3.25 \(\int \frac {(d-c^2 d x^2)^3 (a+b \sin ^{-1}(c x))}{x^2} \, dx\)
Optimal. Leaf size=164 \[ -\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
[Out]
-1/5*b*c*d^3*(-c^2*x^2+1)^(3/2)-1/25*b*c*d^3*(-c^2*x^2+1)^(5/2)-d^3*(a+b*arcsin(c*x))/x-3*c^2*d^3*x*(a+b*arcsi
n(c*x))+c^4*d^3*x^3*(a+b*arcsin(c*x))-1/5*c^6*d^3*x^5*(a+b*arcsin(c*x))-b*c*d^3*arctanh((-c^2*x^2+1)^(1/2))-11
/5*b*c*d^3*(-c^2*x^2+1)^(1/2)
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Rubi [A] time = 0.23, antiderivative size = 164, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used =
{270, 4687, 12, 1799, 1620, 63, 208} \[ -\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
Int[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^2,x]
[Out]
(-11*b*c*d^3*Sqrt[1 - c^2*x^2])/5 - (b*c*d^3*(1 - c^2*x^2)^(3/2))/5 - (b*c*d^3*(1 - c^2*x^2)^(5/2))/25 - (d^3*
(a + b*ArcSin[c*x]))/x - 3*c^2*d^3*x*(a + b*ArcSin[c*x]) + c^4*d^3*x^3*(a + b*ArcSin[c*x]) - (c^6*d^3*x^5*(a +
b*ArcSin[c*x]))/5 - b*c*d^3*ArcTanh[Sqrt[1 - c^2*x^2]]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
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]
Rule 270
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 1620
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]
Rule 1799
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]
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]
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d^3 \left (-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6\right )}{5 x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} \left (b c d^3\right ) \int \frac {-5-15 c^2 x^2+5 c^4 x^4-c^6 x^6}{x \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{10} \left (b c d^3\right ) \operatorname {Subst}\left (\int \frac {-5-15 c^2 x+5 c^4 x^2-c^6 x^3}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{10} \left (b c d^3\right ) \operatorname {Subst}\left (\int \left (-\frac {11 c^2}{\sqrt {1-c^2 x}}-\frac {5}{x \sqrt {1-c^2 x}}-3 c^2 \sqrt {1-c^2 x}-c^2 \left (1-c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )\\ &=-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} \left (b c d^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b d^3\right ) \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}\\ &=-\frac {11}{5} b c d^3 \sqrt {1-c^2 x^2}-\frac {1}{5} b c d^3 \left (1-c^2 x^2\right )^{3/2}-\frac {1}{25} b c d^3 \left (1-c^2 x^2\right )^{5/2}-\frac {d^3 \left (a+b \sin ^{-1}(c x)\right )}{x}-3 c^2 d^3 x \left (a+b \sin ^{-1}(c x)\right )+c^4 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{5} c^6 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-b c d^3 \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )\\ \end {align*}
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Mathematica [A] time = 0.13, size = 166, normalized size = 1.01 \[ -\frac {d^3 \left (5 a c^6 x^6-25 a c^4 x^4+75 a c^2 x^2+25 a+61 b c x \sqrt {1-c^2 x^2}+25 b c x \log \left (\sqrt {1-c^2 x^2}+1\right )+b c^5 x^5 \sqrt {1-c^2 x^2}-7 b c^3 x^3 \sqrt {1-c^2 x^2}+5 b \left (c^6 x^6-5 c^4 x^4+15 c^2 x^2+5\right ) \sin ^{-1}(c x)-25 b c x \log (x)\right )}{25 x} \]
Antiderivative was successfully verified.
