3.9 \(\int \frac{(d-c^2 d x^2) (a+b \sin ^{-1}(c x))}{x^4} \, dx\)

Optimal. Leaf size=81 \[ \frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}+\frac{5}{6} b c^3 d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]

[Out]

-(b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (d*(a + b*ArcSin[c*x]))/(3*x^3) + (c^2*d*(a + b*ArcSin[c*x]))/x + (5*b*c^
3*d*ArcTanh[Sqrt[1 - c^2*x^2]])/6

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Rubi [A]  time = 0.086358, antiderivative size = 81, 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, 78, 63, 208} \[ \frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}+\frac{5}{6} b c^3 d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Int[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^4,x]

[Out]

-(b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (d*(a + b*ArcSin[c*x]))/(3*x^3) + (c^2*d*(a + b*ArcSin[c*x]))/x + (5*b*c^
3*d*ArcTanh[Sqrt[1 - c^2*x^2]])/6

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 78

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

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^4} \, dx &=-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-(b c) \int \frac{d \left (-1+3 c^2 x^2\right )}{3 x^3 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{3} (b c d) \int \frac{-1+3 c^2 x^2}{x^3 \sqrt{1-c^2 x^2}} \, dx\\ &=-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{6} (b c d) \operatorname{Subst}\left (\int \frac{-1+3 c^2 x}{x^2 \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac{1}{12} \left (5 b c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac{1}{6} (5 b c 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 )\\ &=-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}-\frac{d \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac{c^2 d \left (a+b \sin ^{-1}(c x)\right )}{x}+\frac{5}{6} b c^3 d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0395833, size = 93, normalized size = 1.15 \[ \frac{a c^2 d}{x}-\frac{a d}{3 x^3}-\frac{b c d \sqrt{1-c^2 x^2}}{6 x^2}+\frac{5}{6} b c^3 d \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )+\frac{b c^2 d \sin ^{-1}(c x)}{x}-\frac{b d \sin ^{-1}(c x)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d - c^2*d*x^2)*(a + b*ArcSin[c*x]))/x^4,x]

[Out]

-(a*d)/(3*x^3) + (a*c^2*d)/x - (b*c*d*Sqrt[1 - c^2*x^2])/(6*x^2) - (b*d*ArcSin[c*x])/(3*x^3) + (b*c^2*d*ArcSin
[c*x])/x + (5*b*c^3*d*ArcTanh[Sqrt[1 - c^2*x^2]])/6

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Maple [A]  time = 0.011, size = 91, normalized size = 1.1 \begin{align*}{c}^{3} \left ( -da \left ( -{\frac{1}{cx}}+{\frac{1}{3\,{c}^{3}{x}^{3}}} \right ) -db \left ( -{\frac{\arcsin \left ( cx \right ) }{cx}}+{\frac{\arcsin \left ( cx \right ) }{3\,{c}^{3}{x}^{3}}}+{\frac{1}{6\,{c}^{2}{x}^{2}}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{5}{6}{\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^4,x)

[Out]

c^3*(-d*a*(-1/c/x+1/3/c^3/x^3)-d*b*(-1/c/x*arcsin(c*x)+1/3/c^3/x^3*arcsin(c*x)+1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)-
5/6*arctanh(1/(-c^2*x^2+1)^(1/2))))

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Maxima [A]  time = 1.60712, size = 166, normalized size = 2.05 \begin{align*}{\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 c^{2} d - \frac{1}{6} \,{\left ({\left (c^{2} \log \left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac{2}{{\left | x \right |}}\right ) + \frac{\sqrt{-c^{2} x^{2} + 1}}{x^{2}}\right )} c + \frac{2 \, \arcsin \left (c x\right )}{x^{3}}\right )} b d + \frac{a c^{2} d}{x} - \frac{a d}{3 \, x^{3}} \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^4,x, algorithm="maxima")

[Out]

(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*b*c^2*d - 1/6*((c^2*log(2*sqrt(-c^2*x^2 + 1)/a
bs(x) + 2/abs(x)) + sqrt(-c^2*x^2 + 1)/x^2)*c + 2*arcsin(c*x)/x^3)*b*d + a*c^2*d/x - 1/3*a*d/x^3

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Fricas [A]  time = 2.95843, size = 259, normalized size = 3.2 \begin{align*} \frac{5 \, b c^{3} d x^{3} \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) - 5 \, b c^{3} d x^{3} \log \left (\sqrt{-c^{2} x^{2} + 1} - 1\right ) + 12 \, a c^{2} d x^{2} - 2 \, \sqrt{-c^{2} x^{2} + 1} b c d x - 4 \, a d + 4 \,{\left (3 \, b c^{2} d x^{2} - b d\right )} \arcsin \left (c x\right )}{12 \, x^{3}} \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^4,x, algorithm="fricas")

