\(\int \frac {(1-c^2 x^2)^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx\) [338]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=-\frac {c \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b}-\frac {c \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b}-\frac {15 c \log (a+b \arcsin (c x))}{8 b}-\frac {c \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b}-\frac {c \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))},x\right ) \]

[Out]

-c*Ci(2*(a+b*arcsin(c*x))/b)*cos(2*a/b)/b-1/8*c*Ci(4*(a+b*arcsin(c*x))/b)*cos(4*a/b)/b-15/8*c*ln(a+b*arcsin(c*
x))/b-c*Si(2*(a+b*arcsin(c*x))/b)*sin(2*a/b)/b-1/8*c*Si(4*(a+b*arcsin(c*x))/b)*sin(4*a/b)/b+Unintegrable(1/x^2
/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.54 (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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx \]

[In]

Int[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])),x]

[Out]

-((c*Cos[(2*a)/b]*CosIntegral[(2*(a + b*ArcSin[c*x]))/b])/b) - (c*Cos[(4*a)/b]*CosIntegral[(4*(a + b*ArcSin[c*
x]))/b])/(8*b) - (15*c*Log[a + b*ArcSin[c*x]])/(8*b) - (c*Sin[(2*a)/b]*SinIntegral[(2*(a + b*ArcSin[c*x]))/b])
/b - (c*Sin[(4*a)/b]*SinIntegral[(4*(a + b*ArcSin[c*x]))/b])/(8*b) + Defer[Int][1/(x^2*Sqrt[1 - c^2*x^2]*(a +
b*ArcSin[c*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 c^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}+\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}+\frac {3 c^4 x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}-\frac {c^6 x^4}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}\right ) \, dx \\ & = -\left (\left (3 c^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx\right )+\left (3 c^4\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx-c^6 \int \frac {x^4}{\sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {3 c \log (a+b \arcsin (c x))}{b}-\frac {c \text {Subst}\left (\int \frac {\sin ^4\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\frac {(3 c) \text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{b}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {3 c \log (a+b \arcsin (c x))}{b}-\frac {c \text {Subst}\left (\int \left (\frac {3}{8 x}+\frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{8 x}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b}+\frac {(3 c) \text {Subst}\left (\int \left (\frac {1}{2 x}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 x}\right ) \, dx,x,a+b \arcsin (c x)\right )}{b}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {15 c \log (a+b \arcsin (c x))}{8 b}-\frac {c \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}-\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b}+\frac {c \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b}-\frac {(3 c) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {15 c \log (a+b \arcsin (c x))}{8 b}+\frac {\left (c \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b}-\frac {\left (3 c \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b}-\frac {\left (c \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b}+\frac {\left (c \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b}-\frac {\left (3 c \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{2 b}-\frac {\left (c \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 x}{b}\right )}{x} \, dx,x,a+b \arcsin (c x)\right )}{8 b}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ & = -\frac {c \cos \left (\frac {2 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b}-\frac {c \cos \left (\frac {4 a}{b}\right ) \operatorname {CosIntegral}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b}-\frac {15 c \log (a+b \arcsin (c x))}{8 b}-\frac {c \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \arcsin (c x))}{b}\right )}{b}-\frac {c \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \arcsin (c x))}{b}\right )}{8 b}+\int \frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.15 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx \]

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcSin[c*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93

\[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )}d x\]

[In]

int((-c^2*x^2+1)^(5/2)/x^2/(a+b*arcsin(c*x)),x)

[Out]

int((-c^2*x^2+1)^(5/2)/x^2/(a+b*arcsin(c*x)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.75 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(5/2)/x^2/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b*x^2*arcsin(c*x) + a*x^2), x)

Sympy [N/A]

Not integrable

Time = 3.87 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.96 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]

[In]

integrate((-c**2*x**2+1)**(5/2)/x**2/(a+b*asin(c*x)),x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(5/2)/(x**2*(a + b*asin(c*x))), x)

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(5/2)/x^2/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arcsin(c*x) + a)*x^2), x)

Giac [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int { \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}} \,d x } \]

[In]

integrate((-c^2*x^2+1)^(5/2)/x^2/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arcsin(c*x) + a)*x^2), x)

Mupad [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 (a+b \arcsin (c x))} \, dx=\int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \]

[In]

int((1 - c^2*x^2)^(5/2)/(x^2*(a + b*asin(c*x))),x)

[Out]

int((1 - c^2*x^2)^(5/2)/(x^2*(a + b*asin(c*x))), x)