Integrand size = 31, antiderivative size = 1064 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=-\frac {a d (c f-g) (c f+g) \sqrt {d-c^2 d x^2}}{g^3}+\frac {b c d x \sqrt {d-c^2 d x^2}}{3 g \sqrt {1-c^2 x^2}}-\frac {b c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2}}{g^3 \sqrt {1-c^2 x^2}}+\frac {b c^3 d f x^2 \sqrt {d-c^2 d x^2}}{4 g^2 \sqrt {1-c^2 x^2}}-\frac {b c^3 d x^3 \sqrt {d-c^2 d x^2}}{9 g \sqrt {1-c^2 x^2}}-\frac {b d (c f-g) (c f+g) \sqrt {d-c^2 d x^2} \arccos (c x)}{g^3}+\frac {c^2 d f x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{2 g^2}+\frac {d \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} (a+b \arccos (c x))}{3 g}-\frac {c d f \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{4 b g^2 \sqrt {1-c^2 x^2}}+\frac {c d (c f-g) (c f+g) x \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b g^3 \sqrt {1-c^2 x^2}}+\frac {d \left (c^2 f^2-g^2\right )^2 \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c g^4 (f+g x) \sqrt {1-c^2 x^2}}+\frac {d (c f-g) (c f+g) \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \arccos (c x))^2}{2 b c g^2 (f+g x)}+\frac {a d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arctan \left (\frac {g+c^2 f x}{\sqrt {c^2 f^2-g^2} \sqrt {1-c^2 x^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {i b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \arccos (c x) \log \left (1+\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}+\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}}-\frac {b d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4 \sqrt {1-c^2 x^2}} \]
-a*d*(c*f-g)*(c*f+g)*(-c^2*d*x^2+d)^(1/2)/g^3-b*d*(c*f-g)*(c*f+g)*arccos(c *x)*(-c^2*d*x^2+d)^(1/2)/g^3+1/2*c^2*d*f*x*(a+b*arccos(c*x))*(-c^2*d*x^2+d )^(1/2)/g^2+1/3*d*(-c^2*x^2+1)*(a+b*arccos(c*x))*(-c^2*d*x^2+d)^(1/2)/g+1/ 3*b*c*d*x*(-c^2*d*x^2+d)^(1/2)/g/(-c^2*x^2+1)^(1/2)-b*c*d*(c*f-g)*(c*f+g)* x*(-c^2*d*x^2+d)^(1/2)/g^3/(-c^2*x^2+1)^(1/2)+1/4*b*c^3*d*f*x^2*(-c^2*d*x^ 2+d)^(1/2)/g^2/(-c^2*x^2+1)^(1/2)-1/9*b*c^3*d*x^3*(-c^2*d*x^2+d)^(1/2)/g/( -c^2*x^2+1)^(1/2)-1/4*c*d*f*(a+b*arccos(c*x))^2*(-c^2*d*x^2+d)^(1/2)/b/g^2 /(-c^2*x^2+1)^(1/2)+1/2*c*d*(c*f-g)*(c*f+g)*x*(a+b*arccos(c*x))^2*(-c^2*d* x^2+d)^(1/2)/b/g^3/(-c^2*x^2+1)^(1/2)+1/2*d*(c^2*f^2-g^2)^2*(a+b*arccos(c* x))^2*(-c^2*d*x^2+d)^(1/2)/b/c/g^4/(g*x+f)/(-c^2*x^2+1)^(1/2)+a*d*(c^2*f^2 -g^2)^(3/2)*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^4/(-c^2*x^2+1)^(1/2)+I*b*d*(c^2*f^2-g^2)^(3/2)*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^4/(-c^2*x^2+1)^(1/2)-I*b*d*(c^2*f^2-g^2)^(3/2)*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^4/(-c^2*x^2+1)^(1/2)+b*d*(c^2*f^2-g^2)^(3/2)*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^4/( -c^2*x^2+1)^(1/2)-b*d*(c^2*f^2-g^2)^(3/2)*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^4/(-c^2*x^2+1)^( 1/2)+1/2*d*(c*f-g)*(c*f+g)*(a+b*arccos(c*x))^2*(-c^2*x^2+1)^(1/2)*(-c^2...