[Out]

1/12*(5*b*c^3*d*x^3*log(sqrt(-c^2*x^2 + 1) + 1) - 5*b*c^3*d*x^3*log(sqrt(-c^2*x^2 + 1) - 1) + 12*a*c^2*d*x^2 -
2*sqrt(-c^2*x^2 + 1)*b*c*d*x - 4*a*d + 4*(3*b*c^2*d*x^2 - b*d)*arcsin(c*x))/x^3

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Sympy [A]  time = 7.58766, size = 178, normalized size = 2.2 \begin{align*} \frac{a c^{2} d}{x} - \frac{a d}{3 x^{3}} - b c^{3} 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 c^{2} d \operatorname{asin}{\left (c x \right )}}{x} + \frac{b c d \left (\begin{cases} - \frac{c^{2} \operatorname{acosh}{\left (\frac{1}{c x} \right )}}{2} - \frac{c \sqrt{-1 + \frac{1}{c^{2} x^{2}}}}{2 x} & \text{for}\: \frac{1}{\left |{c^{2} x^{2}}\right |} > 1 \\\frac{i c^{2} \operatorname{asin}{\left (\frac{1}{c x} \right )}}{2} - \frac{i c}{2 x \sqrt{1 - \frac{1}{c^{2} x^{2}}}} + \frac{i}{2 c x^{3} \sqrt{1 - \frac{1}{c^{2} x^{2}}}} & \text{otherwise} \end{cases}\right )}{3} - \frac{b d \operatorname{asin}{\left (c x \right )}}{3 x^{3}} \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**4,x)

[Out]

a*c**2*d/x - a*d/(3*x**3) - b*c**3*d*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)), True
)) + b*c**2*d*asin(c*x)/x + b*c*d*Piecewise((-c**2*acosh(1/(c*x))/2 - c*sqrt(-1 + 1/(c**2*x**2))/(2*x), 1/Abs(
c**2*x**2) > 1), (I*c**2*asin(1/(c*x))/2 - I*c/(2*x*sqrt(1 - 1/(c**2*x**2))) + I/(2*c*x**3*sqrt(1 - 1/(c**2*x*
*2))), True))/3 - b*d*asin(c*x)/(3*x**3)

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Giac [B]  time = 22.4812, size = 400, normalized size = 4.94 \begin{align*} -\frac{b c^{6} d x^{3} \arcsin \left (c x\right )}{24 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{3}} - \frac{a c^{6} d x^{3}}{24 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac{b c^{5} d x^{2}}{24 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac{3 \, b c^{4} d x \arcsin \left (c x\right )}{8 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}} + \frac{3 \, a c^{4} d x}{8 \,{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}} - \frac{5}{6} \, b c^{3} d \log \left ({\left | c \right |}{\left | x \right |}\right ) + \frac{5}{6} \, b c^{3} d \log \left (\sqrt{-c^{2} x^{2} + 1} + 1\right ) + \frac{3 \, b c^{2} d{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )} \arcsin \left (c x\right )}{8 \, x} + \frac{3 \, a c^{2} d{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}}{8 \, x} - \frac{b c d{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{2}}{24 \, x^{2}} - \frac{b d{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{3} \arcsin \left (c x\right )}{24 \, x^{3}} - \frac{a d{\left (\sqrt{-c^{2} x^{2} + 1} + 1\right )}^{3}}{24 \, x^{3}} \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^4,x, algorithm="giac")

[Out]

-1/24*b*c^6*d*x^3*arcsin(c*x)/(sqrt(-c^2*x^2 + 1) + 1)^3 - 1/24*a*c^6*d*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + 1/24*
b*c^5*d*x^2/(sqrt(-c^2*x^2 + 1) + 1)^2 + 3/8*b*c^4*d*x*arcsin(c*x)/(sqrt(-c^2*x^2 + 1) + 1) + 3/8*a*c^4*d*x/(s
qrt(-c^2*x^2 + 1) + 1) - 5/6*b*c^3*d*log(abs(c)*abs(x)) + 5/6*b*c^3*d*log(sqrt(-c^2*x^2 + 1) + 1) + 3/8*b*c^2*
d*(sqrt(-c^2*x^2 + 1) + 1)*arcsin(c*x)/x + 3/8*a*c^2*d*(sqrt(-c^2*x^2 + 1) + 1)/x - 1/24*b*c*d*(sqrt(-c^2*x^2
+ 1) + 1)^2/x^2 - 1/24*b*d*(sqrt(-c^2*x^2 + 1) + 1)^3*arcsin(c*x)/x^3 - 1/24*a*d*(sqrt(-c^2*x^2 + 1) + 1)^3/x^
3