[In]
Integrate[((d - c^2*d*x^2)^3*(a + b*ArcSin[c*x]))/x^2,x]
[Out]
-1/25*(d^3*(25*a + 75*a*c^2*x^2 - 25*a*c^4*x^4 + 5*a*c^6*x^6 + 61*b*c*x*Sqrt[1 - c^2*x^2] - 7*b*c^3*x^3*Sqrt[1
- c^2*x^2] + b*c^5*x^5*Sqrt[1 - c^2*x^2] + 5*b*(5 + 15*c^2*x^2 - 5*c^4*x^4 + c^6*x^6)*ArcSin[c*x] - 25*b*c*x*
Log[x] + 25*b*c*x*Log[1 + Sqrt[1 - c^2*x^2]]))/x
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fricas [A] time = 0.57, size = 188, normalized size = 1.15 \[ -\frac {10 \, a c^{6} d^{3} x^{6} - 50 \, a c^{4} d^{3} x^{4} + 150 \, a c^{2} d^{3} x^{2} + 25 \, b c d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} + 1\right ) - 25 \, b c d^{3} x \log \left (\sqrt {-c^{2} x^{2} + 1} - 1\right ) + 50 \, a d^{3} + 10 \, {\left (b c^{6} d^{3} x^{6} - 5 \, b c^{4} d^{3} x^{4} + 15 \, b c^{2} d^{3} x^{2} + 5 \, b d^{3}\right )} \arcsin \left (c x\right ) + 2 \, {\left (b c^{5} d^{3} x^{5} - 7 \, b c^{3} d^{3} x^{3} + 61 \, b c d^{3} x\right )} \sqrt {-c^{2} x^{2} + 1}}{50 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))/x^2,x, algorithm="fricas")
[Out]
-1/50*(10*a*c^6*d^3*x^6 - 50*a*c^4*d^3*x^4 + 150*a*c^2*d^3*x^2 + 25*b*c*d^3*x*log(sqrt(-c^2*x^2 + 1) + 1) - 25
*b*c*d^3*x*log(sqrt(-c^2*x^2 + 1) - 1) + 50*a*d^3 + 10*(b*c^6*d^3*x^6 - 5*b*c^4*d^3*x^4 + 15*b*c^2*d^3*x^2 + 5
*b*d^3)*arcsin(c*x) + 2*(b*c^5*d^3*x^5 - 7*b*c^3*d^3*x^3 + 61*b*c*d^3*x)*sqrt(-c^2*x^2 + 1))/x
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giac [B] time = 48.64, size = 5513, normalized size = 33.62 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))/x^2,x, algorithm="giac")
[Out]
-1/2*b*c^13*d^3*x^12*arcsin(c*x)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^
9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^12) - 1/2*a*c^13*d^3*x^12/((c^11*x^11/(sqr
t(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^
5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^12) + b*c^12*d^3*x^11*log(abs(c)*abs(x))/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^
9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 +
5*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)^11) - b*c^12*d^
3*x^11*log(sqrt(-c^2*x^2 + 1) + 1)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1
)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^11) + 61/25*b*c^12*d^3*x^11/((c^11*x^11/
(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 1
0*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))*(s
qrt(-c^2*x^2 + 1) + 1)^11) - 9*b*c^11*d^3*x^10*arcsin(c*x)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9
/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 +
5*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)^10) - 9*a*c^11*d
^3*x^10/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2
*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^10) + 5*b*c^10*d^3*x^9*log(abs(c)*abs(x))/((c^11*x^11/(sqrt(-c^2*x^
2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sq
rt(-c^2*x^2 + 1) + 1)^5 + 5*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)^9) - 5*b*c^10*d^3*x^9*log(sqrt(-c^2*x^2 + 1) + 1)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^
9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 +
5*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)^9) + 31/5*b*c^1
0*d^3*x^9/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c
^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^9) - 47/2*b*c^9*d^3*x^8*arcsin(c*x)/((c^11*x^11/(sqrt(-c^2*x^2 +
1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-
c^2*x^2 + 1) + 1)^5 + 5*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)^8) - 47/2*a*c^9*d^3*x^8/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 +
10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^8) + 10*b*c^8*d^3*x^7*log(abs(c)*abs(x))/((c^
11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) +
1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^7) - 10*b*c^8*d^3*x^7*log(sqrt(-c^2*x^2 + 1) + 1)/((c^11*x^11/(sqrt(-c^2*x^2 +
1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-
