3.3.63 \(\int \frac {(a+b \text {ArcSin}(c x))^2}{x^4 (d-c^2 d x^2)^{5/2}} \, dx\) [263]

Optimal. Leaf size=538 \[ -\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c (a+b \text {ArcSin}(c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 (a+b \text {ArcSin}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {32 b c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]

[Out]

-1/3*(a+b*arcsin(c*x))^2/d/x^3/(-c^2*d*x^2+d)^(3/2)-2*c^2*(a+b*arcsin(c*x))^2/d/x/(-c^2*d*x^2+d)^(3/2)+8/3*c^4
*x*(a+b*arcsin(c*x))^2/d/(-c^2*d*x^2+d)^(3/2)-1/3*b^2*c^2/d^2/x/(-c^2*d*x^2+d)^(1/2)+2/3*b^2*c^4*x/d^2/(-c^2*d
*x^2+d)^(1/2)+16/3*c^4*x*(a+b*arcsin(c*x))^2/d^2/(-c^2*d*x^2+d)^(1/2)-1/3*b*c*(a+b*arcsin(c*x))/d^2/x^2/(-c^2*
x^2+1)^(1/2)/(-c^2*d*x^2+d)^(1/2)-16/3*I*c^3*(a+b*arcsin(c*x))^2*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-3
2/3*b*c^3*(a+b*arcsin(c*x))*arctanh((I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)+
32/3*b*c^3*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-8/
3*I*b^2*c^3*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)-8/3*I*b^2*c^3
*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))^2)*(-c^2*x^2+1)^(1/2)/d^2/(-c^2*d*x^2+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.72, antiderivative size = 538, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {4789, 4747, 4745, 4765, 3800, 2221, 2317, 2438, 4767, 197, 4793, 4769, 4504, 4268, 277} \begin {gather*} -\frac {b c (a+b \text {ArcSin}(c x))}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {2 c^2 (a+b \text {ArcSin}(c x))^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {(a+b \text {ArcSin}(c x))^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x (a+b \text {ArcSin}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {32 b c^3 \sqrt {1-c^2 x^2} \log \left (1+e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

-1/3*(b^2*c^2)/(d^2*x*Sqrt[d - c^2*d*x^2]) + (2*b^2*c^4*x)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (b*c*(a + b*ArcSin[c*
x]))/(3*d^2*x^2*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2]) - (a + b*ArcSin[c*x])^2/(3*d*x^3*(d - c^2*d*x^2)^(3/2))
- (2*c^2*(a + b*ArcSin[c*x])^2)/(d*x*(d - c^2*d*x^2)^(3/2)) + (8*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d*(d - c^2*d
*x^2)^(3/2)) + (16*c^4*x*(a + b*ArcSin[c*x])^2)/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((16*I)/3)*c^3*Sqrt[1 - c^2*x^2
]*(a + b*ArcSin[c*x])^2)/(d^2*Sqrt[d - c^2*d*x^2]) - (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*ArcTanh[E
^((2*I)*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) + (32*b*c^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])*Log[1 + E
^((2*I)*ArcSin[c*x])])/(3*d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)
*ArcSin[c*x])])/(d^2*Sqrt[d - c^2*d*x^2]) - (((8*I)/3)*b^2*c^3*Sqrt[1 - c^2*x^2]*PolyLog[2, E^((2*I)*ArcSin[c*
x])])/(d^2*Sqrt[d - c^2*d*x^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 277

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 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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 4268

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

Rule 4504

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Dist[
2^n, Int[(c + d*x)^m*Csc[2*a + 2*b*x]^n, x], x] /; FreeQ[{a, b, c, d, m}, x] && IntegerQ[n] && RationalQ[m]

Rule 4745

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

Rule 4747

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

Rule 4765

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Dist[-e^(-1), Subst[In
t[(a + b*x)^n*Tan[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0]

Rule 4767

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

Rule 4769

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Dist[1/d, Subst[Int[(a
+ b*x)^n/(Cos[x]*Sin[x]), x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n
, 0]

Rule 4789

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[c^2*((m + 2*p + 3)/(f^2*(m
+ 1))), Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, x], 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]) /; Free
Q[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && ILtQ[m, -1]

Rule 4793

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/(2*d*f*(p + 1))), x] + (Dist[(m + 2*p + 3)/(2*d*(p
+ 1)), Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Dist[b*c*(n/(2*f*(p + 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]) /; Fre
eQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] &&  !GtQ[m, 1] && (IntegerQ[m] ||
IntegerQ[p] || EqQ[n, 1])

Rubi steps

\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\left (2 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^2 \left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x^3 \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\left (8 c^4\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{x^2 \left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {8 b c^3 \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (16 c^4\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx}{3 d}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{x \left (1-c^2 x^2\right )} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 b c^5 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^4 \sqrt {1-c^2 x^2}\right ) \int \frac {1}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (32 b c^5 \sqrt {1-c^2 x^2}\right ) \int \frac {x \left (a+b \sin ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (32 b c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \tan (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (64 i b c^3 \sqrt {1-c^2 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 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (32 b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 i b^2 c^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}-\frac {2 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {16 i c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {32 b c^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \log \left (1+e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 i b^2 c^3 \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}

