3.2.64 \(\int \frac {(d-c^2 d x^2) (a+b \arcsin (c x))^2}{x^4} \, dx\) [164]

3.2.64.1 Optimal result

Integrand size = 25, antiderivative size = 176 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=-\frac {b^2 c^2 d}{3 x}-\frac {b c d \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 x^2}+\frac {2 c^2 d (a+b \arcsin (c x))^2}{3 x}-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}+\frac {10}{3} b c^3 d (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )-\frac {5}{3} i b^2 c^3 d \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+\frac {5}{3} i b^2 c^3 d \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right ) \]

output

-1/3*b^2*c^2*d/x+2/3*c^2*d*(a+b*arcsin(c*x))^2/x-1/3*d*(-c^2*x^2+1)*(a+b*a 
rcsin(c*x))^2/x^3+10/3*b*c^3*d*(a+b*arcsin(c*x))*arctanh(I*c*x+(-c^2*x^2+1 
)^(1/2))-5/3*I*b^2*c^3*d*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+5/3*I*b^2*c^ 
3*d*polylog(2,I*c*x+(-c^2*x^2+1)^(1/2))-1/3*b*c*d*(a+b*arcsin(c*x))*(-c^2* 
x^2+1)^(1/2)/x^2
 

3.2.64.2 Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.51 \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=\frac {d \left (-a^2+3 a^2 c^2 x^2-b^2 c^2 x^2-a b c x \sqrt {1-c^2 x^2}-2 a b \arcsin (c x)+6 a b c^2 x^2 \arcsin (c x)-b^2 c x \sqrt {1-c^2 x^2} \arcsin (c x)-b^2 \arcsin (c x)^2+3 b^2 c^2 x^2 \arcsin (c x)^2+5 a b c^3 x^3 \text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-5 b^2 c^3 x^3 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+5 b^2 c^3 x^3 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )-5 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+5 i b^2 c^3 x^3 \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )}{3 x^3} \]

input

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

output

(d*(-a^2 + 3*a^2*c^2*x^2 - b^2*c^2*x^2 - a*b*c*x*Sqrt[1 - c^2*x^2] - 2*a*b 
*ArcSin[c*x] + 6*a*b*c^2*x^2*ArcSin[c*x] - b^2*c*x*Sqrt[1 - c^2*x^2]*ArcSi 
n[c*x] - b^2*ArcSin[c*x]^2 + 3*b^2*c^2*x^2*ArcSin[c*x]^2 + 5*a*b*c^3*x^3*A 
rcTanh[Sqrt[1 - c^2*x^2]] - 5*b^2*c^3*x^3*ArcSin[c*x]*Log[1 - E^(I*ArcSin[ 
c*x])] + 5*b^2*c^3*x^3*ArcSin[c*x]*Log[1 + E^(I*ArcSin[c*x])] - (5*I)*b^2* 
c^3*x^3*PolyLog[2, -E^(I*ArcSin[c*x])] + (5*I)*b^2*c^3*x^3*PolyLog[2, E^(I 
*ArcSin[c*x])]))/(3*x^3)
 

3.2.64.3 Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.28, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {5200, 5138, 5196, 15, 5218, 3042, 4671, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx\)

\(\Big \downarrow \) 5200

\(\displaystyle -\frac {2}{3} c^2 d \int \frac {(a+b \arcsin (c x))^2}{x^2}dx+\frac {2}{3} b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^3}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 5138

\(\displaystyle -\frac {2}{3} c^2 d \left (2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d \int \frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{x^3}dx-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 5196

\(\displaystyle -\frac {2}{3} c^2 d \left (2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d \left (-\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx+\frac {1}{2} b c \int \frac {1}{x^2}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 15

\(\displaystyle -\frac {2}{3} c^2 d \left (2 b c \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {(a+b \arcsin (c x))^2}{x}\right )+\frac {2}{3} b c d \left (-\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}}dx-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 5218

\(\displaystyle \frac {2}{3} b c d \left (-\frac {1}{2} c^2 \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )-\frac {2}{3} c^2 d \left (2 b c \int \frac {a+b \arcsin (c x)}{c x}d\arcsin (c x)-\frac {(a+b \arcsin (c x))^2}{x}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {2}{3} b c d \left (-\frac {1}{2} c^2 \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )-\frac {2}{3} c^2 d \left (2 b c \int (a+b \arcsin (c x)) \csc (\arcsin (c x))d\arcsin (c x)-\frac {(a+b \arcsin (c x))^2}{x}\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 4671

