\(\int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx\) [376]

Optimal result

Integrand size = 18, antiderivative size = 55 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \text {Si}(\arcsin (a x))}{4 a}-\frac {3 c \text {Si}(3 \arcsin (a x))}{4 a} \]

[Out]

-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*arcsin(a*x))/a

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4751, 4809, 4491, 3380} \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {3 c \text {Si}(\arcsin (a x))}{4 a}-\frac {3 c \text {Si}(3 \arcsin (a x))}{4 a} \]

[In]

Int[(c - a^2*c*x^2)/ArcSin[a*x]^2,x]

[Out]

-((c*(1 - a^2*x^2)^(3/2))/(a*ArcSin[a*x])) - (3*c*SinIntegral[ArcSin[a*x]])/(4*a) - (3*c*SinIntegral[3*ArcSin[
a*x]])/(4*a)

Rule 3380

Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[SinIntegral[e + f*x]/d, x] /; FreeQ[{c, d,
e, f}, x] && EqQ[d*e - c*f, 0]

Rule 4491

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(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]
&& IGtQ[p, 0]

Rule 4751

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] + Dist[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]

Rule 4809

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(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]

Rubi steps \begin{align*} \text {integral}& = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-(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)}-\frac {(3 c) \text {Subst}\left (\int \frac {\cos ^2(x) \sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{a} \\ & = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {(3 c) \text {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\arcsin (a x)\right )}{a} \\ & = -\frac {c \left (1-a^2 x^2\right )^{3/2}}{a \arcsin (a x)}-\frac {(3 c) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a}-\frac {(3 c) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\arcsin (a x)\right )}{4 a} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.35 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.00 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {c \left (4 \left (1-a^2 x^2\right )^{3/2}+3 \arcsin (a x) \text {Si}(\arcsin (a x))+3 \arcsin (a x) \text {Si}(3 \arcsin (a x))\right )}{4 a \arcsin (a x)} \]

[In]

Integrate[(c - a^2*c*x^2)/ArcSin[a*x]^2,x]

[Out]

-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])

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.07

method	result	size
derivativedivides	\(-\frac {c \left (3 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+\cos \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}\right )}{4 a \arcsin \left (a x \right )}\)	\(59\)
default	\(-\frac {c \left (3 \,\operatorname {Si}\left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+3 \,\operatorname {Si}\left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+\cos \left (3 \arcsin \left (a x \right )\right )+3 \sqrt {-a^{2} x^{2}+1}\right )}{4 a \arcsin \left (a x \right )}\)	\(59\)

[In]

int((-a^2*c*x^2+c)/arcsin(a*x)^2,x,method=_RETURNVERBOSE)

[Out]

-1/4/a*c*(3*Si(arcsin(a*x))*arcsin(a*x)+3*Si(3*arcsin(a*x))*arcsin(a*x)+cos(3*arcsin(a*x))+3*(-a^2*x^2+1)^(1/2
))/arcsin(a*x)

Fricas [F]

\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)/arcsin(a*x)^2,x, algorithm="fricas")

[Out]

integral(-(a^2*c*x^2 - c)/arcsin(a*x)^2, x)

Sympy [F]

\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=- c \left (\int \frac {a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx\right ) \]

[In]

integrate((-a**2*c*x**2+c)/asin(a*x)**2,x)

[Out]

-c*(Integral(a**2*x**2/asin(a*x)**2, x) + Integral(-1/asin(a*x)**2, x))

Maxima [F]

\[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int { -\frac {a^{2} c x^{2} - c}{\arcsin \left (a x\right )^{2}} \,d x } \]

[In]

integrate((-a^2*c*x^2+c)/arcsin(a*x)^2,x, algorithm="maxima")

[Out]

-(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, sqr
t(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)*s
qrt(-a*x + 1)))

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.89 \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=-\frac {3 \, c \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{4 \, a} - \frac {3 \, c \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{4 \, a} - \frac {{\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} c}{a \arcsin \left (a x\right )} \]

[In]

integrate((-a^2*c*x^2+c)/arcsin(a*x)^2,x, algorithm="giac")

[Out]

-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))

Mupad [F(-1)]

Timed out. \[ \int \frac {c-a^2 c x^2}{\arcsin (a x)^2} \, dx=\int \frac {c-a^2\,c\,x^2}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]

[In]

int((c - a^2*c*x^2)/asin(a*x)^2,x)

[Out]

int((c - a^2*c*x^2)/asin(a*x)^2, x)