Integrand size = 22, antiderivative size = 125 \[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=\frac {3 d \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b c}-\frac {d \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b c}-\frac {3 d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )}{4 b c}+\frac {d \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )}{4 b c} \] Output:
3/4*d*Ci((a+b*arccos(c*x))/b)*sin(a/b)/b/c-1/4*d*Ci(3*(a+b*arccos(c*x))/b) *sin(3*a/b)/b/c-3/4*d*cos(a/b)*Si((a+b*arccos(c*x))/b)/b/c+1/4*d*cos(3*a/b )*Si(3*(a+b*arccos(c*x))/b)/b/c
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.74 \[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=\frac {d \left (3 \operatorname {CosIntegral}\left (\frac {a}{b}+\arccos (c x)\right ) \sin \left (\frac {a}{b}\right )-\operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arccos (c x)\right )+\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arccos (c x)\right )\right )\right )}{4 b c} \] Input:
Integrate[(d - c^2*d*x^2)/(a + b*ArcCos[c*x]),x]
Output:
(d*(3*CosIntegral[a/b + ArcCos[c*x]]*Sin[a/b] - CosIntegral[3*(a/b + ArcCo s[c*x])]*Sin[(3*a)/b] - 3*Cos[a/b]*SinIntegral[a/b + ArcCos[c*x]] + Cos[(3 *a)/b]*SinIntegral[3*(a/b + ArcCos[c*x])]))/(4*b*c)
Time = 0.42 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.85, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5169, 25, 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.
\(\displaystyle \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx\) |
\(\Big \downarrow \) 5169 |
\(\displaystyle -\frac {d \int -\frac {\sin ^3\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {d \int \frac {\sin ^3\left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {d \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )^3}{a+b \arccos (c x)}d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle \frac {d \int \left (\frac {3 \sin \left (\frac {a}{b}-\frac {a+b \arccos (c x)}{b}\right )}{4 (a+b \arccos (c x))}-\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arccos (c x))}{b}\right )}{4 (a+b \arccos (c x))}\right )d(a+b \arccos (c x))}{b c}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {d \left (-\frac {3}{4} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arccos (c x)}{b}\right )+\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arccos (c x))}{b}\right )+\frac {3}{4} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arccos (c x)}{b}\right )-\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arccos (c x))}{b}\right )\right )}{b c}\) |
Input:
Int[(d - c^2*d*x^2)/(a + b*ArcCos[c*x]),x]
Output:
-((d*((-3*CosIntegral[(a + b*ArcCos[c*x])/b]*Sin[a/b])/4 + (CosIntegral[(3 *(a + b*ArcCos[c*x]))/b]*Sin[(3*a)/b])/4 + (3*Cos[a/b]*SinIntegral[(a + b* ArcCos[c*x])/b])/4 - (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcCos[c*x]))/b]) /4))/(b*c))
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 = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {d \left (\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )+3 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )\right )}{4 c b}\) | \(94\) |
default | \(\frac {d \left (\operatorname {Si}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-\operatorname {Ci}\left (3 \arccos \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )-3 \,\operatorname {Si}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )+3 \,\operatorname {Ci}\left (\arccos \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )\right )}{4 c b}\) | \(94\) |
Input:
int((-c^2*d*x^2+d)/(a+b*arccos(c*x)),x,method=_RETURNVERBOSE)
Output:
1/4/c*d*(Si(3*arccos(c*x)+3*a/b)*cos(3*a/b)-Ci(3*arccos(c*x)+3*a/b)*sin(3* a/b)-3*Si(arccos(c*x)+a/b)*cos(a/b)+3*Ci(arccos(c*x)+a/b)*sin(a/b))/b
\[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=\int { -\frac {c^{2} d x^{2} - d}{b \arccos \left (c x\right ) + a} \,d x } \] Input:
integrate((-c^2*d*x^2+d)/(a+b*arccos(c*x)),x, algorithm="fricas")
Output:
integral(-(c^2*d*x^2 - d)/(b*arccos(c*x) + a), x)
\[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a + b \operatorname {acos}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a + b \operatorname {acos}{\left (c x \right )}}\right )\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)/(a+b*acos(c*x)),x)
Output:
-d*(Integral(c**2*x**2/(a + b*acos(c*x)), x) + Integral(-1/(a + b*acos(c*x )), x))
\[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=\int { -\frac {c^{2} d x^{2} - d}{b \arccos \left (c x\right ) + a} \,d x } \] Input:
integrate((-c^2*d*x^2+d)/(a+b*arccos(c*x)),x, algorithm="maxima")
Output:
-integrate((c^2*d*x^2 - d)/(b*arccos(c*x) + a), x)
Time = 0.15 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.42 \[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=-\frac {d \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c} + \frac {d \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (c x\right )\right )}{b c} + \frac {d \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c} + \frac {3 \, d \operatorname {Ci}\left (\frac {a}{b} + \arccos \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c} - \frac {3 \, d \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arccos \left (c x\right )\right )}{4 \, b c} - \frac {3 \, d \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arccos \left (c x\right )\right )}{4 \, b c} \] Input:
integrate((-c^2*d*x^2+d)/(a+b*arccos(c*x)),x, algorithm="giac")
Output:
-d*cos(a/b)^2*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b*c) + d*cos(a /b)^3*sin_integral(3*a/b + 3*arccos(c*x))/(b*c) + 1/4*d*cos_integral(3*a/b + 3*arccos(c*x))*sin(a/b)/(b*c) + 3/4*d*cos_integral(a/b + arccos(c*x))*s in(a/b)/(b*c) - 3/4*d*cos(a/b)*sin_integral(3*a/b + 3*arccos(c*x))/(b*c) - 3/4*d*cos(a/b)*sin_integral(a/b + arccos(c*x))/(b*c)
Timed out. \[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=\int \frac {d-c^2\,d\,x^2}{a+b\,\mathrm {acos}\left (c\,x\right )} \,d x \] Input:
int((d - c^2*d*x^2)/(a + b*acos(c*x)),x)
Output:
int((d - c^2*d*x^2)/(a + b*acos(c*x)), x)
\[ \int \frac {d-c^2 d x^2}{a+b \arccos (c x)} \, dx=d \left (-\left (\int \frac {x^{2}}{\mathit {acos} \left (c x \right ) b +a}d x \right ) c^{2}+\int \frac {1}{\mathit {acos} \left (c x \right ) b +a}d x \right ) \] Input:
int((-c^2*d*x^2+d)/(a+b*acos(c*x)),x)
Output:
d*( - int(x**2/(acos(c*x)*b + a),x)*c**2 + int(1/(acos(c*x)*b + a),x))