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Mathematica [A]
time = 2.49, size = 441, normalized size = 0.82 \begin {gather*} \frac {-\frac {a^2 \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right )}{x^3}-\frac {a b \left (2 \left (1+6 c^2 x^2-24 c^4 x^4+16 c^6 x^6\right ) \text {ArcSin}(c x)+c x \sqrt {1-c^2 x^2} \left (1+16 c^2 x^2 \left (-1+c^2 x^2\right ) \log (c x)+8 c^2 x^2 \left (-1+c^2 x^2\right ) \log \left (1-c^2 x^2\right )\right )\right )}{x^3}+b^2 c^3 \left (1-c^2 x^2\right )^{3/2} \left (\frac {c x}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2}}{c x}-\frac {\text {ArcSin}(c x)}{c^2 x^2}+\frac {\text {ArcSin}(c x)}{-1+c^2 x^2}-16 i \text {ArcSin}(c x)^2+\frac {c x \text {ArcSin}(c x)^2}{\left (1-c^2 x^2\right )^{3/2}}+\frac {8 c x \text {ArcSin}(c x)^2}{\sqrt {1-c^2 x^2}}-\frac {\sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2}{c^3 x^3}-\frac {8 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)^2}{c x}+16 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+16 \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-8 i \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )-8 i \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )}{3 d \left (d-c^2 d x^2\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcSin[c*x])^2/(x^4*(d - c^2*d*x^2)^(5/2)),x]

[Out]

(-((a^2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6))/x^3) - (a*b*(2*(1 + 6*c^2*x^2 - 24*c^4*x^4 + 16*c^6*x^6)*Ar
cSin[c*x] + c*x*Sqrt[1 - c^2*x^2]*(1 + 16*c^2*x^2*(-1 + c^2*x^2)*Log[c*x] + 8*c^2*x^2*(-1 + c^2*x^2)*Log[1 - c
^2*x^2])))/x^3 + b^2*c^3*(1 - c^2*x^2)^(3/2)*((c*x)/Sqrt[1 - c^2*x^2] - Sqrt[1 - c^2*x^2]/(c*x) - ArcSin[c*x]/
(c^2*x^2) + ArcSin[c*x]/(-1 + c^2*x^2) - (16*I)*ArcSin[c*x]^2 + (c*x*ArcSin[c*x]^2)/(1 - c^2*x^2)^(3/2) + (8*c
*x*ArcSin[c*x]^2)/Sqrt[1 - c^2*x^2] - (Sqrt[1 - c^2*x^2]*ArcSin[c*x]^2)/(c^3*x^3) - (8*Sqrt[1 - c^2*x^2]*ArcSi
n[c*x]^2)/(c*x) + 16*ArcSin[c*x]*Log[1 - E^((2*I)*ArcSin[c*x])] + 16*ArcSin[c*x]*Log[1 + E^((2*I)*ArcSin[c*x])
] - (8*I)*PolyLog[2, -E^((2*I)*ArcSin[c*x])] - (8*I)*PolyLog[2, E^((2*I)*ArcSin[c*x])]))/(3*d*(d - c^2*d*x^2)^
(3/2))

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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. 5224 vs. \(2 (517 ) = 1034\).
time = 0.61, size = 5225, normalized size = 9.71

method	result	size
default	\(\text {Expression too large to display}\)	\(5225\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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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((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="maxima")

[Out]

1/3*a*b*c*(8*c^2*log(c*x + 1)/d^(5/2) + 8*c^2*log(c*x - 1)/d^(5/2) + 16*c^2*log(x)/d^(5/2) + 1/(c^2*d^(5/2)*x^
4 - d^(5/2)*x^2)) + 2/3*(16*c^4*x/(sqrt(-c^2*d*x^2 + d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c
^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2)*d*x^3))*a*b*arcsin(c*x) + 1/3*(16*c^4*x/(sqrt(-c^2*d*x^2
+ d)*d^2) + 8*c^4*x/((-c^2*d*x^2 + d)^(3/2)*d) - 6*c^2/((-c^2*d*x^2 + d)^(3/2)*d*x) - 1/((-c^2*d*x^2 + d)^(3/2
)*d*x^3))*a^2 + b^2*integrate(arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2/((c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2
*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)), x)/sqrt(d)

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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((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),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)/(c^6*d^3*x^10 - 3*c^4*d^3*x^8 + 3
*c^2*d^3*x^6 - d^3*x^4), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*asin(c*x))**2/x**4/(-c**2*d*x**2+d)**(5/2),x)

[Out]

Integral((a + b*asin(c*x))**2/(x**4*(-d*(c*x - 1)*(c*x + 1))**(5/2)), x)

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Giac [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((a+b*arcsin(c*x))^2/x^4/(-c^2*d*x^2+d)^(5/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2/((-c^2*d*x^2 + d)^(5/2)*x^4), x)

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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}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^(5/2)),x)

[Out]

int((a + b*asin(c*x))^2/(x^4*(d - c^2*d*x^2)^(5/2)), x)

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