Integrand size = 31, antiderivative size = 851 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {d-c^2 d x^2}}{g (f+g x)}-\frac {b \sqrt {d-c^2 d x^2} \arccos (c x)}{g (f+g x)}+\frac {b c^3 f^2 \sqrt {d-c^2 d x^2} \arccos (c x)^2}{2 g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}-\frac {\left (g+c^2 f x\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c (f+g x)^2}-\frac {a c^3 f^2 \sqrt {d-c^2 d x^2} \arcsin (c x)}{g^2 \left (c^2 f^2-g^2\right ) \sqrt {1-c^2 x^2}}+\frac {a c^2 f \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}+\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {i b c^2 f \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \log (f+g x)}{g^2 \sqrt {1-c^2 x^2}}+\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}-\frac {b c^2 f \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2 \sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}} \]
-a*(-c^2*d*x^2+d)^(1/2)/g/(g*x+f)-b*arccos(c*x)*(-c^2*d*x^2+d)^(1/2)/g/(g* x+f)+1/2*b*c^3*f^2*arccos(c*x)^2*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)/(- c^2*x^2+1)^(1/2)-1/2*(c^2*f*x+g)^2*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2 )/b/c/(c^2*f^2-g^2)/(g*x+f)^2/(-c^2*x^2+1)^(1/2)-a*c^3*f^2*arcsin(c*x)*(-c ^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)/(-c^2*x^2+1)^(1/2)-b*c*ln(g*x+f)*(-c^2 *d*x^2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)+a*c^2*f*arctan((c^2*f*x+g)/(c^2*f^2 -g^2)^(1/2)/(-c^2*x^2+1)^(1/2))*(-c^2*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/ 2)/(-c^2*x^2+1)^(1/2)+I*b*c^2*f*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*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2 )/(-c^2*x^2+1)^(1/2)-I*b*c^2*f*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*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2) /(-c^2*x^2+1)^(1/2)+b*c^2*f*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*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^ 2+1)^(1/2)-b*c^2*f*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*d*x^2+d)^(1/2)/g^2/(c^2*f^2-g^2)^(1/2)/(-c^2*x^2+1)^(1/2 )-1/2*(a+b*arccos(c*x))^2*(-c^2*x^2+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)/b/c/(g*x +f)^2
Time = 10.57 (sec) , antiderivative size = 1130, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=-\frac {a \sqrt {-d \left (-1+c^2 x^2\right )}}{g (f+g x)}+\frac {a c \sqrt {d} \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{g^2}+\frac {a c^2 \sqrt {d} f \log (f+g x)}{g^2 \sqrt {-c^2 f^2+g^2}}-\frac {a c^2 \sqrt {d} f \log \left (d g+c^2 d f x+\sqrt {d} \sqrt {-c^2 f^2+g^2} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{g^2 \sqrt {-c^2 f^2+g^2}}-\frac {b c \sqrt {d \left (1-c^2 x^2\right )} \left (\frac {2 g \arccos (c x)}{c f+c g x}-\frac {\arccos (c x)^2}{\sqrt {1-c^2 x^2}}+\frac {2 \log \left (1+\frac {g x}{f}\right )}{\sqrt {1-c^2 x^2}}+\frac {2 c f \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+c 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+c 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 {-c^2 f^2+g^2} \sqrt {1-c^2 x^2}}\right )}{2 g^2} \]
-((a*Sqrt[-(d*(-1 + c^2*x^2))])/(g*(f + g*x))) + (a*c*Sqrt[d]*ArcTan[(c*x* Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/g^2 + (a*c^2*Sqrt[d] *f*Log[f + g*x])/(g^2*Sqrt[-(c^2*f^2) + g^2]) - (a*c^2*Sqrt[d]*f*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(g ^2*Sqrt[-(c^2*f^2) + g^2]) - (b*c*Sqrt[d*(1 - c^2*x^2)]*((2*g*ArcCos[c*x]) /(c*f + c*g*x) - ArcCos[c*x]^2/Sqrt[1 - c^2*x^2] + (2*Log[1 + (g*x)/f])/Sq rt[1 - c^2*x^2] + (2*c*f*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcCos[c*x] /2])/Sqrt[-(c^2*f^2) + g^2]] - 2*ArcCos[-((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)*Arc Tanh[((-(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)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*f + c*g*x]) ] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x]/2])/Sq rt[-(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)*ArcCos[c*x])*Sqrt[-(c^2*f^2) + g^2])/(Sqrt [2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((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[ArcCos[c*x]/2]))] - (ArcCos[-((c*f)/g)] + (2*I)*ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]...
