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

Optimal. Leaf size=69 \[ c^2 (-d) x \left (a+b \sin ^{-1}(c x)\right )-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}-b c d \sqrt{1-c^2 x^2}-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0756378, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {14, 4687, 12, 446, 80, 63, 208} \[ c^2 (-d) x \left (a+b \sin ^{-1}(c x)\right )-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{x}-b c d \sqrt{1-c^2 x^2}-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]]

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 80

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 208

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

Rubi steps

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

Mathematica [A]  time = 0.0355648, size = 78, normalized size = 1.13 \[ -a c^2 d x-\frac{a d}{x}-b c d \sqrt{1-c^2 x^2}-b c d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )-b c^2 d x \sin ^{-1}(c x)-\frac{b d \sin ^{-1}(c x)}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 67, normalized size = 1. \begin{align*} c \left ( -da \left ( cx+{\frac{1}{cx}} \right ) -db \left ( cx\arcsin \left ( cx \right ) +{\frac{\arcsin \left ( cx \right ) }{cx}}+\sqrt{-{c}^{2}{x}^{2}+1}+{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) \right ) \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c*(-d*a*(c*x+1/c/x)-d*b*(c*x*arcsin(c*x)+1/c/x*arcsin(c*x)+(-c^2*x^2+1)^(1/2)+arctanh(1/(-c^2*x^2+1)^(1/2))))

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Maxima [A]  time = 1.56601, size = 111, normalized size = 1.61 \begin{align*} -a c^{2} d x -{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b c d -{\left (c \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\arcsin \left (c x\right )}{x}\right )} b d - \frac{a d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*c^2*d*x - (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*c*d - (c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + a
rcsin(c*x)/x)*b*d - a*d/x

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Fricas [A]  time = 2.56593, size = 236, normalized size = 3.42 \begin{align*} -\frac{2 \, a c^{2} d x^{2} + b c d x \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) - b c d x \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) + 2 \, \sqrt{-c^{2} x^{2} + 1} b c d x + 2 \, a d + 2 \,{\left (b c^{2} d x^{2} + b d\right )} \arcsin \left (c x\right )}{2 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(2*a*c^2*d*x^2 + b*c*d*x*log(sqrt(-c^2*x^2 + 1) + 1) - b*c*d*x*log(sqrt(-c^2*x^2 + 1) - 1) + 2*sqrt(-c^2*
x^2 + 1)*b*c*d*x + 2*a*d + 2*(b*c^2*d*x^2 + b*d)*arcsin(c*x))/x

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Sympy [A]  time = 5.35423, size = 82, normalized size = 1.19 \begin{align*} - a c^{2} d x - \frac{a d}{x} - b c^{2} d \left (\begin{cases} 0 & \text{for}\: c = 0 \\x \operatorname{asin}{\left (c x \right )} + \frac{\sqrt{- c^{2} x^{2} + 1}}{c} & \text{otherwise} \end{cases}\right ) + b c d \left (\begin{cases} - \operatorname{acosh}{\left (\frac{1}{c x} \right )} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname{asin}{\left (\frac{1}{c x} \right )} & \text{otherwise} \end{cases}\right ) - \frac{b d \operatorname{asin}{\left (c x \right )}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*c**2*d*x - a*d/x - b*c**2*d*Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-c**2*x**2 + 1)/c, True)) + b*c*d*
Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b*d*asin(c*x)/x

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Giac [B]  time = 6.15113, size = 1156, normalized size = 16.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*b*c^5*d*x^4*arcsin(c*x)/((c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x
^2 + 1) + 1)^4) - 1/2*a*c^5*d*x^4/((c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-
c^2*x^2 + 1) + 1)^4) + b*c^4*d*x^3*log(abs(c)*abs(x))/((c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^
2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) - b*c^4*d*x^3*log(sqrt(-c^2*x^2 + 1) + 1)/((c^3*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) + b*c^4*d*x^3/((c^3*x^3/(sqrt(-c^2*x^2 +
1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^3) - 3*b*c^3*d*x^2*arcsin(c*x)/((c^3*x^3/(
sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) - 3*a*c^3*d*x^2/((c^3*x^
3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^2) + b*c^2*d*x*log(abs(c
)*abs(x))/((c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)) - b*c
^2*d*x*log(sqrt(-c^2*x^2 + 1) + 1)/((c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(
-c^2*x^2 + 1) + 1)) - b*c^2*d*x/((c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(sqrt(-c^
2*x^2 + 1) + 1)) - 1/2*b*c*d*arcsin(c*x)/(c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1)) -
1/2*a*c*d/(c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))