3.1.18 \(\int \frac {(d-c^2 d x^2)^2 (a+b \text {ArcSin}(c x))}{x^4} \, dx\) [18]

Optimal. Leaf size=128 \[ b c^3 d^2 \sqrt {1-c^2 x^2}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 (a+b \text {ArcSin}(c x))}{3 x^3}+\frac {2 c^2 d^2 (a+b \text {ArcSin}(c x))}{x}+c^4 d^2 x (a+b \text {ArcSin}(c x))+\frac {11}{6} b c^3 d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]

[Out]

-1/3*d^2*(a+b*arcsin(c*x))/x^3+2*c^2*d^2*(a+b*arcsin(c*x))/x+c^4*d^2*x*(a+b*arcsin(c*x))+11/6*b*c^3*d^2*arctan
h((-c^2*x^2+1)^(1/2))+b*c^3*d^2*(-c^2*x^2+1)^(1/2)-1/6*b*c*d^2*(-c^2*x^2+1)^(1/2)/x^2

________________________________________________________________________________________

Rubi [A]
time = 0.11, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {276, 4777, 12, 1265, 911, 1171, 396, 214} \begin {gather*} c^4 d^2 x (a+b \text {ArcSin}(c x))+\frac {2 c^2 d^2 (a+b \text {ArcSin}(c x))}{x}-\frac {d^2 (a+b \text {ArcSin}(c x))}{3 x^3}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}+b c^3 d^2 \sqrt {1-c^2 x^2}+\frac {11}{6} b c^3 d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

b*c^3*d^2*Sqrt[1 - c^2*x^2] - (b*c*d^2*Sqrt[1 - c^2*x^2])/(6*x^2) - (d^2*(a + b*ArcSin[c*x]))/(3*x^3) + (2*c^2
*d^2*(a + b*ArcSin[c*x]))/x + c^4*d^2*x*(a + b*ArcSin[c*x]) + (11*b*c^3*d^2*ArcTanh[Sqrt[1 - c^2*x^2]])/6

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 214

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

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 396

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

Rule 911

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - (2*c*d - b*e)*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1171

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> With[{Qx = PolynomialQ
uotient[(a + b*x^2 + c*x^4)^p, d + e*x^2, x], R = Coeff[PolynomialRemainder[(a + b*x^2 + c*x^4)^p, d + e*x^2,
x], x, 0]}, Simp[(-R)*x*((d + e*x^2)^(q + 1)/(2*d*(q + 1))), x] + Dist[1/(2*d*(q + 1)), Int[(d + e*x^2)^(q + 1
)*ExpandToSum[2*d*(q + 1)*Qx + R*(2*q + 3), x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &&
NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 1265

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

Rule 4777

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]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^2 \left (-1+6 c^2 x^2+3 c^4 x^4\right )}{3 x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} \left (b c d^2\right ) \int \frac {-1+6 c^2 x^2+3 c^4 x^4}{x^3 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-1+6 c^2 x+3 c^4 x^2}{x^2 \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {8-12 x^2+3 x^4}{\left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2} \, dx,x,\sqrt {1-c^2 x^2}\right )}{3 c}\\ &=-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} \left (b c d^2\right ) \text {Subst}\left (\int \frac {-17+6 x^2}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )\\ &=b c^3 d^2 \sqrt {1-c^2 x^2}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{6} \left (11 b c d^2\right ) \text {Subst}\left (\int \frac {1}{\frac {1}{c^2}-\frac {x^2}{c^2}} \, dx,x,\sqrt {1-c^2 x^2}\right )\\ &=b c^3 d^2 \sqrt {1-c^2 x^2}-\frac {b c d^2 \sqrt {1-c^2 x^2}}{6 x^2}-\frac {d^2 \left (a+b \sin ^{-1}(c x)\right )}{3 x^3}+\frac {2 c^2 d^2 \left (a+b \sin ^{-1}(c x)\right )}{x}+c^4 d^2 x \left (a+b \sin ^{-1}(c x)\right )+\frac {11}{6} b c^3 d^2 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 136, normalized size = 1.06 \begin {gather*} \frac {d^2 \left (-2 a+12 a c^2 x^2+6 a c^4 x^4-b c x \sqrt {1-c^2 x^2}+6 b c^3 x^3 \sqrt {1-c^2 x^2}+2 b \left (-1+6 c^2 x^2+3 c^4 x^4\right ) \text {ArcSin}(c x)-11 b c^3 x^3 \log (x)+11 b c^3 x^3 \log \left (1+\sqrt {1-c^2 x^2}\right )\right )}{6 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*(-2*a + 12*a*c^2*x^2 + 6*a*c^4*x^4 - b*c*x*Sqrt[1 - c^2*x^2] + 6*b*c^3*x^3*Sqrt[1 - c^2*x^2] + 2*b*(-1 +
6*c^2*x^2 + 3*c^4*x^4)*ArcSin[c*x] - 11*b*c^3*x^3*Log[x] + 11*b*c^3*x^3*Log[1 + Sqrt[1 - c^2*x^2]]))/(6*x^3)

