Integrand size = 31, antiderivative size = 370 \[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\frac {i \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {c^2 f^2-g^2} \sqrt {d-c^2 d x^2}} \]
I*(a+b*arccos(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^( 1/2)))*(-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)-I*(a+b* arccos(c*x))*ln(1+(c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))* (-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)+b*polylog(2,-( c*x+I*(-c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(-c^2*x^2+1)^(1/2)/ (c^2*f^2-g^2)^(1/2)/(-c^2*d*x^2+d)^(1/2)-b*polylog(2,-(c*x+I*(-c^2*x^2+1)^ (1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))*(-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)^(1/2) /(-c^2*d*x^2+d)^(1/2)
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(930\) vs. \(2(370)=740\).
Time = 2.71 (sec) , antiderivative size = 930, normalized size of antiderivative = 2.51 \[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\frac {\frac {a \log (f+g x)}{\sqrt {d}}-\frac {a \log \left (d \left (g+c^2 f x\right )+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}-\frac {b \sqrt {1-c^2 x^2} \left (2 \arccos (c x) \text {arctanh}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-2 \arccos \left (-\frac {c f}{g}\right ) \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+\left (\arccos \left (-\frac {c f}{g}\right )-2 i \text {arctanh}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )+2 i \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {e^{-\frac {1}{2} i \arccos (c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )+\left (\arccos \left (-\frac {c f}{g}\right )+2 i \left (\text {arctanh}\left (\frac {(c f+g) \cot \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )-\text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right )\right ) \log \left (\frac {e^{\frac {1}{2} i \arccos (c x)} \sqrt {-c^2 f^2+g^2}}{\sqrt {2} \sqrt {g} \sqrt {c (f+g x)}}\right )-\left (\arccos \left (-\frac {c f}{g}\right )-2 i \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (-i c f+i g+\sqrt {-c^2 f^2+g^2}\right ) \left (-i+\tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )-\left (\arccos \left (-\frac {c f}{g}\right )+2 i \text {arctanh}\left (\frac {(-c f+g) \tan \left (\frac {1}{2} \arccos (c x)\right )}{\sqrt {-c^2 f^2+g^2}}\right )\right ) \log \left (\frac {(c f+g) \left (i c f-i g+\sqrt {-c^2 f^2+g^2}\right ) \left (i+\tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )+i \left (\operatorname {PolyLog}\left (2,\frac {\left (c f-i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )-\operatorname {PolyLog}\left (2,\frac {\left (c f+i \sqrt {-c^2 f^2+g^2}\right ) \left (c f+g-\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}{g \left (c f+g+\sqrt {-c^2 f^2+g^2} \tan \left (\frac {1}{2} \arccos (c x)\right )\right )}\right )\right )\right )}{\sqrt {d-c^2 d x^2}}}{\sqrt {-c^2 f^2+g^2}} \]
((a*Log[f + g*x])/Sqrt[d] - (a*Log[d*(g + c^2*f*x) + Sqrt[d]*Sqrt[-(c^2*f^ 2) + g^2]*Sqrt[d - c^2*d*x^2]])/Sqrt[d] - (b*Sqrt[1 - c^2*x^2]*(2*ArcCos[c *x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - 2*Arc Cos[-((c*f)/g)]*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] - (2*I)*ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2 ])/Sqrt[-(c^2*f^2) + g^2]] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2 ])/Sqrt[-(c^2*f^2) + g^2]])*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^((I/2)*A rcCos[c*x])*Sqrt[g]*Sqrt[c*(f + g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(Arc Tanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-( c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^((I/2)*ArcC os[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + g*x)])] - (A rcCos[-((c*f)/g)] - (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[- (c^2*f^2) + g^2]])*Log[((c*f + g)*((-I)*c*f + I*g + Sqrt[-(c^2*f^2) + g^2] )*(-I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcC os[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcC os[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-( c^2*f^2) + g^2])*(I + Tan[ArcCos[c*x]/2]))/(g*(c*f + g + Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^ 2])*(c*f + g - Sqrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))/(g*(c*f + g + S qrt[-(c^2*f^2) + g^2]*Tan[ArcCos[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt...
Time = 1.02 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.75, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.290, Rules used = {5277, 5273, 3042, 3802, 2694, 27, 2620, 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.
