\(\int \frac {(1-c^2 x^2)^{3/2}}{x (a+b \arcsin (c x))} \, dx\) [329]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\frac {5 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b}+\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}-\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b}+\text {Int}\left (\frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))},x\right ) \]

[Out]

-5/4*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b-1/4*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b+5/4*Ci((a+b*arcsin(c*x))/b)
*sin(a/b)/b+1/4*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b+Unintegrable(1/x/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x
)

Rubi [N/A]

Not integrable

Time = 0.44 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx \]

[In]

Int[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])),x]

[Out]

(5*CosIntegral[(a + b*ArcSin[c*x])/b]*Sin[a/b])/(4*b) + (CosIntegral[(3*(a + b*ArcSin[c*x]))/b]*Sin[(3*a)/b])/
(4*b) - (5*Cos[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])/(4*b) - (Cos[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x
]))/b])/(4*b) + Defer[Int][1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}-\frac {2 c^2 x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}+\frac {c^4 x^3}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}\right ) \, dx \\ & = -\left (\left (2 c^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx\right )+c^4 \int \frac {x^3}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {\text {Subst}\left (\int \frac {\sin ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\frac {2 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {\text {Subst}\left (\int \left (-\frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 x}+\frac {3 \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{4 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b}-\frac {\left (2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\frac {\left (2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = \frac {2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b}+\frac {\text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}-\frac {3 \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = \frac {2 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {2 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{b}+\frac {\left (3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}-\frac {\left (3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}+\frac {\sin \left (\frac {3 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{4 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = \frac {5 \operatorname {CosIntegral}\left (\frac {a+b \arcsin (c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{4 b}+\frac {\operatorname {CosIntegral}\left (\frac {3 (a+b \arcsin (c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b}-\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \arcsin (c x)}{b}\right )}{4 b}-\frac {\cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \arcsin (c x))}{b}\right )}{4 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx \]

[In]

Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(3/2)/(x*(a + b*ArcSin[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \arcsin \left (c x \right )\right )}d x\]

[In]

int((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x)

[Out]

int((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(b*x*arcsin(c*x) + a*x), x)

Sympy [N/A]

Not integrable

Time = 2.15 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]

[In]

integrate((-c**2*x**2+1)**(3/2)/x/(a+b*asin(c*x)),x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(3/2)/(x*(a + b*asin(c*x))), x)

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(3/2)/((b*arcsin(c*x) + a)*x), x)

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((-c^2*x^2+1)^(3/2)/x/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \arcsin (c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \]

[In]

int((1 - c^2*x^2)^(3/2)/(x*(a + b*asin(c*x))),x)

[Out]

int((1 - c^2*x^2)^(3/2)/(x*(a + b*asin(c*x))), x)