________________________________________________________________________________________

Maple [A]
time = 0.07, size = 115, normalized size = 0.90

method	result	size
derivativedivides	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(115\)
default	\(c^{3} \left (d^{2} a \left (c x -\frac {1}{3 c^{3} x^{3}}+\frac {2}{c x}\right )+d^{2} b \left (c x \arcsin \left (c x \right )-\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}+\frac {2 \arcsin \left (c x \right )}{c x}+\sqrt {-c^{2} x^{2}+1}-\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}+\frac {11 \arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3*(d^2*a*(c*x-1/3/c^3/x^3+2/c/x)+d^2*b*(c*x*arcsin(c*x)-1/3/c^3/x^3*arcsin(c*x)+2/c/x*arcsin(c*x)+(-c^2*x^2+
1)^(1/2)-1/6/c^2/x^2*(-c^2*x^2+1)^(1/2)+11/6*arctanh(1/(-c^2*x^2+1)^(1/2))))

________________________________________________________________________________________

Maxima [A]
time = 0.49, size = 170, normalized size = 1.33 \begin {gather*} a c^{4} d^{2} x + {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b c^{3} d^{2} + 2 \, {\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^{2} - \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^{2} + \frac {2 \, a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 2.98, size = 162, normalized size = 1.27 \begin {gather*} \frac {12 \, a c^{4} d^{2} x^{4} + 11 \, b c^{3} d^{2} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - 11 \, b c^{3} d^{2} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 24 \, a c^{2} d^{2} x^{2} - 4 \, a d^{2} + 4 \, {\left (3 \, b c^{4} d^{2} x^{4} + 6 \, b c^{2} d^{2} x^{2} - b d^{2}\right )} \arcsin \left (c x\right ) + 2 \, {\left (6 \, b c^{3} d^{2} x^{3} - b c d^{2} x\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(12*a*c^4*d^2*x^4 + 11*b*c^3*d^2*x^3*log(sqrt(-c^2*x^2 + 1) + 1) - 11*b*c^3*d^2*x^3*log(sqrt(-c^2*x^2 + 1
) - 1) + 24*a*c^2*d^2*x^2 - 4*a*d^2 + 4*(3*b*c^4*d^2*x^4 + 6*b*c^2*d^2*x^2 - b*d^2)*arcsin(c*x) + 2*(6*b*c^3*d
^2*x^3 - b*c*d^2*x)*sqrt(-c^2*x^2 + 1))/x^3

