)*ar ctan((c^2*d)^(1/2)*x/(-(x+1/g*f)^2*c^2*d+2*c^2*d*f/g*(x+1/g*f)-d*(c^2*f^2- g^2)/g^2)^(1/2))))+b*(1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x ^2-1)*arcsin(c*x)^2*c/g^2-(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*c*x +c^2*x^2-1)*arcsin(c*x)*(c^2*f*x+g-I*(-c^2*x^2+1)^(1/2)*c*f)/(c^2*x^2-1)/g ^2/(g*x+f)+(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*(arcsin(c*x)*ln((I*f* c+(I*c*x+(-c^2*x^2+1)^(1/2))*g-(-c^2*f^2+g^2)^(1/2))/(I*f*c-(-c^2*f^2+g^2) ^(1/2)))*(-c^2*f^2+g^2)^(1/2)*c*f-arcsin(c*x)*ln((I*f*c+(I*c*x+(-c^2*x^2+1 )^(1/2))*g+(-c^2*f^2+g^2)^(1/2))/(I*f*c+(-c^2*f^2+g^2)^(1/2)))*(-c^2*f^2+g ^2)^(1/2)*c*f-I*dilog((I*f*c+(I*c*x+(-c^2*x^2+1)^(1/2))*g-(-c^2*f^2+g^2)^( 1/2))/(I*f*c-(-c^2*f^2+g^2)^(1/2)))*(-c^2*f^2+g^2)^(1/2)*c*f+I*dilog((I...
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arcsin \left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(g*x+f)^2,x, algorithm="f ricas")
Output:
integral(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/(g^2*x^2 + 2*f*g*x + f^2 ), x)
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \] Input:
integrate((-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))/(g*x+f)**2,x)
Output:
Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))/(f + g*x)**2, x)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(g*x+f)^2,x, algorithm="m axima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor e details)
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/(g*x+f)^2,x, algorithm="g iac")
Output:
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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{{\left (f+g\,x\right )}^2} \,d x \] Input:
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(1/2))/(f + g*x)^2,x)
Output:
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^(1/2))/(f + g*x)^2, x)
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{(f+g x)^2} \, dx=\frac {\sqrt {d}\, \left (-\mathit {asin} \left (c x \right ) a \,c^{3} f^{3}-\mathit {asin} \left (c x \right ) a \,c^{3} f^{2} g x +\mathit {asin} \left (c x \right ) a c f \,g^{2}+\mathit {asin} \left (c x \right ) a c \,g^{3} x +2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2}+2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f g x -\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g +\sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,c^{2} f^{3} g^{2}+\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,c^{2} f^{2} g^{3} x -\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b f \,g^{4}-\left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )}{g^{2} x^{2}+2 f g x +f^{2}}d x \right ) b \,g^{5} x \right )}{g^{2} \left (c^{2} f^{2} g x +c^{2} f^{3}-g^{3} x -f \,g^{2}\right )} \] Input:
int((-c^2*d*x^2+d)^(1/2)*(a+b*asin(c*x))/(g*x+f)^2,x)
Output:
(sqrt(d)*( - asin(c*x)*a*c**3*f**3 - asin(c*x)*a*c**3*f**2*g*x + asin(c*x) *a*c*f*g**2 + asin(c*x)*a*c*g**3*x + 2*sqrt(c**2*f**2 - g**2)*atan((tan(as in(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f**2 + 2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f* g*x - sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g + sqrt( - c**2*x**2 + 1)*a*g**3 + int((sqrt( - c**2*x**2 + 1)*asin(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x)* b*c**2*f**3*g**2 + int((sqrt( - c**2*x**2 + 1)*asin(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x)*b*c**2*f**2*g**3*x - int((sqrt( - c**2*x**2 + 1)*asin(c*x) )/(f**2 + 2*f*g*x + g**2*x**2),x)*b*f*g**4 - int((sqrt( - c**2*x**2 + 1)*a sin(c*x))/(f**2 + 2*f*g*x + g**2*x**2),x)*b*g**5*x))/(g**2*(c**2*f**3 + c* *2*f**2*g*x - f*g**2 - g**3*x))