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

Optimal. Leaf size=137 \[ -\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}-\frac{b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{4 \left (1-c^2\right ) x^2}-\frac{b c d^2 \tanh ^{-1}\left (\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right )}{4 \left (1-c^2\right )^{3/2}} \]

[Out]

-(b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(4*(1 - c^2)*x^2) - (a + b*ArcSin[c + d*x^2])/(4*x^4) - (b*c*d^2*Ar
cTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(4*(1 - c^2)^(3/2))

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Rubi [A]  time = 0.134773, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {4842, 12, 1114, 730, 724, 206} \[ -\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}-\frac{b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{4 \left (1-c^2\right ) x^2}-\frac{b c d^2 \tanh ^{-1}\left (\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right )}{4 \left (1-c^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c + d*x^2])/x^5,x]

[Out]

-(b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(4*(1 - c^2)*x^2) - (a + b*ArcSin[c + d*x^2])/(4*x^4) - (b*c*d^2*Ar
cTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(4*(1 - c^2)^(3/2))

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 1114

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*(a +
b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[(m - 1)/2]

Rule 730

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m + 1)
*(a + b*x + c*x^2)^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Dist[(2*c*d - b*e)/(2*(c*d^2 - b*d*e + a*e
^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c,
0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && EqQ[m + 2*p + 3, 0]

Rule 724

Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[-2, Subst[Int[1/(4*c*d
^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, (2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a,
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[2*c*d - b*e, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{a+b \sin ^{-1}\left (c+d x^2\right )}{x^5} \, dx &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}+\frac{1}{4} b \int \frac{2 d}{x^3 \sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}+\frac{1}{2} (b d) \int \frac{1}{x^3 \sqrt{1-c^2-2 c d x^2-d^2 x^4}} \, dx\\ &=-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}+\frac{1}{4} (b d) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )\\ &=-\frac{b d \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}+\frac{\left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2-2 c d x-d^2 x^2}} \, dx,x,x^2\right )}{4 \left (1-c^2\right )}\\ &=-\frac{b d \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}-\frac{\left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{1}{4 \left (1-c^2\right )-x^2} \, dx,x,\frac{2 \left (1-c^2-c d x^2\right )}{\sqrt{1-c^2-2 c d x^2-d^2 x^4}}\right )}{2 \left (1-c^2\right )}\\ &=-\frac{b d \sqrt{1-c^2-2 c d x^2-d^2 x^4}}{4 \left (1-c^2\right ) x^2}-\frac{a+b \sin ^{-1}\left (c+d x^2\right )}{4 x^4}-\frac{b c d^2 \tanh ^{-1}\left (\frac{1-c^2-c d x^2}{\sqrt{1-c^2} \sqrt{1-c^2-2 c d x^2-d^2 x^4}}\right )}{4 \left (1-c^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.289083, size = 150, normalized size = 1.09 \[ -\frac{a}{4 x^4}+\frac{b d \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}{4 \left (c^2-1\right ) x^2}+\frac{b c d^2 \tanh ^{-1}\left (\frac{-c^2-c d x^2+1}{\sqrt{1-c^2} \sqrt{-c^2-2 c d x^2-d^2 x^4+1}}\right )}{4 (c-1) (c+1) \sqrt{1-c^2}}-\frac{b \sin ^{-1}\left (c+d x^2\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c + d*x^2])/x^5,x]

[Out]

-a/(4*x^4) + (b*d*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])/(4*(-1 + c^2)*x^2) - (b*ArcSin[c + d*x^2])/(4*x^4) + (b
*c*d^2*ArcTanh[(1 - c^2 - c*d*x^2)/(Sqrt[1 - c^2]*Sqrt[1 - c^2 - 2*c*d*x^2 - d^2*x^4])])/(4*(-1 + c)*(1 + c)*S
qrt[1 - c^2])

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Maple [A]  time = 0.015, size = 132, normalized size = 1. \begin{align*} -{\frac{a}{4\,{x}^{4}}}-{\frac{b\arcsin \left ( d{x}^{2}+c \right ) }{4\,{x}^{4}}}-{\frac{bd}{ \left ( -4\,{c}^{2}+4 \right ){x}^{2}}\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1}}-{\frac{b{d}^{2}c}{4}\ln \left ({\frac{1}{{x}^{2}} \left ( -2\,{c}^{2}+2-2\,cd{x}^{2}+2\,\sqrt{-{c}^{2}+1}\sqrt{-{d}^{2}{x}^{4}-2\,cd{x}^{2}-{c}^{2}+1} \right ) } \right ) \left ( -{c}^{2}+1 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(d*x^2+c))/x^5,x)

