\(\int \frac {\arcsin (a+b x)^2}{x^2} \, dx\) [136]

Optimal result

Integrand size = 12, antiderivative size = 230 \[ \int \frac {\arcsin (a+b x)^2}{x^2} \, dx=-\frac {\arcsin (a+b x)^2}{x}-\frac {2 b \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}+\frac {2 b \arcsin (a+b x) \log \left (1-\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}+\frac {2 i b \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a-\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}}-\frac {2 i b \operatorname {PolyLog}\left (2,\frac {e^{i \arcsin (a+b x)}}{i a+\sqrt {1-a^2}}\right )}{\sqrt {1-a^2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4889, 4827, 4857, 3404, 2296, 2221, 2317, 2438} \[ \int \frac {\arcsin (a+b x)^2}{x^2} \, dx=\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}+\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {a^2-1}}\right )}{\sqrt {a^2-1}}-\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{\sqrt {a^2-1}+a}\right )}{\sqrt {a^2-1}}-\frac {\arcsin (a+b x)^2}{x} \]

[In]

Int[ArcSin[a + b*x]^2/x^2,x]

[Out]

-(ArcSin[a + b*x]^2/x) + ((2*I)*b*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/Sqr
t[-1 + a^2] - ((2*I)*b*ArcSin[a + b*x]*Log[1 + (I*E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/Sqrt[-1 + a^2]
+ (2*b*PolyLog[2, ((-I)*E^(I*ArcSin[a + b*x]))/(a - Sqrt[-1 + a^2])])/Sqrt[-1 + a^2] - (2*b*PolyLog[2, ((-I)*
E^(I*ArcSin[a + b*x]))/(a + Sqrt[-1 + a^2])])/Sqrt[-1 + a^2]

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 2296

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[2*(c/q), Int[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Dist[2*(c/q), Int[(f + g
*x)^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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 3404

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[(c + d*x)^m*(E
^(I*(e + f*x))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x)))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4827

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

Rule 4857

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

Rule 4889

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + f*(x/d))^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\arcsin (x)^2}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x\right )}{b} \\ & = -\frac {\arcsin (a+b x)^2}{x}+2 \text {Subst}\left (\int \frac {\arcsin (x)}{\left (-\frac {a}{b}+\frac {x}{b}\right ) \sqrt {1-x^2}} \, dx,x,a+b x\right ) \\ & = -\frac {\arcsin (a+b x)^2}{x}+2 \text {Subst}\left (\int \frac {x}{-\frac {a}{b}+\frac {\sin (x)}{b}} \, dx,x,\arcsin (a+b x)\right ) \\ & = -\frac {\arcsin (a+b x)^2}{x}+4 \text {Subst}\left (\int \frac {e^{i x} x}{\frac {i}{b}-\frac {2 a e^{i x}}{b}-\frac {i e^{2 i x}}{b}} \, dx,x,\arcsin (a+b x)\right ) \\ & = -\frac {\arcsin (a+b x)^2}{x}-\frac {(4 i) \text {Subst}\left (\int \frac {e^{i x} x}{-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}-\frac {2 i e^{i x}}{b}} \, dx,x,\arcsin (a+b x)\right )}{\sqrt {-1+a^2}}+\frac {(4 i) \text {Subst}\left (\int \frac {e^{i x} x}{-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}-\frac {2 i e^{i x}}{b}} \, dx,x,\arcsin (a+b x)\right )}{\sqrt {-1+a^2}} \\ & = -\frac {\arcsin (a+b x)^2}{x}+\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {(2 i b) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x}}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\arcsin (a+b x)\right )}{\sqrt {-1+a^2}}-\frac {(2 i b) \text {Subst}\left (\int \log \left (1-\frac {2 i e^{i x}}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right ) \, dx,x,\arcsin (a+b x)\right )}{\sqrt {-1+a^2}} \\ & = -\frac {\arcsin (a+b x)^2}{x}+\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i x}{\left (-\frac {2 a}{b}-\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \arcsin (a+b x)}\right )}{\sqrt {-1+a^2}}-\frac {(2 b) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i x}{\left (-\frac {2 a}{b}+\frac {2 \sqrt {-1+a^2}}{b}\right ) b}\right )}{x} \, dx,x,e^{i \arcsin (a+b x)}\right )}{\sqrt {-1+a^2}} \\ & = -\frac {\arcsin (a+b x)^2}{x}+\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 i b \arcsin (a+b x) \log \left (1+\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}+\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}}-\frac {2 b \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.90 \[ \int \frac {\arcsin (a+b x)^2}{x^2} \, dx=\frac {-\sqrt {-1+a^2} \arcsin (a+b x)^2+2 i b x \arcsin (a+b x) \left (\log \left (\frac {a-\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a-\sqrt {-1+a^2}}\right )-\log \left (\frac {a+\sqrt {-1+a^2}+i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )\right )+2 b x \operatorname {PolyLog}\left (2,\frac {i e^{i \arcsin (a+b x)}}{-a+\sqrt {-1+a^2}}\right )-2 b x \operatorname {PolyLog}\left (2,-\frac {i e^{i \arcsin (a+b x)}}{a+\sqrt {-1+a^2}}\right )}{\sqrt {-1+a^2} x} \]

