3.385 \(\int \frac {\sqrt {1-c^2 x^2}}{x (a+b \sin ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=105 \[ -\frac {\text {Int}\left (\frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )},x\right )}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2}-\frac {1-c^2 x^2}{b c x \left (a+b \sin ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((1 - c^2*x^2)/(b*c*x*(a + b*ArcSin[c*x]))) - (Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/b^2 - (Sin[a/b]*S
inIntegral[(a + b*ArcSin[c*x])/b])/b^2 - Defer[Int][1/(x^2*(a + b*ArcSin[c*x])), x]/(b*c)

Rubi steps

\begin {align*} \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {1-c^2 x^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac {c \int \frac {1}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {1-c^2 x^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b^2}-\frac {\int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {1-c^2 x^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac {\cos \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b^2}-\frac {\sin \left (\frac {a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \sin ^{-1}(c x)\right )}{b^2}\\ &=-\frac {1-c^2 x^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2}-\frac {\sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{b^2}-\frac {\int \frac {1}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}

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Mathematica [A]  time = 11.11, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1-c^2 x^2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-c^{2} x^{2} + 1}}{b^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b x \arcsin \left (c x\right ) + a^{2} x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.84, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {-c^{2} x^{2}+1}}{x \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {c^{2} x^{2} - \frac {{\left (b^{2} c x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x\right )} {\left (c^{2} \int \frac {x^{2}}{b x^{2} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a x^{2}}\,{d x} + \int \frac {1}{{\left (b \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a\right )} x^{2}}\,{d x}\right )}}{b c} - 1}{b^{2} c x \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) + a b c x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*x^2 - (b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x)*integrate((c^2*x^2 + 1)/(b^2*c*x^2*a
rctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x^2), x) - 1)/(b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x
+ 1)) + a*b*c*x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {1-c^2\,x^2}}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {- \left (c x - 1\right ) \left (c x + 1\right )}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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