3.351 \(\int \frac{a+b \sin ^{-1}(c x^2)}{x^{13}} \, dx\)

Optimal. Leaf size=91 \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}-\frac{2 b c^5 \sqrt{1-c^2 x^4}}{45 x^2}-\frac{b c^3 \sqrt{1-c^2 x^4}}{45 x^6}-\frac{b c \sqrt{1-c^2 x^4}}{60 x^{10}} \]

[Out]

-(b*c*Sqrt[1 - c^2*x^4])/(60*x^10) - (b*c^3*Sqrt[1 - c^2*x^4])/(45*x^6) - (2*b*c^5*Sqrt[1 - c^2*x^4])/(45*x^2)
- (a + b*ArcSin[c*x^2])/(12*x^12)

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Rubi [A]  time = 0.0464586, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {4842, 12, 271, 264} \[ -\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}-\frac{2 b c^5 \sqrt{1-c^2 x^4}}{45 x^2}-\frac{b c^3 \sqrt{1-c^2 x^4}}{45 x^6}-\frac{b c \sqrt{1-c^2 x^4}}{60 x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x^2])/x^13,x]

[Out]

-(b*c*Sqrt[1 - c^2*x^4])/(60*x^10) - (b*c^3*Sqrt[1 - c^2*x^4])/(45*x^6) - (2*b*c^5*Sqrt[1 - c^2*x^4])/(45*x^2)
- (a + b*ArcSin[c*x^2])/(12*x^12)

Rule 4842

Int[((a_.) + ArcSin[u_]*(b_.))*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^(m + 1)*(a + b*ArcSin[
u]))/(d*(m + 1)), x] - Dist[b/(d*(m + 1)), Int[SimplifyIntegrand[((c + d*x)^(m + 1)*D[u, x])/Sqrt[1 - u^2], x]
, x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1] && InverseFunctionFreeQ[u, x] &&  !FunctionOfQ[(c + d*x)^(
m + 1), u, x] &&  !FunctionOfExponentialQ[u, x]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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]

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}\left (c x^2\right )}{x^{13}} \, dx &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac{1}{12} b \int \frac{2 c}{x^{11} \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac{1}{6} (b c) \int \frac{1}{x^{11} \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{b c \sqrt{1-c^2 x^4}}{60 x^{10}}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac{1}{15} \left (2 b c^3\right ) \int \frac{1}{x^7 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{b c \sqrt{1-c^2 x^4}}{60 x^{10}}-\frac{b c^3 \sqrt{1-c^2 x^4}}{45 x^6}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}+\frac{1}{45} \left (4 b c^5\right ) \int \frac{1}{x^3 \sqrt{1-c^2 x^4}} \, dx\\ &=-\frac{b c \sqrt{1-c^2 x^4}}{60 x^{10}}-\frac{b c^3 \sqrt{1-c^2 x^4}}{45 x^6}-\frac{2 b c^5 \sqrt{1-c^2 x^4}}{45 x^2}-\frac{a+b \sin ^{-1}\left (c x^2\right )}{12 x^{12}}\\ \end{align*}

Mathematica [A]  time = 0.0468917, size = 68, normalized size = 0.75 \[ \frac{1}{2} b \left (-\frac{c \sqrt{1-c^2 x^4} \left (8 c^4 x^8+4 c^2 x^4+3\right )}{90 x^{10}}-\frac{\sin ^{-1}\left (c x^2\right )}{6 x^{12}}\right )-\frac{a}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x^2])/x^13,x]

[Out]

-a/(12*x^12) + (b*(-(c*Sqrt[1 - c^2*x^4]*(3 + 4*c^2*x^4 + 8*c^4*x^8))/(90*x^10) - ArcSin[c*x^2]/(6*x^12)))/2

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Maple [A]  time = 0.012, size = 72, normalized size = 0.8 \begin{align*} -{\frac{a}{12\,{x}^{12}}}+b \left ( -{\frac{\arcsin \left ( c{x}^{2} \right ) }{12\,{x}^{12}}}+{\frac{c \left ( c{x}^{2}-1 \right ) \left ( c{x}^{2}+1 \right ) \left ( 8\,{c}^{4}{x}^{8}+4\,{c}^{2}{x}^{4}+3 \right ) }{180\,{x}^{10}}{\frac{1}{\sqrt{-{c}^{2}{x}^{4}+1}}}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x^2))/x^13,x)