[In]

Integrate[ArcSin[a + b*x]^2/x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.31

method	result	size
derivativedivides	\(b \left (-\frac {\arcsin \left (b x +a \right )^{2}}{b x}-\frac {2 \arcsin \left (b x +a \right ) \sqrt {-a^{2}+1}\, \left (\ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )-\ln \left (\frac {-i a +\sqrt {-a^{2}+1}+i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{-i a +\sqrt {-a^{2}+1}}\right )\right )}{a^{2}-1}+\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}-\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-i a +\sqrt {-a^{2}+1}+i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{-i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}\right )\)	\(301\)
default	\(b \left (-\frac {\arcsin \left (b x +a \right )^{2}}{b x}-\frac {2 \arcsin \left (b x +a \right ) \sqrt {-a^{2}+1}\, \left (\ln \left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )-\ln \left (\frac {-i a +\sqrt {-a^{2}+1}+i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{-i a +\sqrt {-a^{2}+1}}\right )\right )}{a^{2}-1}+\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {i a +\sqrt {-a^{2}+1}-i \left (b x +a \right )-\sqrt {1-\left (b x +a \right )^{2}}}{i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}-\frac {2 i \sqrt {-a^{2}+1}\, \operatorname {dilog}\left (\frac {-i a +\sqrt {-a^{2}+1}+i \left (b x +a \right )+\sqrt {1-\left (b x +a \right )^{2}}}{-i a +\sqrt {-a^{2}+1}}\right )}{a^{2}-1}\right )\)	\(301\)

[In]

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

[Out]

b*(-arcsin(b*x+a)^2/b/x-2*arcsin(b*x+a)*(-a^2+1)^(1/2)*(ln((I*a+(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/
(I*a+(-a^2+1)^(1/2)))-ln((-I*a+(-a^2+1)^(1/2)+I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(-I*a+(-a^2+1)^(1/2))))/(a^2-1)+2
*I*(-a^2+1)^(1/2)/(a^2-1)*dilog((I*a+(-a^2+1)^(1/2)-I*(b*x+a)-(1-(b*x+a)^2)^(1/2))/(I*a+(-a^2+1)^(1/2)))-2*I*(
-a^2+1)^(1/2)/(a^2-1)*dilog((-I*a+(-a^2+1)^(1/2)+I*(b*x+a)+(1-(b*x+a)^2)^(1/2))/(-I*a+(-a^2+1)^(1/2))))

Fricas [F]

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

[In]

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

[Out]

integral(arcsin(b*x + a)^2/x^2, x)

Sympy [F]

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

[In]

integrate(asin(b*x+a)**2/x**2,x)

[Out]

Integral(asin(a + b*x)**2/x**2, x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {\arcsin (a+b x)^2}{x^2} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for mor
e details)Is

Giac [F]

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

[In]

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

[Out]

integrate(arcsin(b*x + a)^2/x^2, x)

Mupad [F(-1)]

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

[In]

int(asin(a + b*x)^2/x^2,x)

[Out]

int(asin(a + b*x)^2/x^2, x)