Integrand size = 20, antiderivative size = 67 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=-\frac {35 c^3 \text {Si}(\arccos (a x))}{64 a}+\frac {21 c^3 \text {Si}(3 \arccos (a x))}{64 a}-\frac {7 c^3 \text {Si}(5 \arccos (a x))}{64 a}+\frac {c^3 \text {Si}(7 \arccos (a x))}{64 a} \] Output:
-35/64*c^3*Si(arccos(a*x))/a+21/64*c^3*Si(3*arccos(a*x))/a-7/64*c^3*Si(5*a rccos(a*x))/a+1/64*c^3*Si(7*arccos(a*x))/a
Time = 0.30 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.64 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\frac {c^3 (-35 \text {Si}(\arccos (a x))+21 \text {Si}(3 \arccos (a x))-7 \text {Si}(5 \arccos (a x))+\text {Si}(7 \arccos (a x)))}{64 a} \] Input:
Integrate[(c - a^2*c*x^2)^3/ArcCos[a*x],x]
Output:
(c^3*(-35*SinIntegral[ArcCos[a*x]] + 21*SinIntegral[3*ArcCos[a*x]] - 7*Sin Integral[5*ArcCos[a*x]] + SinIntegral[7*ArcCos[a*x]]))/(64*a)
Time = 0.34 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5169, 3042, 3793, 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.
\(\displaystyle \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx\) |
\(\Big \downarrow \) 5169 |
\(\displaystyle -\frac {c^3 \int \frac {\left (1-a^2 x^2\right )^{7/2}}{\arccos (a x)}d\arccos (a x)}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {c^3 \int \frac {\sin (\arccos (a x))^7}{\arccos (a x)}d\arccos (a x)}{a}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle -\frac {c^3 \int \left (-\frac {21 \sin (3 \arccos (a x))}{64 \arccos (a x)}+\frac {7 \sin (5 \arccos (a x))}{64 \arccos (a x)}-\frac {\sin (7 \arccos (a x))}{64 \arccos (a x)}+\frac {35 \sqrt {1-a^2 x^2}}{64 \arccos (a x)}\right )d\arccos (a x)}{a}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c^3 \left (\frac {35}{64} \text {Si}(\arccos (a x))-\frac {21}{64} \text {Si}(3 \arccos (a x))+\frac {7}{64} \text {Si}(5 \arccos (a x))-\frac {1}{64} \text {Si}(7 \arccos (a x))\right )}{a}\) |
Input:
Int[(c - a^2*c*x^2)^3/ArcCos[a*x],x]
Output:
-((c^3*((35*SinIntegral[ArcCos[a*x]])/64 - (21*SinIntegral[3*ArcCos[a*x]]) /64 + (7*SinIntegral[5*ArcCos[a*x]])/64 - SinIntegral[7*ArcCos[a*x]]/64))/ a)
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Simp[(-(b*c)^(-1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p] Subst[ Int[x^n*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, 0]
Time = 0.17 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {c^{3} \left (21 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )-7 \,\operatorname {Si}\left (5 \arccos \left (a x \right )\right )+\operatorname {Si}\left (7 \arccos \left (a x \right )\right )-35 \,\operatorname {Si}\left (\arccos \left (a x \right )\right )\right )}{64 a}\) | \(42\) |
default | \(\frac {c^{3} \left (21 \,\operatorname {Si}\left (3 \arccos \left (a x \right )\right )-7 \,\operatorname {Si}\left (5 \arccos \left (a x \right )\right )+\operatorname {Si}\left (7 \arccos \left (a x \right )\right )-35 \,\operatorname {Si}\left (\arccos \left (a x \right )\right )\right )}{64 a}\) | \(42\) |
Input:
int((-a^2*c*x^2+c)^3/arccos(a*x),x,method=_RETURNVERBOSE)
Output:
1/64/a*c^3*(21*Si(3*arccos(a*x))-7*Si(5*arccos(a*x))+Si(7*arccos(a*x))-35* Si(arccos(a*x)))
\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\arccos \left (a x\right )} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^3/arccos(a*x),x, algorithm="fricas")
Output:
integral(-(a^6*c^3*x^6 - 3*a^4*c^3*x^4 + 3*a^2*c^3*x^2 - c^3)/arccos(a*x), x)
\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=- c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {acos}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {acos}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {acos}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {acos}{\left (a x \right )}}\right )\, dx\right ) \] Input:
integrate((-a**2*c*x**2+c)**3/acos(a*x),x)
Output:
-c**3*(Integral(3*a**2*x**2/acos(a*x), x) + Integral(-3*a**4*x**4/acos(a*x ), x) + Integral(a**6*x**6/acos(a*x), x) + Integral(-1/acos(a*x), x))
\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\int { -\frac {{\left (a^{2} c x^{2} - c\right )}^{3}}{\arccos \left (a x\right )} \,d x } \] Input:
integrate((-a^2*c*x^2+c)^3/arccos(a*x),x, algorithm="maxima")
Output:
-integrate((a^2*c*x^2 - c)^3/arccos(a*x), x)
Time = 0.14 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\frac {c^{3} \operatorname {Si}\left (7 \, \arccos \left (a x\right )\right )}{64 \, a} - \frac {7 \, c^{3} \operatorname {Si}\left (5 \, \arccos \left (a x\right )\right )}{64 \, a} + \frac {21 \, c^{3} \operatorname {Si}\left (3 \, \arccos \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (\arccos \left (a x\right )\right )}{64 \, a} \] Input:
integrate((-a^2*c*x^2+c)^3/arccos(a*x),x, algorithm="giac")
Output:
1/64*c^3*sin_integral(7*arccos(a*x))/a - 7/64*c^3*sin_integral(5*arccos(a* x))/a + 21/64*c^3*sin_integral(3*arccos(a*x))/a - 35/64*c^3*sin_integral(a rccos(a*x))/a
Timed out. \[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=\int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{\mathrm {acos}\left (a\,x\right )} \,d x \] Input:
int((c - a^2*c*x^2)^3/acos(a*x),x)
Output:
int((c - a^2*c*x^2)^3/acos(a*x), x)
\[ \int \frac {\left (c-a^2 c x^2\right )^3}{\arccos (a x)} \, dx=c^{3} \left (-\left (\int \frac {x^{6}}{\mathit {acos} \left (a x \right )}d x \right ) a^{6}+3 \left (\int \frac {x^{4}}{\mathit {acos} \left (a x \right )}d x \right ) a^{4}-3 \left (\int \frac {x^{2}}{\mathit {acos} \left (a x \right )}d x \right ) a^{2}+\int \frac {1}{\mathit {acos} \left (a x \right )}d x \right ) \] Input:
int((-a^2*c*x^2+c)^3/acos(a*x),x)
Output:
c**3*( - int(x**6/acos(a*x),x)*a**6 + 3*int(x**4/acos(a*x),x)*a**4 - 3*int (x**2/acos(a*x),x)*a**2 + int(1/acos(a*x),x))