3.4.94 \(\int \frac {(1-c^2 x^2)^{3/2}}{x (a+b \text {ArcSin}(c x))^2} \, dx\) [394]

Optimal. Leaf size=177 \[ -\frac {\left (1-c^2 x^2\right )^2}{b c x (a+b \text {ArcSin}(c x))}-\frac {9 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{4 b^2}-\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{4 b^2}-\frac {9 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{4 b^2}-\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{4 b^2}-\frac {\text {Int}\left (\frac {1-c^2 x^2}{x^2 (a+b \text {ArcSin}(c x))},x\right )}{b c} \]

[Out]

-(-c^2*x^2+1)^2/b/c/x/(a+b*arcsin(c*x))-9/4*Ci((a+b*arcsin(c*x))/b)*cos(a/b)/b^2-3/4*Ci(3*(a+b*arcsin(c*x))/b)
*cos(3*a/b)/b^2-9/4*Si((a+b*arcsin(c*x))/b)*sin(a/b)/b^2-3/4*Si(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b^2-Unintegr
able((-c^2*x^2+1)/x^2/(a+b*arcsin(c*x)),x)/b/c

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Rubi [A]
time = 0.27, 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 = {} \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {ArcSin}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

-((1 - c^2*x^2)^2/(b*c*x*(a + b*ArcSin[c*x]))) - (9*Cos[a/b]*CosIntegral[(a + b*ArcSin[c*x])/b])/(4*b^2) - (3*
Cos[(3*a)/b]*CosIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2) - (9*Sin[a/b]*SinIntegral[(a + b*ArcSin[c*x])/b])
/(4*b^2) - (3*Sin[(3*a)/b]*SinIntegral[(3*(a + b*ArcSin[c*x]))/b])/(4*b^2) - Defer[Int][(1 - c^2*x^2)/(x^2*(a
+ b*ArcSin[c*x])), x]/(b*c)

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x \left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\int \frac {1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac {(3 c) \int \frac {1-c^2 x^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {3 \text {Subst}\left (\int \frac {\cos ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {\int \frac {1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {3 \text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 (a+b x)}+\frac {\cos (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {\int \frac {1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac {9 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac {\int \frac {1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ &=-\frac {\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {\int \frac {1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}-\frac {\left (9 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac {\left (9 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b}\\ &=-\frac {\left (1-c^2 x^2\right )^2}{b c x \left (a+b \sin ^{-1}(c x)\right )}-\frac {9 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b^2}-\frac {3 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2}-\frac {9 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{4 b^2}-\frac {3 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{4 b^2}-\frac {\int \frac {1-c^2 x^2}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx}{b c}\\ \end {align*}

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Mathematica [A]
time = 7.33, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x (a+b \text {ArcSin}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.77, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^4*x^4 - 2*c^2*x^2 - (b^2*c*x*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)) + a*b*c*x)*integrate((3*c^4*x^4 -
2*c^2*x^2 - 1)/(b^2*c*x^2*arctan2(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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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(3/2)/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,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-c^2\,x^2\right )}^{3/2}}{x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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