\(\displaystyle \frac {2}{3} b c d \left (-\frac {1}{2} c^2 \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )-\frac {2}{3} c^2 d \left (-\frac {(a+b \arcsin (c x))^2}{x}+2 b c \left (-b \int \log \left (1-e^{i \arcsin (c x)}\right )d\arcsin (c x)+b \int \log \left (1+e^{i \arcsin (c x)}\right )d\arcsin (c x)-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {2}{3} b c d \left (-\frac {1}{2} c^2 \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )-\frac {2}{3} c^2 d \left (-\frac {(a+b \arcsin (c x))^2}{x}+2 b c \left (i b \int e^{-i \arcsin (c x)} \log \left (1-e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-i b \int e^{-i \arcsin (c x)} \log \left (1+e^{i \arcsin (c x)}\right )de^{i \arcsin (c x)}-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))\right )\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {2}{3} b c d \left (-\frac {1}{2} c^2 \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 x^2}-\frac {b c}{2 x}\right )-\frac {2}{3} c^2 d \left (-\frac {(a+b \arcsin (c x))^2}{x}+2 b c \left (-2 \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))+i b \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )-i b \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )\right )\right )-\frac {d \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 x^3}\)

input

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

output

-1/3*(d*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/x^3 - (2*c^2*d*(-((a + b*ArcS 
in[c*x])^2/x) + 2*b*c*(-2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + 
 I*b*PolyLog[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))) 
/3 + (2*b*c*d*(-1/2*(b*c)/x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*x 
^2) - (c^2*(-2*(a + b*ArcSin[c*x])*ArcTanh[E^(I*ArcSin[c*x])] + I*b*PolyLo 
g[2, -E^(I*ArcSin[c*x])] - I*b*PolyLog[2, E^(I*ArcSin[c*x])]))/2))/3
 

3.2.64.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4671

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] + (-Simp[d*(m/f)   Int[(c + 
d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Simp[d*(m/f)   Int[(c + d*x 
)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IG 
tQ[m, 0]
 

rule 5138

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
:> Simp[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^n/(d*(m + 1))), x] - Simp[b*c*(n 
/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2 
*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 5196

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

rule 5200

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*((a + b*ArcS 
in[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1)))   Int[(f*x)^(m + 
 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[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} 
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
 

rule 5218

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

3.2.64.4 Maple [A] (verified)

Time = 0.18 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.48

method	result	size
parts	\(-d \,a^{2} \left (-\frac {c^{2}}{x}+\frac {1}{3 x^{3}}\right )-d \,b^{2} c^{3} \left (-\frac {3 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {5 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )-2 d a b \,c^{3} \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\)	\(261\)
derivativedivides	\(c^{3} \left (-d \,a^{2} \left (\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )-d \,b^{2} \left (-\frac {3 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {5 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )-2 d a b \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(262\)
default	\(c^{3} \left (-d \,a^{2} \left (\frac {1}{3 c^{3} x^{3}}-\frac {1}{c x}\right )-d \,b^{2} \left (-\frac {3 \arcsin \left (c x \right )^{2} x^{2} c^{2}-\sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x c -\arcsin \left (c x \right )^{2}-c^{2} x^{2}}{3 c^{3} x^{3}}-\frac {5 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 i \operatorname {polylog}\left (2, -i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}+\frac {5 \arcsin \left (c x \right ) \ln \left (1-i c x -\sqrt {-c^{2} x^{2}+1}\right )}{3}-\frac {5 i \operatorname {polylog}\left (2, i c x +\sqrt {-c^{2} x^{2}+1}\right )}{3}\right )-2 d a b \left (\frac {\arcsin \left (c x \right )}{3 c^{3} x^{3}}-\frac {\arcsin \left (c x \right )}{c x}+\frac {\sqrt {-c^{2} x^{2}+1}}{6 c^{2} x^{2}}-\frac {5 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {-c^{2} x^{2}+1}}\right )}{6}\right )\right )\)	\(262\)

input

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

output

-d*a^2*(-c^2/x+1/3/x^3)-d*b^2*c^3*(-1/3*(3*arcsin(c*x)^2*x^2*c^2-(-c^2*x^2 
+1)^(1/2)*arcsin(c*x)*x*c-arcsin(c*x)^2-c^2*x^2)/c^3/x^3-5/3*arcsin(c*x)*l 
n(1+I*c*x+(-c^2*x^2+1)^(1/2))+5/3*I*polylog(2,-I*c*x-(-c^2*x^2+1)^(1/2))+5 
/3*arcsin(c*x)*ln(1-I*c*x-(-c^2*x^2+1)^(1/2))-5/3*I*polylog(2,I*c*x+(-c^2* 
x^2+1)^(1/2)))-2*d*a*b*c^3*(1/3/c^3/x^3*arcsin(c*x)-1/c/x*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)))
 

3.2.64.5 Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]

input

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

output

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

3.2.64.6 Sympy [F]

\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=- d \left (\int \left (- \frac {a^{2}}{x^{4}}\right )\, dx + \int \frac {a^{2} c^{2}}{x^{2}}\, dx + \int \left (- \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \left (- \frac {2 a b \operatorname {asin}{\left (c x \right )}}{x^{4}}\right )\, dx + \int \frac {b^{2} c^{2} \operatorname {asin}^{2}{\left (c x \right )}}{x^{2}}\, dx + \int \frac {2 a b c^{2} \operatorname {asin}{\left (c x \right )}}{x^{2}}\, dx\right ) \]

input

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

output

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

3.2.64.7 Maxima [F]

\[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=\int { -\frac {{\left (c^{2} d x^{2} - d\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{x^{4}} \,d x } \]

input

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

output

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

3.2.64.8 Giac [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=\text {Timed out} \]

input

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

output

Timed out
 

3.2.64.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right ) (a+b \arcsin (c x))^2}{x^4} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\left (d-c^2\,d\,x^2\right )}{x^4} \,d x \]

input

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

output

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