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(3034\) vs. \(2(1064)=2128\).
Time = 15.83 (sec) , antiderivative size = 3034, normalized size of antiderivative = 2.85 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Result too large to show} \]
Sqrt[-(d*(-1 + c^2*x^2))]*((a*d*(-3*c^2*f^2 + 4*g^2))/(3*g^3) + (a*c^2*d*f *x)/(2*g^2) - (a*c^2*d*x^2)/(3*g)) + (a*c*d^(3/2)*f*(2*c^2*f^2 - 3*g^2)*Ar cTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*g^4) + (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*d^(3/2)*(-(c^2* f^2) + g^2)^(3/2)*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqr t[-(d*(-1 + c^2*x^2))]])/g^4 - (b*d*Sqrt[d*(1 - c^2*x^2)]*((-2*c*g*x)/Sqrt [1 - c^2*x^2] - 2*g*ArcCos[c*x] + (c*f*ArcCos[c*x]^2)/Sqrt[1 - c^2*x^2] + (2*(-(c*f) + g)*(c*f + g)*(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)*Ar cTanh[((-(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])/S qrt[-(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])/(Sqr t[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)...
Time = 2.40 (sec) , antiderivative size = 700, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {5277, 5267, 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.
| \(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx\) |
| \(\Big \downarrow \) 5277 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x}dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 5267 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \int \left (-\frac {x \sqrt {1-c^2 x^2} (a+b \arccos (c x)) c^2}{g}+\frac {f \sqrt {1-c^2 x^2} (a+b \arccos (c x)) c^2}{g^2}+\frac {\left (g^2-c^2 f^2\right ) \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{g^2 (f+g x)}\right )dx}{\sqrt {1-c^2 x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {d-c^2 d x^2} \left (\frac {\left (1-c^2 x^2\right ) \left (c^2 f^2-g^2\right ) (a+b \arccos (c x))^2}{2 b c g^2 (f+g x)}+\frac {\left (c^2 f^2-g^2\right )^2 (a+b \arccos (c x))^2}{2 b c g^4 (f+g x)}+\frac {c x \left (c^2 f^2-g^2\right ) (a+b \arccos (c x))^2}{2 b g^3}+\frac {c^2 f x \sqrt {1-c^2 x^2} (a+b \arccos (c x))}{2 g^2}+\frac {\left (1-c^2 x^2\right )^{3/2} (a+b \arccos (c x))}{3 g}-\frac {c f (a+b \arccos (c x))^2}{4 b g^2}+\frac {a \left (c^2 f^2-g^2\right )^{3/2} \arctan \left (\frac {c^2 f x+g}{\sqrt {1-c^2 x^2} \sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {a \sqrt {1-c^2 x^2} (c f-g) (c f+g)}{g^3}+\frac {b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {b \left (c^2 f^2-g^2\right )^{3/2} \operatorname {PolyLog}\left (2,-\frac {e^{i \arccos (c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^4}+\frac {i b \arccos (c x) \left (c^2 f^2-g^2\right )^{3/2} \log \left (1+\frac {g e^{i \arccos (c x)}}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^4}-\frac {i b \arccos (c x) \left (c^2 f^2-g^2\right )^{3/2} \log \left (1+\frac {g e^{i \arccos (c x)}}{\sqrt {c^2 f^2-g^2}+c f}\right )}{g^4}-\frac {b \sqrt {1-c^2 x^2} \arccos (c x) (c f-g) (c f+g)}{g^3}+\frac {b c^3 f x^2}{4 g^2}-\frac {b c^3 x^3}{9 g}-\frac {b c x \left (c^2 f^2-g^2\right )}{g^3}+\frac {b c x}{3 g}\right )}{\sqrt {1-c^2 x^2}}\) |