[Out]

-1/4*a/x^4-1/4*b/x^4*arcsin(d*x^2+c)-1/4*b*d*(-d^2*x^4-2*c*d*x^2-c^2+1)^(1/2)/(-c^2+1)/x^2-1/4*b*d^2*c/(-c^2+1
)^(3/2)*ln((-2*c^2+2-2*c*d*x^2+2*(-c^2+1)^(1/2)*(-d^2*x^4-2*c*d*x^2-c^2+1)^(1/2))/x^2)

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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((a+b*arcsin(d*x^2+c))/x^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 3.51549, size = 884, normalized size = 6.45 \begin{align*} \left [-\frac{\sqrt{-c^{2} + 1} b c d^{2} x^{4} \log \left (\frac{{\left (2 \, c^{2} - 1\right )} d^{2} x^{4} + 2 \, c^{4} + 4 \,{\left (c^{3} - c\right )} d x^{2} - 2 \, \sqrt{-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}{\left (c d x^{2} + c^{2} - 1\right )} \sqrt{-c^{2} + 1} - 4 \, c^{2} + 2}{x^{4}}\right ) + 2 \, a c^{4} - 2 \, \sqrt{-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}{\left (b c^{2} - b\right )} d x^{2} - 4 \, a c^{2} + 2 \,{\left (b c^{4} - 2 \, b c^{2} + b\right )} \arcsin \left (d x^{2} + c\right ) + 2 \, a}{8 \,{\left (c^{4} - 2 \, c^{2} + 1\right )} x^{4}}, -\frac{\sqrt{c^{2} - 1} b c d^{2} x^{4} \arctan \left (\frac{\sqrt{-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}{\left (c d x^{2} + c^{2} - 1\right )} \sqrt{c^{2} - 1}}{{\left (c^{2} - 1\right )} d^{2} x^{4} + c^{4} + 2 \,{\left (c^{3} - c\right )} d x^{2} - 2 \, c^{2} + 1}\right ) + a c^{4} - \sqrt{-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1}{\left (b c^{2} - b\right )} d x^{2} - 2 \, a c^{2} +{\left (b c^{4} - 2 \, b c^{2} + b\right )} \arcsin \left (d x^{2} + c\right ) + a}{4 \,{\left (c^{4} - 2 \, c^{2} + 1\right )} x^{4}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x^2+c))/x^5,x, algorithm="fricas")

[Out]

[-1/8*(sqrt(-c^2 + 1)*b*c*d^2*x^4*log(((2*c^2 - 1)*d^2*x^4 + 2*c^4 + 4*(c^3 - c)*d*x^2 - 2*sqrt(-d^2*x^4 - 2*c
*d*x^2 - c^2 + 1)*(c*d*x^2 + c^2 - 1)*sqrt(-c^2 + 1) - 4*c^2 + 2)/x^4) + 2*a*c^4 - 2*sqrt(-d^2*x^4 - 2*c*d*x^2
- c^2 + 1)*(b*c^2 - b)*d*x^2 - 4*a*c^2 + 2*(b*c^4 - 2*b*c^2 + b)*arcsin(d*x^2 + c) + 2*a)/((c^4 - 2*c^2 + 1)*
x^4), -1/4*(sqrt(c^2 - 1)*b*c*d^2*x^4*arctan(sqrt(-d^2*x^4 - 2*c*d*x^2 - c^2 + 1)*(c*d*x^2 + c^2 - 1)*sqrt(c^2
- 1)/((c^2 - 1)*d^2*x^4 + c^4 + 2*(c^3 - c)*d*x^2 - 2*c^2 + 1)) + a*c^4 - sqrt(-d^2*x^4 - 2*c*d*x^2 - c^2 + 1
)*(b*c^2 - b)*d*x^2 - 2*a*c^2 + (b*c^4 - 2*b*c^2 + b)*arcsin(d*x^2 + c) + a)/((c^4 - 2*c^2 + 1)*x^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{a + b \operatorname{asin}{\left (c + d x^{2} \right )}}{x^{5}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(d*x**2+c))/x**5,x)

[Out]

Integral((a + b*asin(c + d*x**2))/x**5, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \arcsin \left (d x^{2} + c\right ) + a}{x^{5}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arcsin(d*x^2+c))/x^5,x, algorithm="giac")

[Out]

integrate((b*arcsin(d*x^2 + c) + a)/x^5, x)