Integrand size = 18, antiderivative size = 54 \[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arccos (a x)}-\frac {3 c \operatorname {CosIntegral}(\arccos (a x))}{4 a}+\frac {3 c \operatorname {CosIntegral}(3 \arccos (a x))}{4 a} \] Output:
c*(-a^2*x^2+1)^(3/2)/a/arccos(a*x)-3/4*c*Ci(arccos(a*x))/a+3/4*c*Ci(3*arcc os(a*x))/a
Time = 0.14 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.02 \[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=\frac {c \left (4 \left (1-a^2 x^2\right )^{3/2}-3 \arccos (a x) \operatorname {CosIntegral}(\arccos (a x))+3 \arccos (a x) \operatorname {CosIntegral}(3 \arccos (a x))\right )}{4 a \arccos (a x)} \] Input:
Integrate[(c - a^2*c*x^2)/ArcCos[a*x]^2,x]
Output:
(c*(4*(1 - a^2*x^2)^(3/2) - 3*ArcCos[a*x]*CosIntegral[ArcCos[a*x]] + 3*Arc Cos[a*x]*CosIntegral[3*ArcCos[a*x]]))/(4*a*ArcCos[a*x])
Time = 0.41 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5167, 5225, 4906, 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.
\(\displaystyle \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx\) |
\(\Big \downarrow \) 5167 |
\(\displaystyle 3 a c \int \frac {x \sqrt {1-a^2 x^2}}{\arccos (a x)}dx+\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arccos (a x)}\) |
\(\Big \downarrow \) 5225 |
\(\displaystyle \frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arccos (a x)}-\frac {3 c \int \frac {a x \left (1-a^2 x^2\right )}{\arccos (a x)}d\arccos (a x)}{a}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arccos (a x)}-\frac {3 c \int \left (\frac {a x}{4 \arccos (a x)}-\frac {\cos (3 \arccos (a x))}{4 \arccos (a x)}\right )d\arccos (a x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arccos (a x)}-\frac {3 c \left (\frac {1}{4} \operatorname {CosIntegral}(\arccos (a x))-\frac {1}{4} \operatorname {CosIntegral}(3 \arccos (a x))\right )}{a}\) |
Input:
Int[(c - a^2*c*x^2)/ArcCos[a*x]^2,x]
Output:
(c*(1 - a^2*x^2)^(3/2))/(a*ArcCos[a*x]) - (3*c*(CosIntegral[ArcCos[a*x]]/4 - CosIntegral[3*ArcCos[a*x]]/4))/a
Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b _.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin[a + b*x ]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IG tQ[p, 0]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_)^2)^(p_.), x_ Symbol] :> Simp[(-Sqrt[1 - c^2*x^2])*(d + e*x^2)^p*((a + b*ArcCos[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c*((2*p + 1)/(b*(n + 1)))*Simp[(d + e*x^2)^p /(1 - c^2*x^2)^p] Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcCos[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && LtQ[n, - 1]
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(-(b*c^(m + 1))^(-1))*Simp[(d + e*x^2)^p/(1 - c ^2*x^2)^p] Subst[Int[x^n*Cos[-a/b + x/b]^m*Sin[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCos[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e , 0] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {c \left (3 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-3 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right ) \arccos \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}-\sin \left (3 \arccos \left (a x \right )\right )\right )}{4 a \arccos \left (a x \right )}\) | \(61\) |
default | \(\frac {c \left (3 \,\operatorname {Ci}\left (3 \arccos \left (a x \right )\right ) \arccos \left (a x \right )-3 \,\operatorname {Ci}\left (\arccos \left (a x \right )\right ) \arccos \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}-\sin \left (3 \arccos \left (a x \right )\right )\right )}{4 a \arccos \left (a x \right )}\) | \(61\) |
Input:
int((-a^2*c*x^2+c)/arccos(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/4/a*c*(3*Ci(3*arccos(a*x))*arccos(a*x)-3*Ci(arccos(a*x))*arccos(a*x)+3*( -a^2*x^2+1)^(1/2)-sin(3*arccos(a*x)))/arccos(a*x)
\[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arccos \left (a x\right )^{2}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)/arccos(a*x)^2,x, algorithm="fricas")
Output:
integral(-(a^2*c*x^2 - c)/arccos(a*x)^2, x)
\[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=- c \left (\int \frac {a^{2} x^{2}}{\operatorname {acos}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acos}^{2}{\left (a x \right )}}\right )\, dx\right ) \] Input:
integrate((-a**2*c*x**2+c)/acos(a*x)**2,x)
Output:
-c*(Integral(a**2*x**2/acos(a*x)**2, x) + Integral(-1/acos(a*x)**2, x))
\[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arccos \left (a x\right )^{2}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)/arccos(a*x)^2,x, algorithm="maxima")
Output:
(3*a^2*c*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x)*integrate(sqrt(a*x + 1 )*sqrt(-a*x + 1)*x/arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x), x) - (a^2*c *x^2 - c)*sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x))
Time = 0.15 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.35 \[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=-\frac {\sqrt {-a^{2} x^{2} + 1} a c x^{2}}{\arccos \left (a x\right )} + \frac {3 \, c \operatorname {Ci}\left (3 \, \arccos \left (a x\right )\right )}{4 \, a} - \frac {3 \, c \operatorname {Ci}\left (\arccos \left (a x\right )\right )}{4 \, a} + \frac {\sqrt {-a^{2} x^{2} + 1} c}{a \arccos \left (a x\right )} \] Input:
integrate((-a^2*c*x^2+c)/arccos(a*x)^2,x, algorithm="giac")
Output:
-sqrt(-a^2*x^2 + 1)*a*c*x^2/arccos(a*x) + 3/4*c*cos_integral(3*arccos(a*x) )/a - 3/4*c*cos_integral(arccos(a*x))/a + sqrt(-a^2*x^2 + 1)*c/(a*arccos(a *x))
Timed out. \[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=\int \frac {c-a^2\,c\,x^2}{{\mathrm {acos}\left (a\,x\right )}^2} \,d x \] Input:
int((c - a^2*c*x^2)/acos(a*x)^2,x)
Output:
int((c - a^2*c*x^2)/acos(a*x)^2, x)
\[ \int \frac {c-a^2 c x^2}{\arccos (a x)^2} \, dx=c \left (-\left (\int \frac {x^{2}}{\mathit {acos} \left (a x \right )^{2}}d x \right ) a^{2}+\int \frac {1}{\mathit {acos} \left (a x \right )^{2}}d x \right ) \] Input:
int((-a^2*c*x^2+c)/acos(a*x)^2,x)
Output:
c*( - int(x**2/acos(a*x)**2,x)*a**2 + int(1/acos(a*x)**2,x))