[Out]

-1/12*a/x^12+b*(-1/12/x^12*arcsin(c*x^2)+1/180*c*(c*x^2-1)*(c*x^2+1)*(8*c^4*x^8+4*c^2*x^4+3)/x^10/(-c^2*x^4+1)
^(1/2))

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Maxima [A]  time = 1.42715, size = 111, normalized size = 1.22 \begin{align*} -\frac{1}{180} \,{\left ({\left (\frac{15 \, \sqrt{-c^{2} x^{4} + 1} c^{4}}{x^{2}} + \frac{10 \,{\left (-c^{2} x^{4} + 1\right )}^{\frac{3}{2}} c^{2}}{x^{6}} + \frac{3 \,{\left (-c^{2} x^{4} + 1\right )}^{\frac{5}{2}}}{x^{10}}\right )} c + \frac{15 \, \arcsin \left (c x^{2}\right )}{x^{12}}\right )} b - \frac{a}{12 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^13,x, algorithm="maxima")

[Out]

-1/180*((15*sqrt(-c^2*x^4 + 1)*c^4/x^2 + 10*(-c^2*x^4 + 1)^(3/2)*c^2/x^6 + 3*(-c^2*x^4 + 1)^(5/2)/x^10)*c + 15
*arcsin(c*x^2)/x^12)*b - 1/12*a/x^12

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Fricas [A]  time = 3.69557, size = 154, normalized size = 1.69 \begin{align*} \frac{15 \, a x^{12} - 15 \, b \arcsin \left (c x^{2}\right ) -{\left (8 \, b c^{5} x^{10} + 4 \, b c^{3} x^{6} + 3 \, b c x^{2}\right )} \sqrt{-c^{2} x^{4} + 1} - 15 \, a}{180 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^13,x, algorithm="fricas")

[Out]

1/180*(15*a*x^12 - 15*b*arcsin(c*x^2) - (8*b*c^5*x^10 + 4*b*c^3*x^6 + 3*b*c*x^2)*sqrt(-c^2*x^4 + 1) - 15*a)/x^
12

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Sympy [A]  time = 56.1371, size = 170, normalized size = 1.87 \begin{align*} - \frac{a}{12 x^{12}} + \frac{b c \left (\begin{cases} - \frac{4 c^{5} \sqrt{-1 + \frac{1}{c^{2} x^{4}}}}{15} - \frac{2 c^{3} \sqrt{-1 + \frac{1}{c^{2} x^{4}}}}{15 x^{4}} - \frac{c \sqrt{-1 + \frac{1}{c^{2} x^{4}}}}{10 x^{8}} & \text{for}\: \frac{1}{\left |{c^{2} x^{4}}\right |} > 1 \\- \frac{4 i c^{5} \sqrt{1 - \frac{1}{c^{2} x^{4}}}}{15} - \frac{2 i c^{3} \sqrt{1 - \frac{1}{c^{2} x^{4}}}}{15 x^{4}} - \frac{i c \sqrt{1 - \frac{1}{c^{2} x^{4}}}}{10 x^{8}} & \text{otherwise} \end{cases}\right )}{6} - \frac{b \operatorname{asin}{\left (c x^{2} \right )}}{12 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x**2))/x**13,x)

[Out]

-a/(12*x**12) + b*c*Piecewise((-4*c**5*sqrt(-1 + 1/(c**2*x**4))/15 - 2*c**3*sqrt(-1 + 1/(c**2*x**4))/(15*x**4)
- c*sqrt(-1 + 1/(c**2*x**4))/(10*x**8), 1/Abs(c**2*x**4) > 1), (-4*I*c**5*sqrt(1 - 1/(c**2*x**4))/15 - 2*I*c*
*3*sqrt(1 - 1/(c**2*x**4))/(15*x**4) - I*c*sqrt(1 - 1/(c**2*x**4))/(10*x**8), True))/6 - b*asin(c*x**2)/(12*x*
*12)

