3.1.18 \(\int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx\) [18]

3.1.18.1 Optimal result

Integrand size = 31, antiderivative size = 496 \[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\frac {g \left (1-c^2 x^2\right ) (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {i c^2 f \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {1-c^2 x^2} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b c^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c^2 f \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \]

output

g*(-c^2*x^2+1)*(a+b*arccos(c*x))/(c^2*f^2-g^2)/(g*x+f)/(-c^2*d*x^2+d)^(1/2 
)+b*c*ln(g*x+f)*(-c^2*x^2+1)^(1/2)/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2)+I*c^ 
2*f*(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)^(3/2)/(-c^2*d*x^2+d)^(1/2)-I*c^2 
*f*(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)^(3/2)/(-c^2*d*x^2+d)^(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 
*x^2+1)^(1/2)/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(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*x^2+1)^(1/2) 
/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)
 

3.1.18.2 Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1108\) vs. \(2(496)=992\).

Time = 5.90 (sec) , antiderivative size = 1108, normalized size of antiderivative = 2.23 \[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {a g \sqrt {d-c^2 d x^2}}{d \left (-c^2 f^2+g^2\right ) (f+g x)}-\frac {a c^2 f \log (f+g x)}{\sqrt {d} \left (-c^2 f^2+g^2\right )^{3/2}}-\frac {a c^2 f \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} (c f-g) (c f+g) \sqrt {-c^2 f^2+g^2}}-\frac {b c \sqrt {1-c^2 x^2} \left (-\frac {g \sqrt {1-c^2 x^2} \arccos (c x)}{(c f-g) (c f+g) (c f+c g x)}-\frac {\log \left (1+\frac {g x}{f}\right )}{c^2 f^2-g^2}-\frac {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+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 )}{\left (-c^2 f^2+g^2\right )^{3/2}}\right )}{\sqrt {d-c^2 d x^2}} \]

input

Integrate[(a + b*ArcCos[c*x])/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]
 

output

-((a*g*Sqrt[d - c^2*d*x^2])/(d*(-(c^2*f^2) + g^2)*(f + g*x))) - (a*c^2*f*L 
og[f + g*x])/(Sqrt[d]*(-(c^2*f^2) + g^2)^(3/2)) - (a*c^2*f*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]*(c*f 
- g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) - (b*c*Sqrt[1 - c^2*x^2]*(-((g*Sqrt 
[1 - c^2*x^2]*ArcCos[c*x])/((c*f - g)*(c*f + g)*(c*f + c*g*x))) - Log[1 + 
(g*x)/f]/(c^2*f^2 - g^2) - (c*f*(2*ArcCos[c*x]*ArcTanh[((c*f + g)*Cot[ArcC 
os[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)*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)*ArcCos[c*x])*Sqrt[g]*Sqrt[c*(f + 
 g*x)])] + (ArcCos[-((c*f)/g)] + (2*I)*(ArcTanh[((c*f + g)*Cot[ArcCos[c*x] 
/2])/Sqrt[-(c^2*f^2) + g^2]] - ArcTanh[((-(c*f) + g)*Tan[ArcCos[c*x]/2])/S 
qrt[-(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 + g*x)])] - (ArcCos[-((c*f)/g)] - (2*I)*ArcTa 
nh[((-(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]])*Log[((c*f + g)*(I*c*f - I*g + Sqrt[-(c^2*f^2) + g^2])*(I + Tan[Ar...
 

3.1.18.3 Rubi [A] (verified)

Time = 1.36 (sec) , antiderivative size = 374, normalized size of antiderivative = 0.75, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.452, Rules used = {5277, 5273, 3042, 3805, 25, 3042, 3147, 16, 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.

\(\displaystyle \int \frac {a+b \arccos (c x)}{\sqrt {d-c^2 d x^2} (f+g x)^2} \, dx\)

\(\Big \downarrow \) 5277

\(\displaystyle \frac {\sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {1-c^2 x^2}}dx}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 5273

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{(c f+c g x)^2}d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \int \frac {a+b \arccos (c x)}{\left (c f+g \sin \left (\arccos (c x)+\frac {\pi }{2}\right )\right )^2}d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3805

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {c f \int \frac {a+b \arccos (c x)}{c f+c g x}d\arccos (c x)}{c^2 f^2-g^2}-\frac {b g \int -\frac {\sqrt {1-c^2 x^2}}{c f+c g x}d\arccos (c x)}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {c f \int \frac {a+b \arccos (c x)}{c f+c g x}d\arccos (c x)}{c^2 f^2-g^2}+\frac {b g \int \frac {\sqrt {1-c^2 x^2}}{c f+c g x}d\arccos (c x)}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {c f \int \frac {a+b \arccos (c x)}{c f+g \sin \left (\arccos (c x)+\frac {\pi }{2}\right )}d\arccos (c x)}{c^2 f^2-g^2}+\frac {b g \int \frac {\cos \left (\arccos (c x)-\frac {\pi }{2}\right )}{c f-g \sin \left (\arccos (c x)-\frac {\pi }{2}\right )}d\arccos (c x)}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3147

