3.60 \(\int \frac{\sqrt{d-c^2 d x^2} (a+b \sin ^{-1}(c x))}{x^6} \, dx\)
Optimal. Leaf size=187 \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 d x^3}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}} \]
[Out]
-(b*c*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[1 - c^2*x^2]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[1 - c^2*x^2])
- ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5*d*x^5) - (2*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(
15*d*x^3) - (2*b*c^5*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[1 - c^2*x^2])
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Rubi [A] time = 0.134158, antiderivative size = 187, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used =
{271, 264, 4691, 12, 14} \[ -\frac{2 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 d x^3}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{1-c^2 x^2}}-\frac{b c \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{1-c^2 x^2}}-\frac{2 b c^5 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Int[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^6,x]
[Out]
-(b*c*Sqrt[d - c^2*d*x^2])/(20*x^4*Sqrt[1 - c^2*x^2]) + (b*c^3*Sqrt[d - c^2*d*x^2])/(30*x^2*Sqrt[1 - c^2*x^2])
- ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(5*d*x^5) - (2*c^2*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(
15*d*x^3) - (2*b*c^5*Sqrt[d - c^2*d*x^2]*Log[x])/(15*Sqrt[1 - c^2*x^2])
Rule 271
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x^(m + 1)*(a + b*x^n)^(p + 1))/(a*(m + 1)), x]
- Dist[(b*(m + n*(p + 1) + 1))/(a*(m + 1)), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Rule 264
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Rule 4691
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(x_)^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = IntHide[x^
m*(1 - c^2*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], Int[x^m*(d + e*x^2)^p, x], x] - Dist[(b*c*d^(p - 1/2)*Sqrt[d +
e*x^2])/Sqrt[1 - c^2*x^2], Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x
] && EqQ[c^2*d + e, 0] && IGtQ[p + 1/2, 0] && (IGtQ[(m + 1)/2, 0] || ILtQ[(m + 2*p + 3)/2, 0])
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 14
Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&& !LinearQ[u, x] && !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]
Rubi steps
\begin{align*} \int \frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{-3+c^2 x^2+2 c^4 x^4}{15 x^5} \, dx}{\sqrt{1-c^2 x^2}}+\left (a+b \sin ^{-1}(c x)\right ) \int \frac{\sqrt{d-c^2 d x^2}}{x^6} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \frac{-3+c^2 x^2+2 c^4 x^4}{x^5} \, dx}{15 \sqrt{1-c^2 x^2}}+\frac{1}{5} \left (2 c^2 \left (a+b \sin ^{-1}(c x)\right )\right ) \int \frac{\sqrt{d-c^2 d x^2}}{x^4} \, dx\\ &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 d x^3}-\frac{\left (b c \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{3}{x^5}+\frac{c^2}{x^3}+\frac{2 c^4}{x}\right ) \, dx}{15 \sqrt{1-c^2 x^2}}\\ &=-\frac{b c \sqrt{d-c^2 d x^2}}{20 x^4 \sqrt{1-c^2 x^2}}+\frac{b c^3 \sqrt{d-c^2 d x^2}}{30 x^2 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{5 d x^5}-\frac{2 c^2 \left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{15 d x^3}-\frac{2 b c^5 \sqrt{d-c^2 d x^2} \log (x)}{15 \sqrt{1-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.137784, size = 162, normalized size = 0.87 \[ \frac{\sqrt{d-c^2 d x^2} \left (12 a \left (2 c^2 x^2+3\right ) \left (c^2 x^2-1\right )^2+b c x \sqrt{1-c^2 x^2} \left (-50 c^4 x^4-6 c^2 x^2+9\right )+12 b \left (2 c^2 x^2+3\right ) \left (c^2 x^2-1\right )^2 \sin ^{-1}(c x)\right )}{180 x^5 \left (c^2 x^2-1\right )}-\frac{2 b c^5 \log (x) \sqrt{d-c^2 d x^2}}{15 \sqrt{1-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/x^6,x]
[Out]
(Sqrt[d - c^2*d*x^2]*(12*a*(-1 + c^2*x^2)^2*(3 + 2*c^2*x^2) + b*c*x*Sqrt[1 - c^2*x^2]*(9 - 6*c^2*x^2 - 50*c^4*
x^4) + 12*b*(-1 + c^2*x^2)^2*(3 + 2*c^2*x^2)*ArcSin[c*x]))/(180*x^5*(-1 + c^2*x^2)) - (2*b*c^5*Sqrt[d - c^2*d*
x^2]*Log[x])/(15*Sqrt[1 - c^2*x^2])
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Maple [C] time = 0.302, size = 1902, normalized size = 10.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^6,x)
[Out]
-2/15*a*c^2/d/x^3*(-c^2*d*x^2+d)^(3/2)+2/15*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^7
/(c^2*x^2-1)*(-c^2*x^2+1)*c^12+3/10*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x/(c^2*x^2-
1)*(-c^2*x^2+1)*c^6-2/15*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^5/(c^2*x^2-1)*(-c^2*
x^2+1)*c^10-3/10*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c
^8+6/5*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/
2)*c^5+9/20*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c-
4*I*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)*arcsin(c*x)*c^5/(15*c^2*x^2-15)+2/15*I*b*(-d*(c^2*x^2-1))^(1/2
)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^9/(c^2*x^2-1)*c^14-4/15*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x
