Integrand size = 29, antiderivative size = 257 \[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=-\frac {2 \sqrt {1-c^2 x^2} (a+b \arccos (c x))^2 \text {arctanh}\left (e^{i \arccos (c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,-e^{i \arccos (c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \arccos (c x)) \operatorname {PolyLog}\left (2,e^{i \arccos (c x)}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,-e^{i \arccos (c x)}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (3,e^{i \arccos (c x)}\right )}{\sqrt {d-c^2 d x^2}} \] Output:
-2*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^2*arctanh(c*x+I*(-c^2*x^2+1)^(1/2) )/(-c^2*d*x^2+d)^(1/2)+2*I*b*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))*polylog( 2,-c*x-I*(-c^2*x^2+1)^(1/2))/(-c^2*d*x^2+d)^(1/2)-2*I*b*(-c^2*x^2+1)^(1/2) *(a+b*arccos(c*x))*polylog(2,c*x+I*(-c^2*x^2+1)^(1/2))/(-c^2*d*x^2+d)^(1/2 )-2*b^2*(-c^2*x^2+1)^(1/2)*polylog(3,-c*x-I*(-c^2*x^2+1)^(1/2))/(-c^2*d*x^ 2+d)^(1/2)+2*b^2*(-c^2*x^2+1)^(1/2)*polylog(3,c*x+I*(-c^2*x^2+1)^(1/2))/(- c^2*d*x^2+d)^(1/2)
Time = 0.64 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.06 \[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {a^2 \log (c x)}{\sqrt {d}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )}{\sqrt {d}}+\frac {2 i a b \sqrt {1-c^2 x^2} \left (2 \arccos (c x) \arctan \left (e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+\operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )\right )}{\sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \left (i \arccos (c x)^2 \arctan \left (e^{i \arccos (c x)}\right )-i \arccos (c x) \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )+i \arccos (c x) \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )+\operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )}{\sqrt {d-c^2 d x^2}} \] Input:
Integrate[(a + b*ArcCos[c*x])^2/(x*Sqrt[d - c^2*d*x^2]),x]
Output:
(a^2*Log[c*x])/Sqrt[d] - (a^2*Log[d + Sqrt[d]*Sqrt[d - c^2*d*x^2]])/Sqrt[d ] + ((2*I)*a*b*Sqrt[1 - c^2*x^2]*(2*ArcCos[c*x]*ArcTan[E^(I*ArcCos[c*x])] - PolyLog[2, (-I)*E^(I*ArcCos[c*x])] + PolyLog[2, I*E^(I*ArcCos[c*x])]))/S qrt[d - c^2*d*x^2] + (2*b^2*Sqrt[1 - c^2*x^2]*(I*ArcCos[c*x]^2*ArcTan[E^(I *ArcCos[c*x])] - I*ArcCos[c*x]*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] + I*ArcC os[c*x]*PolyLog[2, I*E^(I*ArcCos[c*x])] + PolyLog[3, (-I)*E^(I*ArcCos[c*x] )] - PolyLog[3, I*E^(I*ArcCos[c*x])]))/Sqrt[d - c^2*d*x^2]
Time = 0.68 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.62, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {5219, 3042, 4669, 3011, 2720, 7143}
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.
\(\displaystyle \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 5219 |
\(\displaystyle -\frac {\sqrt {1-c^2 x^2} \int \frac {(a+b \arccos (c x))^2}{c x}d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\sqrt {1-c^2 x^2} \int (a+b \arccos (c x))^2 \csc \left (\arccos (c x)+\frac {\pi }{2}\right )d\arccos (c x)}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 4669 |
\(\displaystyle -\frac {\sqrt {1-c^2 x^2} \left (-2 b \int (a+b \arccos (c x)) \log \left (1-i e^{i \arccos (c x)}\right )d\arccos (c x)+2 b \int (a+b \arccos (c x)) \log \left (1+i e^{i \arccos (c x)}\right )d\arccos (c x)-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 3011 |
\(\displaystyle -\frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-i b \int \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )d\arccos (c x)\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle -\frac {\sqrt {1-c^2 x^2} \left (2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \int e^{-i \arccos (c x)} \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right )de^{i \arccos (c x)}\right )-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2\right )}{\sqrt {d-c^2 d x^2}}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle -\frac {\sqrt {1-c^2 x^2} \left (-2 i \arctan \left (e^{i \arccos (c x)}\right ) (a+b \arccos (c x))^2+2 b \left (i \operatorname {PolyLog}\left (2,-i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,-i e^{i \arccos (c x)}\right )\right )-2 b \left (i \operatorname {PolyLog}\left (2,i e^{i \arccos (c x)}\right ) (a+b \arccos (c x))-b \operatorname {PolyLog}\left (3,i e^{i \arccos (c x)}\right )\right )\right )}{\sqrt {d-c^2 d x^2}}\) |
Input:
Int[(a + b*ArcCos[c*x])^2/(x*Sqrt[d - c^2*d*x^2]),x]
Output:
-((Sqrt[1 - c^2*x^2]*((-2*I)*(a + b*ArcCos[c*x])^2*ArcTan[E^(I*ArcCos[c*x] )] + 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, (-I)*E^(I*ArcCos[c*x])] - b*Pol yLog[3, (-I)*E^(I*ArcCos[c*x])]) - 2*b*(I*(a + b*ArcCos[c*x])*PolyLog[2, I *E^(I*ArcCos[c*x])] - b*PolyLog[3, I*E^(I*ArcCos[c*x])])))/Sqrt[d - c^2*d* x^2])
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol ] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si mp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x ))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[(((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)* (x_)^2], x_Symbol] :> Simp[(-(c^(m + 1))^(-1))*Simp[Sqrt[1 - c^2*x^2]/Sqrt[ d + e*x^2]] Subst[Int[(a + b*x)^n*Cos[x]^m, x], x, ArcCos[c*x]], x] /; Fr eeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 0.80 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.53
method | result | size |
default | \(-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (i \arccos \left (c x \right )^{2} \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}-i \arccos \left (c x \right )^{2} \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}+2 i \arccos \left (c x \right ) \ln \left (1-i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b -2 i \arccos \left (c x \right ) \ln \left (1+i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b +2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}-2 \arccos \left (c x \right ) \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}+2 i \operatorname {polylog}\left (3, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}-2 i \operatorname {polylog}\left (3, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) b^{2}+2 \operatorname {polylog}\left (2, i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b -2 \operatorname {polylog}\left (2, -i \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right ) a b +2 a^{2} \arctan \left (c x +i \sqrt {-c^{2} x^{2}+1}\right )\right )}{d \left (c^{2} x^{2}-1\right )}\) | \(394\) |
Input:
int((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
-I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(I*arccos(c*x)^2*ln(1-I*(c*x+ I*(-c^2*x^2+1)^(1/2)))*b^2-I*arccos(c*x)^2*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2 )))*b^2+2*I*arccos(c*x)*ln(1-I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a*b-2*I*arccos( c*x)*ln(1+I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a*b+2*arccos(c*x)*polylog(2,I*(c*x +I*(-c^2*x^2+1)^(1/2)))*b^2-2*arccos(c*x)*polylog(2,-I*(c*x+I*(-c^2*x^2+1) ^(1/2)))*b^2+2*I*polylog(3,I*(c*x+I*(-c^2*x^2+1)^(1/2)))*b^2-2*I*polylog(3 ,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))*b^2+2*polylog(2,I*(c*x+I*(-c^2*x^2+1)^(1/2 )))*a*b-2*polylog(2,-I*(c*x+I*(-c^2*x^2+1)^(1/2)))*a*b+2*a^2*arctan(c*x+I* (-c^2*x^2+1)^(1/2)))/d/(c^2*x^2-1)
\[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas" )
Output:
integral(-sqrt(-c^2*d*x^2 + d)*(b^2*arccos(c*x)^2 + 2*a*b*arccos(c*x) + a^ 2)/(c^2*d*x^3 - d*x), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acos}{\left (c x \right )}\right )^{2}}{x \sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate((a+b*acos(c*x))**2/x/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral((a + b*acos(c*x))**2/(x*sqrt(-d*(c*x - 1)*(c*x + 1))), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \arccos \left (c x\right ) + a\right )}^{2}}{\sqrt {-c^{2} d x^{2} + d} x} \,d x } \] Input:
integrate((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima" )
Output:
-a^2*log(2*sqrt(-c^2*d*x^2 + d)*sqrt(d)/abs(x) + 2*d/abs(x))/sqrt(d) - sqr t(d)*integrate((b^2*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^2 + 2*a*b*a rctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c ^2*d*x^3 - d*x), x)
Exception generated. \[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+b*arccos(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\int \frac {{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^2}{x\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((a + b*acos(c*x))^2/(x*(d - c^2*d*x^2)^(1/2)),x)
Output:
int((a + b*acos(c*x))^2/(x*(d - c^2*d*x^2)^(1/2)), x)
\[ \int \frac {(a+b \arccos (c x))^2}{x \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \left (\int \frac {\mathit {acos} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) a b +\left (\int \frac {\mathit {acos} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, x}d x \right ) b^{2}+\mathrm {log}\left (\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right )\right ) a^{2}}{\sqrt {d}} \] Input:
int((a+b*acos(c*x))^2/x/(-c^2*d*x^2+d)^(1/2),x)
Output:
(2*int(acos(c*x)/(sqrt( - c**2*x**2 + 1)*x),x)*a*b + int(acos(c*x)**2/(sqr t( - c**2*x**2 + 1)*x),x)*b**2 + log(tan(asin(c*x)/2))*a**2)/sqrt(d)