3.435 \(\int \frac {d-c^2 d x^2}{x (a+b \sin ^{-1}(c x))^{3/2}} \, dx\)

Optimal. Leaf size=171 \[ -\frac {2 d \text {Int}\left (\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}},x\right )}{b c}-\frac {2 \sqrt {\pi } d \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 \sqrt {\pi } d \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}} \]

[Out]

-2*d*cos(2*a/b)*FresnelC(2*(a+b*arcsin(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(3/2)-2*d*FresnelS(2*(a+b*arcs
in(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(3/2)-2*d*(-c^2*x^2+1)^(3/2)/b/c/x/(a+b*arcsin(c*x))^(1
/2)-2*d*Unintegrable(1/x^2/(-c^2*x^2+1)^(1/2)/(a+b*arcsin(c*x))^(1/2),x)/b/c

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Rubi [A]  time = 0.78, 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 = {} \[ \int \frac {d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

(-2*d*(1 - c^2*x^2)^(3/2))/(b*c*x*Sqrt[a + b*ArcSin[c*x]]) - (2*d*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b
*ArcSin[c*x]])/(Sqrt[b]*Sqrt[Pi])])/b^(3/2) - (2*d*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c*x]])/(Sqrt[b]*Sqrt
[Pi])]*Sin[(2*a)/b])/b^(3/2) - (2*d*Defer[Int][1/(x^2*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcSin[c*x]]), x])/(b*c)

Rubi steps

\begin {align*} \int \frac {d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \int \frac {\sqrt {1-c^2 x^2}}{x^2 \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {(4 c d) \int \frac {\sqrt {1-c^2 x^2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(4 d) \operatorname {Subst}\left (\int \frac {\cos ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \left (-\frac {c^2}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}+\frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}}\right ) \, dx}{b c}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(4 d) \operatorname {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}+\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}+\frac {(2 c d) \int \frac {1}{\sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \operatorname {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {\left (2 d \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}-\frac {\left (2 d \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}-\frac {\left (4 d \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2}-\frac {\left (4 d \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{b^2}\\ &=-\frac {2 d \left (1-c^2 x^2\right )^{3/2}}{b c x \sqrt {a+b \sin ^{-1}(c x)}}-\frac {2 d \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2}}-\frac {2 d \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2}}-\frac {(2 d) \int \frac {1}{x^2 \sqrt {1-c^2 x^2} \sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b c}\\ \end {align*}

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Mathematica [A]  time = 2.02, size = 0, normalized size = 0.00 \[ \int \frac {d-c^2 d x^2}{x \left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [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*d*x^2+d)/x/(a+b*arcsin(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {-c^{2} d \,x^{2}+d}{x \left (a +b \arcsin \left (c x \right )\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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