(d*Sqrt[d - c^2*d*x^2]*((b*c*x)/(3*g) - (b*c*(c^2*f^2 - g^2)*x)/g^3 + (b*c ^3*f*x^2)/(4*g^2) - (b*c^3*x^3)/(9*g) - (a*(c*f - g)*(c*f + g)*Sqrt[1 - c^ 2*x^2])/g^3 - (b*(c*f - g)*(c*f + g)*Sqrt[1 - c^2*x^2]*ArcCos[c*x])/g^3 + (c^2*f*x*Sqrt[1 - c^2*x^2]*(a + b*ArcCos[c*x]))/(2*g^2) + ((1 - c^2*x^2)^( 3/2)*(a + b*ArcCos[c*x]))/(3*g) - (c*f*(a + b*ArcCos[c*x])^2)/(4*b*g^2) + (c*(c^2*f^2 - g^2)*x*(a + b*ArcCos[c*x])^2)/(2*b*g^3) + ((c^2*f^2 - g^2)^2 *(a + b*ArcCos[c*x])^2)/(2*b*c*g^4*(f + g*x)) + ((c^2*f^2 - g^2)*(1 - c^2* x^2)*(a + b*ArcCos[c*x])^2)/(2*b*c*g^2*(f + g*x)) + (a*(c^2*f^2 - g^2)^(3/ 2)*ArcTan[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqrt[1 - c^2*x^2])])/g^4 + (I *b*(c^2*f^2 - g^2)^(3/2)*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g^4 - (I*b*(c^2*f^2 - g^2)^(3/2)*ArcCos[c*x]*Log[1 + (E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/g^4 + (b*(c^2*f^2 - g^2)^(3/2)*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f - Sqrt[c^2*f^2 - g^2])) ])/g^4 - (b*(c^2*f^2 - g^2)^(3/2)*PolyLog[2, -((E^(I*ArcCos[c*x])*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/g^4))/Sqrt[1 - c^2*x^2]
3.1.9.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_ ) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[Sqrt[d + e*x^2]*(a + b*ArcCos[c*x])^n, (f + g*x)^m*(d + e*x^2)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && IGtQ[p + 1/2, 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]
Time = 3.01 (sec) , antiderivative size = 1559, normalized size of antiderivative = 1.47
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1559\) |
| parts | \(\text {Expression too large to display}\) | \(1559\) |
a/g*(1/3*(-(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*d*f/g*(-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)))-d*(c^2*f^2-g^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))))+b*(-1/4*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/(c^2*x^ 2-1)*arccos(c*x)^2*f*(2*c^2*f^2-3*g^2)*c*d/g^4-1/72*(-d*(c^2*x^2-1))^(1/2) *(4*I*c^3*x^3*(-c^2*x^2+1)^(1/2)+4*c^4*x^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^ 2*x^2+1)*(I+3*arccos(c*x))*d/(c^2*x^2-1)/g+1/16*(-d*(c^2*x^2-1))^(1/2)*(2* I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3-I*(-c^2*x^2+1)^(1/2)-2*c*x)*f*(I+2* arccos(c*x))*c*d/(c^2*x^2-1)/g^2-1/8*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1 )^(1/2)*x*c+c^2*x^2-1)*(4*arccos(c*x)*c^2*f^2+4*I*c^2*f^2-5*arccos(c*x)*g^ 2-5*I*g^2)*d/(c^2*x^2-1)/g^3-1/8*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x ^2+1)^(1/2)*x*c-1)*(4*arccos(c*x)*c^2*f^2-4*I*c^2*f^2-5*arccos(c*x)*g^2+5* I*g^2)*d/(c^2*x^2-1)/g^3+1/16*(-d*(c^2*x^2-1))^(1/2)*(-2*I*(-c^2*x^2+1)...
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \arccos \left (c x\right ) + a\right )}}{g x + f} \,d x } \]
integral(-(a*c^2*d*x^2 - a*d + (b*c^2*d*x^2 - b*d)*arccos(c*x))*sqrt(-c^2* d*x^2 + d)/(g*x + f), x)
\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acos}{\left (c x \right )}\right )}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \arccos (c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{f+g\,x} \,d x \]