3.3.17 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{x^4} \, dx\) [217]

Optimal. Leaf size=314 \[ -\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \text {ArcSin}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 x^2}+\frac {i c^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 \sqrt {1-c^2 x^2}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{3 d x^3}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x)) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {i b^2 c^3 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {1-c^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 314, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.310, Rules used = {4771, 4775, 283, 222, 4721, 3798, 2221, 2317, 2438} \begin {gather*} -\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))}{3 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {ArcSin}(c x))^2}{3 d x^3}+\frac {i c^3 \sqrt {d-c^2 d x^2} (a+b \text {ArcSin}(c x))^2}{3 \sqrt {1-c^2 x^2}}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} \log \left (1-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 \sqrt {1-c^2 x^2}}+\frac {i b^2 c^3 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b^2 c^3 \text {ArcSin}(c x) \sqrt {d-c^2 d x^2}}{3 \sqrt {1-c^2 x^2}}-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/3*(b^2*c^2*Sqrt[d - c^2*d*x^2])/x - (b^2*c^3*Sqrt[d - c^2*d*x^2]*ArcSin[c*x])/(3*Sqrt[1 - c^2*x^2]) - (b*c*
Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*x^2) + ((I/3)*c^3*Sqrt[d - c^2*d*x^2]*(a + b*Arc
Sin[c*x])^2)/Sqrt[1 - c^2*x^2] - ((d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x])^2)/(3*d*x^3) - (2*b*c^3*Sqrt[d - c
^2*d*x^2]*(a + b*ArcSin[c*x])*Log[1 - E^((2*I)*ArcSin[c*x])])/(3*Sqrt[1 - c^2*x^2]) + ((I/3)*b^2*c^3*Sqrt[d -
c^2*d*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*x])])/Sqrt[1 - c^2*x^2]

Rule 222

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

Rule 283

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^p/(c*(m + 1
))), x] - Dist[b*n*(p/(c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2317

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 4721

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n*Cot[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4771

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

Rule 4775

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

Rubi steps

\begin {align*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{x^4} \, dx &=-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac {\left (2 b c \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{x^3} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}+\frac {\left (b^2 c^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {1-c^2 x^2}}{x^2} \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (2 b c^3 \sqrt {d-c^2 d x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}-\frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3}-\frac {\left (2 b c^3 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \cot (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^4 \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {i c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \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 )^2}{3 d x^3}+\frac {\left (4 i b c^3 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)}{1-e^{2 i x}} \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {i c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \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 )^2}{3 d x^3}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {\left (2 b^2 c^3 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {i c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \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 )^2}{3 d x^3}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}-\frac {\left (i b^2 c^3 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {b^2 c^2 \sqrt {d-c^2 d x^2}}{3 x}-\frac {b^2 c^3 \sqrt {d-c^2 d x^2} \sin ^{-1}(c x)}{3 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 x^2}+\frac {i c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 \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 )^2}{3 d x^3}-\frac {2 b c^3 \sqrt {d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1-e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}+\frac {i b^2 c^3 \sqrt {d-c^2 d x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.84, size = 248, normalized size = 0.79 \begin {gather*} \frac {\sqrt {d-c^2 d x^2} \left (2 b^2 \left (i c^3 x^3-\sqrt {1-c^2 x^2}+c^2 x^2 \sqrt {1-c^2 x^2}\right ) \text {ArcSin}(c x)^2-b \text {ArcSin}(c x) \left (2 b c x+3 a \sqrt {1-c^2 x^2}+a \cos (3 \text {ArcSin}(c x))+4 b c^3 x^3 \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )\right )-2 \left (a b c x+b^2 c^2 x^2 \sqrt {1-c^2 x^2}+a^2 \left (1-c^2 x^2\right )^{3/2}+2 a b c^3 x^3 \log (c x)\right )+2 i b^2 c^3 x^3 \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )}{6 x^3 \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[d - c^2*d*x^2]*(2*b^2*(I*c^3*x^3 - Sqrt[1 - c^2*x^2] + c^2*x^2*Sqrt[1 - c^2*x^2])*ArcSin[c*x]^2 - b*ArcS
in[c*x]*(2*b*c*x + 3*a*Sqrt[1 - c^2*x^2] + a*Cos[3*ArcSin[c*x]] + 4*b*c^3*x^3*Log[1 - E^((2*I)*ArcSin[c*x])])
- 2*(a*b*c*x + b^2*c^2*x^2*Sqrt[1 - c^2*x^2] + a^2*(1 - c^2*x^2)^(3/2) + 2*a*b*c^3*x^3*Log[c*x]) + (2*I)*b^2*c
^3*x^3*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(6*x^3*Sqrt[1 - c^2*x^2])

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3016 vs. \(2 (294 ) = 588\).
time = 0.45, size = 3017, normalized size = 9.61

method	result	size
default	\(\text {Expression too large to display}\)	\(3017\)

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))^2/x^4,x,method=_RETURNVERBOSE)