| \(\displaystyle \int \frac {a+b \arccos (c x)}{\sqrt {d-c^2 d x^2} (f+g x)} \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {1-c^2 x^2}}dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 5273 |
| \(\displaystyle -\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{c f+c g x}d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{c f+g \sin \left (\arccos (c x)+\frac {\pi }{2}\right )}d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 3802 |
| \(\displaystyle -\frac {2 \sqrt {1-c^2 x^2} \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{2 c e^{i \arccos (c x)} f+e^{2 i \arccos (c x)} g+g}d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2694 |
| \(\displaystyle -\frac {2 \sqrt {1-c^2 x^2} \left (\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{2 \left (c f+e^{i \arccos (c x)} g-\sqrt {c^2 f^2-g^2}\right )}d\arccos (c x)}{\sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{2 \left (c f+e^{i \arccos (c x)} g+\sqrt {c^2 f^2-g^2}\right )}d\arccos (c x)}{\sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \sqrt {1-c^2 x^2} \left (\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{c f+e^{i \arccos (c x)} g-\sqrt {c^2 f^2-g^2}}d\arccos (c x)}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{c f+e^{i \arccos (c x)} g+\sqrt {c^2 f^2-g^2}}d\arccos (c x)}{2 \sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2620 |
| \(\displaystyle -\frac {2 \sqrt {1-c^2 x^2} \left (\frac {g \left (\frac {i b \int \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )d\arccos (c x)}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {i b \int \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )d\arccos (c x)}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {2 \sqrt {1-c^2 x^2} \left (\frac {g \left (\frac {b \int e^{-i \arccos (c x)} \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )de^{i \arccos (c x)}}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {b \int e^{-i \arccos (c x)} \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )de^{i \arccos (c x)}}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {2 \sqrt {1-c^2 x^2} \left (\frac {g \left (-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{\sqrt {d-c^2 d x^2}}\) |
(-2*Sqrt[1 - c^2*x^2]*((g*(((-I)*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[ c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g - (b*PolyLog[2, -((E^(I*ArcCos[c* x])*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g))/(2*Sqrt[c^2*f^2 - g^2]) - (g*((( -I)*(a + b*ArcCos[c*x])*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g - (b*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/g))/(2*Sqrt[c^2*f^2 - g^2])))/Sqrt[d - c^2*d*x^2]
3.1.17.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ ((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp [((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si mp[d*(m/(b*f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x )))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) *(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q) Int [(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q) Int[(f + g*x) ^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 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[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (f_.)*( x_)]), x_Symbol] :> Simp[2 Int[(c + d*x)^m*E^(I*Pi*(k - 1/2))*(E^(I*(e + f*x))/(b + 2*a*E^(I*Pi*(k - 1/2))*E^(I*(e + f*x)) - b*E^(2*I*k*Pi)*E^(2*I*( e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[2*k] && NeQ [a^2 - b^2, 0] && IGtQ[m, 0]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sq rt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[-(c^(m + 1)*Sqrt[d])^(-1) Subs t[Int[(a + b*x)^n*(c*f + g*Cos[x])^m, x], x, ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] & & (GtQ[m, 0] || IGtQ[n, 0])
Int[((a_.) + ArcCos[(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*ArcCos[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]
Time = 1.23 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.32
| method | result | size |
| default | \(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-i \arccos \left (c x \right ) \ln \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{\sqrt {c^{2} f^{2}-g^{2}}\, d \left (c^{2} x^{2}-1\right )}\) | \(487\) |
| parts | \(-\frac {a \ln \left (\frac {-\frac {2 d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}+\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}+2 \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}\, \sqrt {-\left (x +\frac {f}{g}\right )^{2} c^{2} d +\frac {2 c^{2} d f \left (x +\frac {f}{g}\right )}{g}-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}{x +\frac {f}{g}}\right )}{g \sqrt {-\frac {d \left (c^{2} f^{2}-g^{2}\right )}{g^{2}}}}-\frac {b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \left (i \arccos \left (c x \right ) \ln \left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-i \arccos \left (c x \right ) \ln \left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )+\operatorname {dilog}\left (\frac {-\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g -c f +\sqrt {c^{2} f^{2}-g^{2}}}{-c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )-\operatorname {dilog}\left (\frac {\left (c x +i \sqrt {-c^{2} x^{2}+1}\right ) g +c f +\sqrt {c^{2} f^{2}-g^{2}}}{c f +\sqrt {c^{2} f^{2}-g^{2}}}\right )\right )}{\sqrt {c^{2} f^{2}-g^{2}}\, d \left (c^{2} x^{2}-1\right )}\) | \(487\) |
-a/g/(-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+f/g)+2*(-d*(c^2*f^2-g^2)/g^2)^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g )-d*(c^2*f^2-g^2)/g^2)^(1/2))/(x+f/g))-b*(-d*(c^2*x^2-1))^(1/2)/(c^2*f^2-g ^2)^(1/2)*(-c^2*x^2+1)^(1/2)*(I*arccos(c*x)*ln((-(c*x+I*(-c^2*x^2+1)^(1/2) )*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))-I*arccos(c*x)*ln( ((c*x+I*(-c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^ (1/2)))+dilog((-(c*x+I*(-c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c* f+(c^2*f^2-g^2)^(1/2)))-dilog(((c*x+I*(-c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2-g ^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2))))/d/(c^2*x^2-1)
\[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^2*d*g*x^3 + c^2*d*f* x^2 - d*g*x - d*f), x)
\[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acos}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )}\, dx \]
\[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}} \,d x } \]
Exception generated. \[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E rror: Bad Argument Value
Timed out. \[ \int \frac {a+b \arccos (c x)}{(f+g x) \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{\left (f+g\,x\right )\,\sqrt {d-c^2\,d\,x^2}} \,d x \]