3.4.30 \(\int \frac {(1-c^2 x^2)^{3/2}}{x^2 (a+b \text {ArcSin}(c x))} \, dx\) [330]

Optimal. Leaf size=107 \[ -\frac {c \cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{2 b}-\frac {3 c \log (a+b \text {ArcSin}(c x))}{2 b}-\frac {c \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{2 b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.37, 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^2 (a+b \text {ArcSin}(c x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.85, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{3/2}}{x^2 (a+b \text {ArcSin}(c x))} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.38, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

[Out]

integral((-c^2*x^2 + 1)^(3/2)/(b*x^2*arcsin(c*x) + a*x^2), 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^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [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^2/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

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

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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^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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