3.338 \(\int \frac {(1-c^2 x^2)^{5/2}}{x^2 (a+b \sin ^{-1}(c x))} \, dx\)

Optimal. Leaf size=161 \[ \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )},x\right )-\frac {c \cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b}-\frac {c \cos \left (\frac {4 a}{b}\right ) \text {Ci}\left (\frac {4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 b}-\frac {c \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{b}-\frac {c \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{8 b}-\frac {15 c \log \left (a+b \sin ^{-1}(c x)\right )}{8 b} \]

[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)

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Rubi [A]  time = 0.93, 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 = {} \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[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 + 2*ArcSin[c*x]])/b) - (c*Cos[(4*a)/b]*CosIntegral[(4*a)/b + 4*ArcSin[c*
x]])/(8*b) - (15*c*Log[a + b*ArcSin[c*x]])/(8*b) - (c*Sin[(2*a)/b]*SinIntegral[(2*a)/b + 2*ArcSin[c*x]])/b - (
c*Sin[(4*a)/b]*SinIntegral[(4*a)/b + 4*ArcSin[c*x]])/(8*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 )^{5/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \left (-\frac {3 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 {3 c^4 x^2}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}-\frac {c^6 x^4}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}\right ) \, dx\\ &=-\left (\left (3 c^2\right ) \int \frac {1}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\right )+\left (3 c^4\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx-c^6 \int \frac {x^4}{\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 {3 c \log \left (a+b \sin ^{-1}(c x)\right )}{b}-c \operatorname {Subst}\left (\int \frac {\sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+(3 c) \operatorname {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 {3 c \log \left (a+b \sin ^{-1}(c x)\right )}{b}-c \operatorname {Subst}\left (\int \left (\frac {3}{8 (a+b x)}-\frac {\cos (2 x)}{2 (a+b x)}+\frac {\cos (4 x)}{8 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )+(3 c) \operatorname {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 {15 c \log \left (a+b \sin ^{-1}(c x)\right )}{8 b}-\frac {1}{8} c \operatorname {Subst}\left (\int \frac {\cos (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{2} c \operatorname {Subst}\left (\int \frac {\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{2} (3 c) \operatorname {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 {15 c \log \left (a+b \sin ^{-1}(c x)\right )}{8 b}+\frac {1}{2} \left (c \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {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 (3 c \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {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}{8} \left (c \cos \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 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 ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{2} \left (3 c \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{8} \left (c \sin \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 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 )}{b}-\frac {c \cos \left (\frac {4 a}{b}\right ) \text {Ci}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b}-\frac {15 c \log \left (a+b \sin ^{-1}(c x)\right )}{8 b}-\frac {c \sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{b}-\frac {c \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 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 = 1.43, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[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]

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fricas [A]  time = 1.15, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-c^{2} x^{2} + 1}}{b x^{2} \arcsin \left (c x\right ) + a x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \arcsin \left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(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)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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