c^2*x^2 + 1) + 1)^5 + 5*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)^7) + 22/5*b*c^8*d^3*x^7/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 +
10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^7) - 182/5*b*c^7*d^3*x^6*arcsin(c*x)/((c^11*x
^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7
+ 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^6) - 182/5*a*c^7*d^3*x^6/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(
-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^6) + 10*b*c^6*d^3*x^5*l
og(abs(c)*abs(x))/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/
(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^5) - 10*b*c^6*d^3*x^5*log(sqrt(-c^2*x^2 + 1) + 1)/((c^11*
x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^
7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^5) - 22/5*b*c^6*d^3*x^5/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(
-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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)^5) - 47/2*b*c^5*d^3*x^4
*arcsin(c*x)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt
(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sq
rt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)^4) - 47/2*a*c^5*d^3*x^4/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^1
1 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 +
1) + 1)^5 + 5*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)
+ 5*b*c^4*d^3*x^3*log(abs(c)*abs(x))/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) +
1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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) - 5*b*c^4*d^3*x^3*log(sqrt(-c^2*x^2
+ 1) + 1)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-
c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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) - 31/5*b*c^4*d^3*x^3/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11
+ 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1
) + 1)^5 + 5*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) -
9*b*c^3*d^3*x^2*arcsin(c*x)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 1
0*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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) - 9*a*c^3*d^3*x^2/((c^11*x^11/(sqrt(-c^2*x^2
+ 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqr
t(-c^2*x^2 + 1) + 1)^5 + 5*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^3*x*log(abs(c)*abs(x))/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^
2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^3*x^3/(sqr
t(-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^3*x*log(sqrt(-c^2*x
^2 + 1) + 1)/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt
(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^3*x^3/(sqrt(-c^2*x^2 + 1) + 1)^3 + c*x/(sq
rt(-c^2*x^2 + 1) + 1))*(sqrt(-c^2*x^2 + 1) + 1)) - 61/25*b*c^2*d^3*x/((c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 +
5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1)
+ 1)^5 + 5*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^3*arcsin(c*x)/(c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2 + 1) + 1)^9 + 10*c^7*x^
7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*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^3/(c^11*x^11/(sqrt(-c^2*x^2 + 1) + 1)^11 + 5*c^9*x^9/(sqrt(-c^2*x^2
+ 1) + 1)^9 + 10*c^7*x^7/(sqrt(-c^2*x^2 + 1) + 1)^7 + 10*c^5*x^5/(sqrt(-c^2*x^2 + 1) + 1)^5 + 5*c^3*x^3/(sqrt(
-c^2*x^2 + 1) + 1)^3 + c*x/(sqrt(-c^2*x^2 + 1) + 1))
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maple [A] time = 0.01, size = 155, normalized size = 0.95 \[ c \left (-d^{3} a \left (\frac {c^{5} x^{5}}{5}-c^{3} x^{3}+3 c x +\frac {1}{c x}\right )-d^{3} b \left (\frac {\arcsin \left (c x \right ) c^{5} x^{5}}{5}-c^{3} x^{3} \arcsin \left (c x \right )+3 c x \arcsin \left (c x \right )+\frac {\arcsin \left (c x \right )}{c x}+\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{25}-\frac {7 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{25}+\frac {61 \sqrt {-c^{2} x^{2}+1}}{25}+\arctanh \left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )\right )\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))/x^2,x)