________________________________________________________________________________________

Sympy [A]
time = 4.38, size = 233, normalized size = 1.82 \begin {gather*} a c^{4} d^{2} x + \frac {2 a c^{2} d^{2}}{x} - \frac {a d^{2}}{3 x^{3}} + b c^{4} d^{2} \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 ) - 2 b c^{3} d^{2} \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 {2 b c^{2} d^{2} \operatorname {asin}{\left (c x \right )}}{x} + \frac {b c d^{2} \left (\begin {cases} - \frac {c^{2} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{2} + \frac {c}{2 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} - \frac {1}{2 c x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \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 \sqrt {1 - \frac {1}{c^{2} x^{2}}}}{2 x} & \text {otherwise} \end {cases}\right )}{3} - \frac {b d^{2} \operatorname {asin}{\left (c x \right )}}{3 x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**4*d**2*x + 2*a*c**2*d**2/x - a*d**2/(3*x**3) + b*c**4*d**2*Piecewise((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-
c**2*x**2 + 1)/c, True)) - 2*b*c**3*d**2*Piecewise((-acosh(1/(c*x)), 1/Abs(c**2*x**2) > 1), (I*asin(1/(c*x)),
True)) + 2*b*c**2*d**2*asin(c*x)/x + b*c*d**2*Piecewise((-c**2*acosh(1/(c*x))/2 + c/(2*x*sqrt(-1 + 1/(c**2*x**
2))) - 1/(2*c*x**3*sqrt(-1 + 1/(c**2*x**2))), 1/Abs(c**2*x**2) > 1), (I*c**2*asin(1/(c*x))/2 - I*c*sqrt(1 - 1/
(c**2*x**2))/(2*x), True))/3 - b*d**2*asin(c*x)/(3*x**3)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1409 vs. \(2 (116) = 232\).
time = 58.26, size = 1409, normalized size = 11.01 \begin {gather*} -\frac {b c^{11} d^{2} x^{8} \arcsin \left (c x\right )}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{8}} - \frac {a c^{11} d^{2} x^{8}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{8}} + \frac {b c^{10} d^{2} x^{7}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{7}} + \frac {5 \, b c^{9} d^{2} x^{6} \arcsin \left (c x\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{6}} + \frac {5 \, a c^{9} d^{2} x^{6}}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{6}} - \frac {11 \, b c^{8} d^{2} x^{5} \log \left ({\left | c \right |} {\left | x \right |}\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {11 \, b c^{8} d^{2} x^{5} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} - \frac {23 \, b c^{8} d^{2} x^{5}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {15 \, b c^{7} d^{2} x^{4} \arcsin \left (c x\right )}{4 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} + \frac {15 \, a c^{7} d^{2} x^{4}}{4 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{4}} - \frac {11 \, b c^{6} d^{2} x^{3} \log \left ({\left | c \right |} {\left | x \right |}\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {11 \, b c^{6} d^{2} x^{3} \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {23 \, b c^{6} d^{2} x^{3}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}} + \frac {5 \, b c^{5} d^{2} x^{2} \arcsin \left (c x\right )}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} + \frac {5 \, a c^{5} d^{2} x^{2}}{6 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{2}} - \frac {b c^{4} d^{2} x}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )} {\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}} - \frac {b c^{3} d^{2} \arcsin \left (c x\right )}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )}} - \frac {a c^{3} d^{2}}{24 \, {\left (\frac {c^{5} x^{5}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{5}} + \frac {c^{3} x^{3}}{{\left (\sqrt {-c^{2} x^{2} + 1} + 1\right )}^{3}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/24*b*c^11*d^2*x^8*arcsin(c*x)/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(s
qrt(-c^2*x^2 + 1) + 1)^8) - 1/24*a*c^11*d^2*x^8/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2
+ 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^8) + 1/24*b*c^10*d^2*x^7/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/
(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^7) + 5/6*b*c^9*d^2*x^6*arcsin(c*x)/((c^5*x^5/(sqrt(-c^2*x
^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) + 5/6*a*c^9*d^2*x^6/((c^5*x^5
/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^6) - 11/6*b*c^8*d^2
*x^5*log(abs(c)*abs(x))/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*
x^2 + 1) + 1)^5) + 11/6*b*c^8*d^2*x^5*log(sqrt(-c^2*x^2 + 1) + 1)/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x
^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) - 23/24*b*c^8*d^2*x^5/((c^5*x^5/(sqrt(-c^2*x^2 + 1)
+ 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^5) + 15/4*b*c^7*d^2*x^4*arcsin(c*x)/((c
^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^4) + 15/4*a*c
^7*d^2*x^4/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)
^4) - 11/6*b*c^6*d^2*x^3*log(abs(c)*abs(x))/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1)
+ 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) + 11/6*b*c^6*d^2*x^3*log(sqrt(-c^2*x^2 + 1) + 1)/((c^5*x^5/(sqrt(-c^2*x^2
+ 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) + 23/24*b*c^6*d^2*x^3/((c^5*x^5
/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1) + 1)^3) + 5/6*b*c^5*d^2*
x^2*arcsin(c*x)/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(-c^2*x^2 + 1)
+ 1)^2) + 5/6*a*c^5*d^2*x^2/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3)*(sqrt(
-c^2*x^2 + 1) + 1)^2) - 1/24*b*c^4*d^2*x/((c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) +
1)^3)*(sqrt(-c^2*x^2 + 1) + 1)) - 1/24*b*c^3*d^2*arcsin(c*x)/(c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sq
rt(-c^2*x^2 + 1) + 1)^3) - 1/24*a*c^3*d^2/(c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + c^3*x^3/(sqrt(-c^2*x^2 + 1) +
1)^3)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________