Time = 2.63 (sec) , antiderivative size = 623, 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 = {5277, 5265, 27, 5255, 27, 5299, 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.
| \(\displaystyle \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \int \frac {\sqrt {1-c^2 x^2} (a+b \arccos (c x))}{(f+g x)^2}dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5265 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (\frac {\int -\frac {2 \left (f x c^2+g\right ) (a+b \arccos (c x))^2}{(f+g x)^3}dx}{2 b c}-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {\int \frac {\left (f x c^2+g\right ) (a+b \arccos (c x))^2}{(f+g x)^3}dx}{b c}-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5255 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {2 b c \int \frac {\left (f x c^2+g\right )^2 (a+b \arccos (c x))}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2 \sqrt {1-c^2 x^2}}dx+\frac {\left (c^2 f x+g\right )^2 (a+b \arccos (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}}{b c}-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {\frac {b c \int \frac {\left (f x c^2+g\right )^2 (a+b \arccos (c x))}{(f+g x)^2 \sqrt {1-c^2 x^2}}dx}{c^2 f^2-g^2}+\frac {\left (c^2 f x+g\right )^2 (a+b \arccos (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}}{b c}-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5299 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {\frac {b c \int \left (\frac {b \arccos (c x) \left (f x c^2+g\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}+\frac {a \left (f x c^2+g\right )^2}{(f+g x)^2 \sqrt {1-c^2 x^2}}\right )dx}{c^2 f^2-g^2}+\frac {\left (c^2 f x+g\right )^2 (a+b \arccos (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}}{b c}-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}\right )}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {d-c^2 d x^2} \left (-\frac {\left (1-c^2 x^2\right ) (a+b \arccos (c x))^2}{2 b c (f+g x)^2}-\frac {\frac {\left (c^2 f x+g\right )^2 (a+b \arccos (c x))^2}{2 \left (c^2 f^2-g^2\right ) (f+g x)^2}+\frac {b c \left (\frac {a c^3 f^2 \arcsin (c x)}{g^2}-\frac {a c^2 f \sqrt {c^2 f^2-g^2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {a \sqrt {1-c^2 x^2} (c f-g) (c f+g)}{g (f+g x)}-\frac {b c^3 f^2 \arccos (c x)^2}{2 g^2}-\frac {b c^2 f \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {b c^2 f \sqrt {c^2 f^2-g^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^2}-\frac {i b c^2 f \arccos (c x) \sqrt {c^2 f^2-g^2} \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^2}+\frac {i b c^2 f \arccos (c x) \sqrt {c^2 f^2-g^2} \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^2}+\frac {b \sqrt {1-c^2 x^2} \arccos (c x) (c f-g) (c f+g)}{g (f+g x)}-b c \left (1-\frac {c^2 f^2}{g^2}\right ) \log (f+g x)\right )}{c^2 f^2-g^2}}{b c}\right )}{\sqrt {1-c^2 x^2}}\) |
(Sqrt[d - c^2*d*x^2]*(-1/2*((1 - c^2*x^2)*(a + b*ArcCos[c*x])^2)/(b*c*(f + g*x)^2) - (((g + c^2*f*x)^2*(a + b*ArcCos[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)) + (b*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/(g*(f + g*x)) - (b *c^3*f^2*ArcCos[c*x]^2)/(2*g^2) + (a*c^3*f^2*ArcSin[c*x])/g^2 - (a*c^2*f*S qrt[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]*ArcCos[c*x]*Log[1 + (E^(I*Arc Cos[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]*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[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, -((E^(I*ArcCos[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, -((E^(I*ArcCos[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]
3.1.5.3.1 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[((a_.) + ArcCos[(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*ArcCos[c*x])^n u, x] + Simp[b*c*n Int[SimplifyInteg rand[u*((a + b*ArcCos[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_.) + ArcCos[(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* ArcCos[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*ArcC os[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_.) + 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]
Int[(ArcCos[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p _), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p, RFx*(a + b*ArcCos[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1938 vs. \(2 (813 ) = 1626\).
Time = 2.81 (sec) , antiderivative size = 1939, normalized size of antiderivative = 2.28
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1939\) |
| parts | \(\text {Expression too large to display}\) | \(1939\) |
a/g^2*(1/d/(c^2*f^2-g^2)*g^2/(x+f/g)*(-(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)^(3/2)-c^2*f*g/(c^2*f^2-g^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)+c^2*d*f/g/(c^2*d)^(1/2)*arctan(( c^2*d)^(1/2)*x/(-(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))+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+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)))+2*c^2/(c^2*f ^2-g^2)*g^2*(-1/4*(-2*(x+f/g)*c^2*d+2*c^2*d*f/g)/c^2/d*(-(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)-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)*arctan((c^2*d)^(1/2)*x/(-(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))))-1/2*b*(-d*(c^ 2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*arccos(c*x)^2*c/g^2-b*(-d*( c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g^2/(g*x+f)*(-c^2*x^2+1)*x*c^2*f -b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g^2/(g*x+f)*x^3*c^4*f+I* b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g^2/(g*x+f)*(-c^2*x^2+1)^ (1/2)*c*f-b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g/(g*x+f)*x^2*c ^2+I*b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g/(g*x+f)*(-c^2*x^2+ 1)^(1/2)*x*c+b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g^2/(g*x+f)* x*c^2*f+b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/(c^2*x^2-1)/g/(g*x+f)+I*b*c^2 *(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)^(1/2)/(c^2*x^2...
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\int { \frac {\sqrt {-c^{2} d x^{2} + d} {\left (b \arccos \left (c x\right ) + a\right )}}{{\left (g x + f\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{\left (f + g x\right )^{2}}\, dx \]
Exception generated. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{(f+g x)^2} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,\sqrt {d-c^2\,d\,x^2}}{{\left (f+g\,x\right )}^2} \,d x \]