Integrand size = 22, antiderivative size = 158 \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=-\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arcsin (c x))}+\frac {3 d \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b^2 c}+\frac {3 d \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^2 c}-\frac {3 d \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b^2 c}-\frac {3 d \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b^2 c} \] Output:
-d*(-c^2*x^2+1)^(3/2)/b/c/(a+b*arcsin(c*x))+3/4*d*Ci((a+b*arcsin(c*x))/b)* sin(a/b)/b^2/c+3/4*d*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2/c-3/4*d*cos( a/b)*Si((a+b*arcsin(c*x))/b)/b^2/c-3/4*d*cos(3*a/b)*Si(3*(a+b*arcsin(c*x)) /b)/b^2/c
Time = 0.31 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.77 \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=\frac {d \left (-\frac {4 b \left (1-c^2 x^2\right )^{3/2}}{a+b \arcsin (c x)}+3 \left (\operatorname {CosIntegral}\left (\frac {a}{b}+\arcsin (c x)\right ) \sin \left (\frac {a}{b}\right )+\operatorname {CosIntegral}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-\cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\arcsin (c x)\right )-\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\arcsin (c x)\right )\right )\right )\right )}{4 b^2 c} \] Input:
Integrate[(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^2,x]
Output:
(d*((-4*b*(1 - c^2*x^2)^(3/2))/(a + b*ArcSin[c*x]) + 3*(CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b] + CosIntegral[3*(a/b + ArcSin[c*x])]*Sin[(3*a)/b] - Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]] - Cos[(3*a)/b]*SinIntegral[3*(a/b + ArcSin[c*x])])))/(4*b^2*c)
Time = 0.61 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.89, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {5166, 5224, 25, 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.
\(\displaystyle \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx\) |
\(\Big \downarrow \) 5166 |
\(\displaystyle -\frac {3 c d \int \frac {x \sqrt {1-c^2 x^2}}{a+b \arcsin (c x)}dx}{b}-\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {3 d \int -\frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c}-\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {3 d \int \frac {\cos ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{a+b \arcsin (c x)}d(a+b \arcsin (c x))}{b^2 c}-\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle \frac {3 d \int \left (\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 (a+b \arcsin (c x))}+\frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c x)}{b}\right )}{4 (a+b \arcsin (c x))}\right )d(a+b \arcsin (c x))}{b^2 c}-\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arcsin (c x))}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {3 d \left (-\frac {1}{4} \sin \left (\frac {a}{b}\right ) \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right )-\frac {1}{4} \sin \left (\frac {3 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )+\frac {1}{4} \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )+\frac {1}{4} \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )\right )}{b^2 c}-\frac {d \left (1-c^2 x^2\right )^{3/2}}{b c (a+b \arcsin (c x))}\) |
Input:
Int[(d - c^2*d*x^2)/(a + b*ArcSin[c*x])^2,x]
Output:
-((d*(1 - c^2*x^2)^(3/2))/(b*c*(a + b*ArcSin[c*x]))) - (3*d*(-1/4*(CosInte gral[(a + b*ArcSin[c*x])/b]*Sin[a/b]) - (CosIntegral[(3*(a + b*ArcSin[c*x] ))/b]*Sin[(3*a)/b])/4 + (Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/4 + (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/4))/(b^2*c)
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_.) + ArcSin[(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*ArcSin[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*ArcSin[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_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x ^2)^p] Subst[Int[x^n*Sin[-a/b + x/b]^m*Cos[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSin[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 = 0.07 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.50
method | result | size |
derivativedivides | \(-\frac {d \left (3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -3 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +3 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -3 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +3 \sqrt {-c^{2} x^{2}+1}\, b +\cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{4 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(237\) |
