Integrand size = 18, antiderivative size = 55 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \text {Si}(\arcsin (a x))}{4 a}-\frac {3 c \text {Si}(3 \arcsin (a x))}{4 a} \] Output:
-c*(-a^2*x^2+1)^(3/2)/a/arcsin(a*x)-3/4*c*Si(arcsin(a*x))/a-3/4*c*Si(3*arc sin(a*x))/a
Time = 0.29 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (4 \left (1-a^2 x^2\right )^{3/2}+3 \arcsin (a x) \text {Si}(\arcsin (a x))+3 \arcsin (a x) \text {Si}(3 \arcsin (a x))\right )}{4 a \arcsin (a x)} \] Input:
Integrate[(c - a^2*c*x^2)/ArcSin[a*x]^2,x]
Output:
-1/4*(c*(4*(1 - a^2*x^2)^(3/2) + 3*ArcSin[a*x]*SinIntegral[ArcSin[a*x]] + 3*ArcSin[a*x]*SinIntegral[3*ArcSin[a*x]]))/(a*ArcSin[a*x])
Time = 0.40 (sec) , antiderivative size = 54, 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 = {5166, 5224, 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.
\(\displaystyle \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx\) |
\(\Big \downarrow \) 5166 |
\(\displaystyle -3 a c \int \frac {x \sqrt {1-a^2 x^2}}{\arcsin (a x)}dx-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}\) |
\(\Big \downarrow \) 5224 |
\(\displaystyle -\frac {3 c \int \frac {a x \left (1-a^2 x^2\right )}{\arcsin (a x)}d\arcsin (a x)}{a}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}\) |
\(\Big \downarrow \) 4906 |
\(\displaystyle -\frac {3 c \int \left (\frac {a x}{4 \arcsin (a x)}+\frac {\sin (3 \arcsin (a x))}{4 \arcsin (a x)}\right )d\arcsin (a x)}{a}-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \left (\frac {1}{4} \text {Si}(\arcsin (a x))+\frac {1}{4} \text {Si}(3 \arcsin (a x))\right )}{a}\) |
Input:
Int[(c - a^2*c*x^2)/ArcSin[a*x]^2,x]
Output:
-((c*(1 - a^2*x^2)^(3/2))/(a*ArcSin[a*x])) - (3*c*(SinIntegral[ArcSin[a*x] ]/4 + SinIntegral[3*ArcSin[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_.) + 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 = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(-\frac {c \left (3 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}+\cos \left (3 \arcsin \left (a x \right )\right )\right )}{4 a \arcsin \left (a x \right )}\) | \(59\) |
default | \(-\frac {c \left (3 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \sqrt {-a^{2} x^{2}+1}+\cos \left (3 \arcsin \left (a x \right )\right )\right )}{4 a \arcsin \left (a x \right )}\) | \(59\) |
Input:
int((-a^2*c*x^2+c)/arcsin(a*x)^2,x,method=_RETURNVERBOSE)
Output:
-1/4/a*c*(3*Si(arcsin(a*x))*arcsin(a*x)+3*Si(3*arcsin(a*x))*arcsin(a*x)+3* (-a^2*x^2+1)^(1/2)+cos(3*arcsin(a*x)))/arcsin(a*x)
\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)/arcsin(a*x)^2,x, algorithm="fricas")
Output:
integral(-(a^2*c*x^2 - c)/arcsin(a*x)^2, x)
\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=- c \left (\int \frac {a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx\right ) \] Input:
integrate((-a**2*c*x**2+c)/asin(a*x)**2,x)
Output:
-c*(Integral(a**2*x**2/asin(a*x)**2, x) + Integral(-1/asin(a*x)**2, x))
\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}} \,d x } \] Input:
integrate((-a^2*c*x^2+c)/arcsin(a*x)^2,x, algorithm="maxima")
Output:
-(3*a^2*c*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))*integrate(sqrt(a*x + 1)*sqrt(-a*x + 1)*x/arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)), x) - (a^2* c*x^2 - c)*sqrt(a*x + 1)*sqrt(-a*x + 1))/(a*arctan2(a*x, sqrt(a*x + 1)*sqr t(-a*x + 1)))
Time = 0.13 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {3 \, c \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a} - \frac {3 \, c \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{4 \, a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a \arcsin \left (a x\right )} \] Input:
integrate((-a^2*c*x^2+c)/arcsin(a*x)^2,x, algorithm="giac")
Output:
-3/4*c*sin_integral(3*arcsin(a*x))/a - 3/4*c*sin_integral(arcsin(a*x))/a - (-a^2*x^2 + 1)^(3/2)*c/(a*arcsin(a*x))
Timed out. \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int \frac {c-a^2\,c\,x^2}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \] Input:
int((c - a^2*c*x^2)/asin(a*x)^2,x)
Output:
int((c - a^2*c*x^2)/asin(a*x)^2, x)
\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=c \left (-\left (\int \frac {x^{2}}{\mathit {asin} \left (a x \right )^{2}}d x \right ) a^{2}+\int \frac {1}{\mathit {asin} \left (a x \right )^{2}}d x \right ) \] Input:
int((-a^2*c*x^2+c)/asin(a*x)^2,x)
Output:
c*( - int(x**2/asin(a*x)**2,x)*a**2 + int(1/asin(a*x)**2,x))