[Out]
c*(-d^3*a*(1/5*c^5*x^5-c^3*x^3+3*c*x+1/c/x)-d^3*b*(1/5*arcsin(c*x)*c^5*x^5-c^3*x^3*arcsin(c*x)+3*c*x*arcsin(c*
x)+1/c/x*arcsin(c*x)+1/25*c^4*x^4*(-c^2*x^2+1)^(1/2)-7/25*c^2*x^2*(-c^2*x^2+1)^(1/2)+61/25*(-c^2*x^2+1)^(1/2)+
arctanh(1/(-c^2*x^2+1)^(1/2))))
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maxima [A] time = 0.48, size = 250, normalized size = 1.52 \[ -\frac {1}{5} \, a c^{6} d^{3} x^{5} - \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b c^{6} d^{3} + a c^{4} d^{3} x^{3} + \frac {1}{3} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b c^{4} d^{3} - 3 \, a c^{2} d^{3} x - 3 \, {\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b c d^{3} - {\left (c \log \left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{{\left | x \right |}} + \frac {2}{{\left | x \right |}}\right ) + \frac {\arcsin \left (c x\right )}{x}\right )} b d^{3} - \frac {a d^{3}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c^2*d*x^2+d)^3*(a+b*arcsin(c*x))/x^2,x, algorithm="maxima")
[Out]
-1/5*a*c^6*d^3*x^5 - 1/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 +
8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*c^6*d^3 + a*c^4*d^3*x^3 + 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c
^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*c^4*d^3 - 3*a*c^2*d^3*x - 3*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*c*d^3 -
(c*log(2*sqrt(-c^2*x^2 + 1)/abs(x) + 2/abs(x)) + arcsin(c*x)/x)*b*d^3 - a*d^3/x
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^3)/x^2,x)
[Out]
int(((a + b*asin(c*x))*(d - c^2*d*x^2)^3)/x^2, x)
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sympy [A] time = 8.12, size = 287, normalized size = 1.75 \[ - \frac {a c^{6} d^{3} x^{5}}{5} + a c^{4} d^{3} x^{3} - 3 a c^{2} d^{3} x - \frac {a d^{3}}{x} + \frac {b c^{7} d^{3} \left (\begin {cases} - \frac {x^{4} \sqrt {- c^{2} x^{2} + 1}}{5 c^{2}} - \frac {4 x^{2} \sqrt {- c^{2} x^{2} + 1}}{15 c^{4}} - \frac {8 \sqrt {- c^{2} x^{2} + 1}}{15 c^{6}} & \text {for}\: c \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right )}{5} - \frac {b c^{6} d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} - b c^{5} d^{3} \left (\begin {cases} - \frac {x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c^{2}} - \frac {2 \sqrt {- c^{2} x^{2} + 1}}{3 c^{4}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4} & \text {otherwise} \end {cases}\right ) + b c^{4} d^{3} x^{3} \operatorname {asin}{\left (c x \right )} - 3 b c^{2} d^{3} \left (\begin {cases} 0 & \text {for}\: c = 0 \\x \operatorname {asin}{\left (c x \right )} + \frac {\sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right ) + b c d^{3} \left (\begin {cases} - \operatorname {acosh}{\left (\frac {1}{c x} \right )} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\i \operatorname {asin}{\left (\frac {1}{c x} \right )} & \text {otherwise} \end {cases}\right ) - \frac {b d^{3} \operatorname {asin}{\left (c x \right )}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((-c**2*d*x**2+d)**3*(a+b*asin(c*x))/x**2,x)
[Out]
-a*c**6*d**3*x**5/5 + a*c**4*d**3*x**3 - 3*a*c**2*d**3*x - a*d**3/x + b*c**7*d**3*Piecewise((-x**4*sqrt(-c**2*
x**2 + 1)/(5*c**2) - 4*x**2*sqrt(-c**2*x**2 + 1)/(15*c**4) - 8*sqrt(-c**2*x**2 + 1)/(15*c**6), Ne(c, 0)), (x**
6/6, True))/5 - b*c**6*d**3*x**5*asin(c*x)/5 - b*c**5*d**3*Piecewise((-x**2*sqrt(-c**2*x**2 + 1)/(3*c**2) - 2*
sqrt(-c**2*x**2 + 1)/(3*c**4), Ne(c, 0)), (x**4/4, True)) + b*c**4*d**3*x**3*asin(c*x) - 3*b*c**2*d**3*Piecewi
se((0, Eq(c, 0)), (x*asin(c*x) + sqrt(-c**2*x**2 + 1)/c, True)) + b*c*d**3*Piecewise((-acosh(1/(c*x)), 1/Abs(c
**2*x**2) > 1), (I*asin(1/(c*x)), True)) - b*d**3*asin(c*x)/x
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