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Giac [B]  time = 1.25528, size = 680, normalized size = 7.47 \begin{align*} -\frac{\frac{15 \, b c^{13} x^{12} \arcsin \left (c x^{2}\right )}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{6}} + \frac{15 \, a c^{13} x^{12}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{6}} - \frac{6 \, b c^{12} x^{10}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{5}} + \frac{90 \, b c^{11} x^{8} \arcsin \left (c x^{2}\right )}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4}} + \frac{90 \, a c^{11} x^{8}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4}} - \frac{50 \, b c^{10} x^{6}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{3}} + \frac{225 \, b c^{9} x^{4} \arcsin \left (c x^{2}\right )}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}} + \frac{225 \, a c^{9} x^{4}}{{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}} - \frac{300 \, b c^{8} x^{2}}{\sqrt{-c^{2} x^{4} + 1} + 1} + 300 \, b c^{7} \arcsin \left (c x^{2}\right ) + 300 \, a c^{7} + \frac{300 \, b c^{6}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}}{x^{2}} + \frac{225 \, b c^{5}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2} \arcsin \left (c x^{2}\right )}{x^{4}} + \frac{225 \, a c^{5}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{2}}{x^{4}} + \frac{50 \, b c^{4}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{3}}{x^{6}} + \frac{90 \, b c^{3}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4} \arcsin \left (c x^{2}\right )}{x^{8}} + \frac{90 \, a c^{3}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{4}}{x^{8}} + \frac{6 \, b c^{2}{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{5}}{x^{10}} + \frac{15 \, b c{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{6} \arcsin \left (c x^{2}\right )}{x^{12}} + \frac{15 \, a c{\left (\sqrt{-c^{2} x^{4} + 1} + 1\right )}^{6}}{x^{12}}}{11520 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(c*x^2))/x^13,x, algorithm="giac")

[Out]

-1/11520*(15*b*c^13*x^12*arcsin(c*x^2)/(sqrt(-c^2*x^4 + 1) + 1)^6 + 15*a*c^13*x^12/(sqrt(-c^2*x^4 + 1) + 1)^6
- 6*b*c^12*x^10/(sqrt(-c^2*x^4 + 1) + 1)^5 + 90*b*c^11*x^8*arcsin(c*x^2)/(sqrt(-c^2*x^4 + 1) + 1)^4 + 90*a*c^1
1*x^8/(sqrt(-c^2*x^4 + 1) + 1)^4 - 50*b*c^10*x^6/(sqrt(-c^2*x^4 + 1) + 1)^3 + 225*b*c^9*x^4*arcsin(c*x^2)/(sqr
t(-c^2*x^4 + 1) + 1)^2 + 225*a*c^9*x^4/(sqrt(-c^2*x^4 + 1) + 1)^2 - 300*b*c^8*x^2/(sqrt(-c^2*x^4 + 1) + 1) + 3
00*b*c^7*arcsin(c*x^2) + 300*a*c^7 + 300*b*c^6*(sqrt(-c^2*x^4 + 1) + 1)/x^2 + 225*b*c^5*(sqrt(-c^2*x^4 + 1) +
1)^2*arcsin(c*x^2)/x^4 + 225*a*c^5*(sqrt(-c^2*x^4 + 1) + 1)^2/x^4 + 50*b*c^4*(sqrt(-c^2*x^4 + 1) + 1)^3/x^6 +
90*b*c^3*(sqrt(-c^2*x^4 + 1) + 1)^4*arcsin(c*x^2)/x^8 + 90*a*c^3*(sqrt(-c^2*x^4 + 1) + 1)^4/x^8 + 6*b*c^2*(sqr
t(-c^2*x^4 + 1) + 1)^5/x^10 + 15*b*c*(sqrt(-c^2*x^4 + 1) + 1)^6*arcsin(c*x^2)/x^12 + 15*a*c*(sqrt(-c^2*x^4 + 1
) + 1)^6/x^12)/c