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {c f \int \frac {a+b \arccos (c x)}{c f+g \sin \left (\arccos (c x)+\frac {\pi }{2}\right )}d\arccos (c x)}{c^2 f^2-g^2}-\frac {b \int \frac {1}{c f+c g x}d(c g x)}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 16

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {c f \int \frac {a+b \arccos (c x)}{c f+g \sin \left (\arccos (c x)+\frac {\pi }{2}\right )}d\arccos (c x)}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3802

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{2 c e^{i \arccos (c x)} f+e^{2 i \arccos (c x)} g+g}d\arccos (c x)}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2694

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{2 \left (c f+e^{i \arccos (c x)} g-\sqrt {c^2 f^2-g^2}\right )}d\arccos (c x)}{\sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{2 \left (c f+e^{i \arccos (c x)} g+\sqrt {c^2 f^2-g^2}\right )}d\arccos (c x)}{\sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{c f+e^{i \arccos (c x)} g-\sqrt {c^2 f^2-g^2}}d\arccos (c x)}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{i \arccos (c x)} (a+b \arccos (c x))}{c f+e^{i \arccos (c x)} g+\sqrt {c^2 f^2-g^2}}d\arccos (c x)}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {g \left (\frac {i b \int \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )d\arccos (c x)}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {i b \int \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )d\arccos (c x)}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {g \left (\frac {b \int e^{-i \arccos (c x)} \log \left (\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )de^{i \arccos (c x)}}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {b \int e^{-i \arccos (c x)} \log \left (\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )de^{i \arccos (c x)}}{g}-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {c \sqrt {1-c^2 x^2} \left (\frac {2 c f \left (\frac {g \left (-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}-\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (-\frac {i (a+b \arccos (c x)) \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g}-\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}-\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

input

Int[(a + b*ArcCos[c*x])/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]
 

output

-((c*Sqrt[1 - c^2*x^2]*(-((g*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/((c^2* 
f^2 - g^2)*(c*f + c*g*x))) - (b*Log[c*f + c*g*x])/(c^2*f^2 - g^2) + (2*c*f 
*((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*Poly 
Log[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])))/(c^2*f^2 - g^2)))/Sqrt[d - c^2*d*x^2])
 

3.1.18.3.1 Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

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]
 

rule 2694

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]
 

rule 2715

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]
 

rule 2838

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3802

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]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 5273

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])
 

rule 5277

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]
 

3.1.18.4 Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1621 vs. \(2 (492 ) = 984\).

Time = 2.68 (sec) , antiderivative size = 1622, normalized size of antiderivative = 3.27

method	result	size
default	\(\text {Expression too large to display}\)	\(1622\)
parts	\(\text {Expression too large to display}\)	\(1622\)

input

int((a+b*arccos(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBO 
SE)
 

output

a/d/(c^2*f^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)^(1/2)-a/g*c^2*f/(c^2*f^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)) 
+b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-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)/d/(c^2*x^2-1)/(c 
^2*f^2-g^2)/(g*x+f)*x^3*c^4*f+I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2 
)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^(3/2)*c^2*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)))*f+b*(-d*(c^ 
2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x^2*c^2*g- 
I*b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-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)/d/(c^2*x^2-1) 
/(c^2*f^2-g^2)/(g*x+f)*x*c^2*f-b*(-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2 
*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*g-I*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1 
/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^(3/2)*c^2*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)))*f-I*b*( 
-d*(c^2*x^2-1))^(1/2)*arccos(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*(-c^ 
2*x^2+1)^(1/2)*x*c*g+2*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2* 
x^2-1)/(c^2*f^2-g^2)^2*c^3*ln(c*x+I*(-c^2*x^2+1)^(1/2))*f^2-b*(-c^2*x^2+1) 
^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d/(c^2*x^2-1)/(c^2*f^2-g^2)^2*c^3*ln((c*x...
 

3.1.18.5 Fricas [F]

\[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="f 
ricas")
 

output

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccos(c*x) + a)/(c^2*d*g^2*x^4 + 2*c^2* 
d*f*g*x^3 - 2*d*f*g*x - d*f^2 + (c^2*d*f^2 - d*g^2)*x^2), x)
 

3.1.18.6 Sympy [F]

\[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acos}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, dx \]

input

integrate((a+b*acos(c*x))/(g*x+f)**2/(-c**2*d*x**2+d)**(1/2),x)
 

output

Integral((a + b*acos(c*x))/(sqrt(-d*(c*x - 1)*(c*x + 1))*(f + g*x)**2), x)
 

3.1.18.7 Maxima [F]

\[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="m 
axima")
 

output

integrate((b*arccos(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*(g*x + f)^2), x)
 

3.1.18.8 Giac [F]

\[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \arccos \left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \]

input

integrate((a+b*arccos(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="g 
iac")
 

output

integrate((b*arccos(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*(g*x + f)^2), x)
 

3.1.18.9 Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \arccos (c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \]

input

int((a + b*acos(c*x))/((f + g*x)^2*(d - c^2*d*x^2)^(1/2)),x)
 

output

int((a + b*acos(c*x))/((f + g*x)^2*(d - c^2*d*x^2)^(1/2)), x)