default | \(-\frac {d \left (3 \arcsin \left (c x \right ) \operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b -3 \arcsin \left (c x \right ) \operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b +3 \arcsin \left (c x \right ) \operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b -3 \arcsin \left (c x \right ) \operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b +3 \,\operatorname {Si}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a -3 \,\operatorname {Ci}\left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a +3 \,\operatorname {Si}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a -3 \,\operatorname {Ci}\left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a +3 \sqrt {-c^{2} x^{2}+1}\, b +\cos \left (3 \arcsin \left (c x \right )\right ) b \right )}{4 c \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(237\) |
Input:
int((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^2,x,method=_RETURNVERBOSE)
Output:
-1/4/c*d*(3*arcsin(c*x)*Si(3*arcsin(c*x)+3*a/b)*cos(3*a/b)*b-3*arcsin(c*x) *Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*b+3*arcsin(c*x)*Si(arcsin(c*x)+a/b)*co s(a/b)*b-3*arcsin(c*x)*Ci(arcsin(c*x)+a/b)*sin(a/b)*b+3*Si(3*arcsin(c*x)+3 *a/b)*cos(3*a/b)*a-3*Ci(3*arcsin(c*x)+3*a/b)*sin(3*a/b)*a+3*Si(arcsin(c*x) +a/b)*cos(a/b)*a-3*Ci(arcsin(c*x)+a/b)*sin(a/b)*a+3*(-c^2*x^2+1)^(1/2)*b+c os(3*arcsin(c*x))*b)/(a+b*arcsin(c*x))/b^2
\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^2,x, algorithm="fricas")
Output:
integral(-(c^2*d*x^2 - d)/(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2), x )
\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=- d \left (\int \frac {c^{2} x^{2}}{a^{2} + 2 a b \operatorname {asin}{\left (c x \right )} + b^{2} \operatorname {asin}^{2}{\left (c x \right )}}\, dx + \int \left (- \frac {1}{a^{2} + 2 a b \operatorname {asin}{\left (c x \right )} + b^{2} \operatorname {asin}^{2}{\left (c x \right )}}\right )\, dx\right ) \] Input:
integrate((-c**2*d*x**2+d)/(a+b*asin(c*x))**2,x)
Output:
-d*(Integral(c**2*x**2/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), x) + Integral(-1/(a**2 + 2*a*b*asin(c*x) + b**2*asin(c*x)**2), x))
\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=\int { -\frac {c^{2} d x^{2} - d}{{\left (b \arcsin \left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^2,x, algorithm="maxima")
Output:
((c^2*d*x^2 - d)*sqrt(c*x + 1)*sqrt(-c*x + 1) - 3*(b^2*c^2*d*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c^2*d)*integrate(sqrt(c*x + 1)*sqrt(-c *x + 1)*x/(b^2*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b), x))/(b^2 *c*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c)
Leaf count of result is larger than twice the leaf count of optimal. 571 vs. \(2 (148) = 296\).
Time = 0.21 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.61 \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx =\text {Too large to display} \] Input:
integrate((-c^2*d*x^2+d)/(a+b*arcsin(c*x))^2,x, algorithm="giac")
Output:
3*b*d*arcsin(c*x)*cos(a/b)^2*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/ (b^3*c*arcsin(c*x) + a*b^2*c) - 3*b*d*arcsin(c*x)*cos(a/b)^3*sin_integral( 3*a/b + 3*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) + 3*a*d*cos(a/b)^2*co s_integral(3*a/b + 3*arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) - 3*a*d*cos(a/b)^3*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 3/4*b*d*arcsin(c*x)*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/ b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 3/4*b*d*arcsin(c*x)*cos_integral(a/b + arcsin(c*x))*sin(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 9/4*b*d*arcsin(c*x)* cos(a/b)*sin_integral(3*a/b + 3*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 3/4*b*d*arcsin(c*x)*cos(a/b)*sin_integral(a/b + arcsin(c*x))/(b^3*c*arc sin(c*x) + a*b^2*c) - 3/4*a*d*cos_integral(3*a/b + 3*arcsin(c*x))*sin(a/b) /(b^3*c*arcsin(c*x) + a*b^2*c) + 3/4*a*d*cos_integral(a/b + arcsin(c*x))*s in(a/b)/(b^3*c*arcsin(c*x) + a*b^2*c) + 9/4*a*d*cos(a/b)*sin_integral(3*a/ b + 3*arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - 3/4*a*d*cos(a/b)*sin_in tegral(a/b + arcsin(c*x))/(b^3*c*arcsin(c*x) + a*b^2*c) - (-c^2*x^2 + 1)^( 3/2)*b*d/(b^3*c*arcsin(c*x) + a*b^2*c)
Timed out. \[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=\int \frac {d-c^2\,d\,x^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \] Input:
int((d - c^2*d*x^2)/(a + b*asin(c*x))^2,x)
Output:
int((d - c^2*d*x^2)/(a + b*asin(c*x))^2, x)
\[ \int \frac {d-c^2 d x^2}{(a+b \arcsin (c x))^2} \, dx=d \left (-\left (\int \frac {x^{2}}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) c^{2}+\int \frac {1}{\mathit {asin} \left (c x \right )^{2} b^{2}+2 \mathit {asin} \left (c x \right ) a b +a^{2}}d x \right ) \] Input:
int((-c^2*d*x^2+d)/(a+b*asin(c*x))^2,x)
Output:
d*( - int(x**2/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x)*c**2 + int( 1/(asin(c*x)**2*b**2 + 2*asin(c*x)*a*b + a**2),x))