[Out]

2*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^7-2*I*
a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^5-b^2*(-d*
(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+b^2*(-d*(c^2*x^2-1))
^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)^2*c^8-3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2
*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)^2*c^6+10/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1
)*arcsin(c*x)^2*c^4-5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x/(c^2*x^2-1)*arcsin(c*x)^2*c^2+2*b
^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1+I*c*x+(-c^2*x^2+1)^(1/2))+2*b^
2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3/(3*c^2*x^2-3)*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))+1/3*b
^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6-1/3*I*b^2*(-d*(c^2*x^2-1))^
(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3-2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(
1/2)*c^3/(3*c^2*x^2-3)*arcsin(c*x)^2-2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3/(3*c^2*x^2-3)*polyl
og(2,-I*c*x-(-c^2*x^2+1)^(1/2))-2*I*b^2*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*c^3/(3*c^2*x^2-3)*polylog(2,
I*c*x+(-c^2*x^2+1)^(1/2))-1/3*a^2/d/x^3*(-c^2*d*x^2+d)^(3/2)-2/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x
^2+1)*x^5/(c^2*x^2-1)*c^8+5/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*c^6-4/3*b^2*(
-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*c^4+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^
2*x^2+1)/x/(c^2*x^2-1)*c^2+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^3/(c^2*x^2-1)*arcsin(c*x)^
2-1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*(-c^2*x^2+1)*c^6+2/3*I*a*b*(-d*(c^2
*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c^3+1/3*I*a*b*(-d*(c^2*x^2-1
))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*(-c^2*x^2+1)*c^4-a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^
2+1)/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^3+2/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^3/(c^2*x^2-1)
*arcsin(c*x)+2/3*a*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^3+20/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4-1/3*I*a*b*(-d*(c^2*x
^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*c^4-10/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)
/x/(c^2*x^2-1)*arcsin(c*x)*c^2+1/3*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^2/(c^2*x^2-1)*(-c^2*x^
2+1)^(1/2)*c+2/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*c^6+2*a*b*(-d*(c^2*x^2-1
))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8-1/3*I*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3
*c^2*x^2+1)*x^5/(c^2*x^2-1)*c^8-6*a*b*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*
x)*c^6-4*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*arcsin(c*x)*c^3/(3*c^2*x^2-3)+a*b*(-d*(c^2*x^2-1))^(1
/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4
-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^6-I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1
)*x^2/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^5+1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1
)*x/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)*c^4+I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c^2*x^2
-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^7+1/3*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)/x^2/(c^2*x^2-1
)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c+2/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^3/(c^2*x^2-1)*ar
csin(c*x)*c^6+b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*arcsin(c*x)*(-c^2*x^2+1)^(1/2
)*c^5+I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^2/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^5-1/3*I*b^2*(-
d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x/(c^2*x^2-1)*arcsin(c*x)*c^4+1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3
*c^4*x^4-3*c^2*x^2+1)/(c^2*x^2-1)*arcsin(c*x)^2*(-c^2*x^2+1)^(1/2)*c^3-1/3*I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4
*x^4-3*c^2*x^2+1)*x^5/(c^2*x^2-1)*arcsin(c*x)*c^8-I*b^2*(-d*(c^2*x^2-1))^(1/2)/(3*c^4*x^4-3*c^2*x^2+1)*x^4/(c^
2*x^2-1)*(-c^2*x^2+1)^(1/2)*c^7

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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))^2/x^4,x, algorithm="maxima")

[Out]

1/3*((-1)^(-2*c^2*d*x^2 + 2*d)*c^2*d^(3/2)*log(-2*c^2*d + 2*d/x^2) + c^2*d^(3/2)*log(x^2 - 1/c^2) - sqrt(c^4*d
*x^4 - 2*c^2*d*x^2 + d)*d/x^2)*a*b*c/d - 2/3*(-c^2*d*x^2 + d)^(3/2)*a*b*arcsin(c*x)/(d*x^3) + 1/3*((c^2*x^2 -
1)*sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(d)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 - sqrt(d)*x^3*integrate(2
*(c^3*x^2 - c)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))/x^3, x))*b^2/x^3 - 1/3*(-c^2*d*x^2 + d)^(3/2)*a^2/(d
*x^3)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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))^2/x^4,x, algorithm="fricas")

[Out]

integral(sqrt(-c^2*d*x^2 + d)*(b^2*arcsin(c*x)^2 + 2*a*b*arcsin(c*x) + a^2)/x^4, x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4}}\, dx \end {gather*}

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))**2/x**4,x)

[Out]

Integral(sqrt(-d*(c*x - 1)*(c*x + 1))*(a + b*asin(c*x))**2/x**4, x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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))^2/x^4,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d-c^2\,d\,x^2}}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(1/2))/x^4,x)

[Out]

int(((a + b*asin(c*x))^2*(d - c^2*d*x^2)^(1/2))/x^4, x)

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