^4-15*c^2*x^2+9)*x^7/(c^2*x^2-1)*c^12-1/6*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^5/(
c^2*x^2-1)*c^10+3/5*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^3/(c^2*x^2-1)*c^8-3/10*I*
b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x/(c^2*x^2-1)*c^6+2*b*(-d*(c^2*x^2-1))^(1/2)/(15*
c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^7/(c^2*x^2-1)*arcsin(c*x)*c^12-5/3*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^
4*x^4-15*c^2*x^2+9)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^10-1/2*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2
*x^2+9)*x^4/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^9-17/3*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+
9)*x^3/(c^2*x^2-1)*arcsin(c*x)*c^8+11/12*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^2/(c^2
*x^2-1)*c^7*(-c^2*x^2+1)^(1/2)+98/15*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x/(c^2*x^2-1
)*arcsin(c*x)*c^6+12/5*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x/(c^2*x^2-1)*arcsin(c*x)*
c^4-21/20*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^2/(c^2*x^2-1)*c^3*(-c^2*x^2+1)^(1/2)-
27/5*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^3/(c^2*x^2-1)*arcsin(c*x)*c^2+9/5*b*(-d*(c
^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)/x^5/(c^2*x^2-1)*arcsin(c*x)+2/15*b*(-d*(c^2*x^2-1))^(1/2)
*(-c^2*x^2+1)^(1/2)/(c^2*x^2-1)*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2-1)*c^5+1/4*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^
6-5*c^4*x^4-15*c^2*x^2+9)/(c^2*x^2-1)*c^5*(-c^2*x^2+1)^(1/2)-1/5*a/d/x^5*(-c^2*d*x^2+d)^(3/2)+2*I*b*(-d*(c^2*x
^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^6/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^11-2/3*I*b*(
-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^9-2
*I*b*(-d*(c^2*x^2-1))^(1/2)/(15*c^6*x^6-5*c^4*x^4-15*c^2*x^2+9)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
*c^7
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^6,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 2.71307, size = 1056, normalized size = 5.65 \begin{align*} \left [\frac{4 \,{\left (b c^{7} x^{7} - b c^{5} x^{5}\right )} \sqrt{d} \log \left (\frac{c^{2} d x^{6} + c^{2} d x^{2} - d x^{4} + \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{4} - 1\right )} \sqrt{d} - d}{c^{2} x^{4} - x^{2}}\right ) -{\left (2 \, b c^{3} x^{3} -{\left (2 \, b c^{3} - 3 \, b c\right )} x^{5} - 3 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 4 \,{\left (2 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} +{\left (2 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 3 \, b\right )} \arcsin \left (c x\right ) + 3 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{60 \,{\left (c^{2} x^{7} - x^{5}\right )}}, -\frac{8 \,{\left (b c^{7} x^{7} - b c^{5} x^{5}\right )} \sqrt{-d} \arctan \left (\frac{\sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1}{\left (x^{2} + 1\right )} \sqrt{-d}}{c^{2} d x^{4} -{\left (c^{2} + 1\right )} d x^{2} + d}\right ) +{\left (2 \, b c^{3} x^{3} -{\left (2 \, b c^{3} - 3 \, b c\right )} x^{5} - 3 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} - 4 \,{\left (2 \, a c^{6} x^{6} - a c^{4} x^{4} - 4 \, a c^{2} x^{2} +{\left (2 \, b c^{6} x^{6} - b c^{4} x^{4} - 4 \, b c^{2} x^{2} + 3 \, b\right )} \arcsin \left (c x\right ) + 3 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{60 \,{\left (c^{2} x^{7} - x^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^6,x, algorithm="fricas")
[Out]
[1/60*(4*(b*c^7*x^7 - b*c^5*x^5)*sqrt(d)*log((c^2*d*x^6 + c^2*d*x^2 - d*x^4 + sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x
^2 + 1)*(x^4 - 1)*sqrt(d) - d)/(c^2*x^4 - x^2)) - (2*b*c^3*x^3 - (2*b*c^3 - 3*b*c)*x^5 - 3*b*c*x)*sqrt(-c^2*d*
x^2 + d)*sqrt(-c^2*x^2 + 1) + 4*(2*a*c^6*x^6 - a*c^4*x^4 - 4*a*c^2*x^2 + (2*b*c^6*x^6 - b*c^4*x^4 - 4*b*c^2*x^
2 + 3*b)*arcsin(c*x) + 3*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^7 - x^5), -1/60*(8*(b*c^7*x^7 - b*c^5*x^5)*sqrt(-d)*a
rctan(sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1)*(x^2 + 1)*sqrt(-d)/(c^2*d*x^4 - (c^2 + 1)*d*x^2 + d)) + (2*b*c^3
*x^3 - (2*b*c^3 - 3*b*c)*x^5 - 3*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) - 4*(2*a*c^6*x^6 - a*c^4*x^4 -
4*a*c^2*x^2 + (2*b*c^6*x^6 - b*c^4*x^4 - 4*b*c^2*x^2 + 3*b)*arcsin(c*x) + 3*a)*sqrt(-c^2*d*x^2 + d))/(c^2*x^7
- x^5)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{x^{6}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c**2*d*x**2+d)**(1/2)*(a+b*asin(c*x))/x**6,x)
[Out]
Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))/x**6, x)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-c^{2} d x^{2} + d}{\left (b \arcsin \left (c x\right ) + a\right )}}{x^{6}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))/x^6,x, algorithm="giac")
[Out]
integrate(sqrt(-c^2*d*x^2 + d)*(b*arcsin(c*x) + a)/x^6, x)