3.80 \(\int x^4 \sin ^{-1}(a x)^{3/2} \, dx\)

Optimal. Leaf size=214 \[ -\frac{3 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{16 a^5}+\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{32 a^5}-\frac{3 \sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2} \]

[Out]

(4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^3) + (3*x
^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(50*a) + (x^5*ArcSin[a*x]^(3/2))/5 - (3*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]
*Sqrt[ArcSin[a*x]]])/(16*a^5) + (Sqrt[Pi/6]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(32*a^5) - (3*Sqrt[Pi/10]*
FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(800*a^5)

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Rubi [A]  time = 0.532559, antiderivative size = 282, normalized size of antiderivative = 1.32, number of steps used = 23, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4629, 4707, 4677, 4623, 3304, 3352, 4635, 4406} \[ -\frac{2 \sqrt{2 \pi } \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{25 a^5}-\frac{11 \sqrt{\frac{\pi }{2}} \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{400 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}+\frac{\sqrt{\frac{\pi }{6}} \text{FresnelC}\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{50 a^5}-\frac{3 \sqrt{\frac{\pi }{10}} \text{FresnelC}\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcSin[a*x]^(3/2),x]

[Out]

(4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^5) + (2*x^2*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(25*a^3) + (3*x
^4*Sqrt[1 - a^2*x^2]*Sqrt[ArcSin[a*x]])/(50*a) + (x^5*ArcSin[a*x]^(3/2))/5 - (11*Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi
]*Sqrt[ArcSin[a*x]]])/(400*a^5) - (2*Sqrt[2*Pi]*FresnelC[Sqrt[2/Pi]*Sqrt[ArcSin[a*x]]])/(25*a^5) + (Sqrt[Pi/6]
*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/(50*a^5) + (3*Sqrt[(3*Pi)/2]*FresnelC[Sqrt[6/Pi]*Sqrt[ArcSin[a*x]]])/
(800*a^5) - (3*Sqrt[Pi/10]*FresnelC[Sqrt[10/Pi]*Sqrt[ArcSin[a*x]]])/(800*a^5)

Rule 4629

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

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(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[m, 1] && IntegerQ[m]

Rule 4677

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[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 4623

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Dist[1/(b*c), Subst[Int[x^n*Cos[a/b - x/b], x], x, a
+ b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x]

Rule 3304

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[(f*x^2)/d],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3352

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]*FresnelC[Sqrt[2/Pi]*Rt[d, 2]*(e + f*x)])/
(f*Rt[d, 2]), x] /; FreeQ[{d, e, f}, x]

Rule 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 0]

Rule 4406

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

Rubi steps

\begin{align*} \int x^4 \sin ^{-1}(a x)^{3/2} \, dx &=\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{1}{10} (3 a) \int \frac{x^5 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx\\ &=\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{3}{100} \int \frac{x^4}{\sqrt{\sin ^{-1}(a x)}} \, dx-\frac{6 \int \frac{x^3 \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{25 a}\\ &=\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^4(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{100 a^5}-\frac{4 \int \frac{x \sqrt{\sin ^{-1}(a x)}}{\sqrt{1-a^2 x^2}} \, dx}{25 a^3}-\frac{\int \frac{x^2}{\sqrt{\sin ^{-1}(a x)}} \, dx}{25 a^2}\\ &=\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{8 \sqrt{x}}-\frac{3 \cos (3 x)}{16 \sqrt{x}}+\frac{\cos (5 x)}{16 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{100 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{25 a^5}-\frac{2 \int \frac{1}{\sqrt{\sin ^{-1}(a x)}} \, dx}{25 a^4}\\ &=\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (5 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{1600 a^5}-\frac{3 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{800 a^5}+\frac{9 \operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{1600 a^5}-\frac{\operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 \sqrt{x}}-\frac{\cos (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{25 a^5}-\frac{2 \operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{25 a^5}\\ &=\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{3 \operatorname{Subst}\left (\int \cos \left (5 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}-\frac{3 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{400 a^5}-\frac{\operatorname{Subst}\left (\int \frac{\cos (x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{100 a^5}+\frac{\operatorname{Subst}\left (\int \frac{\cos (3 x)}{\sqrt{x}} \, dx,x,\sin ^{-1}(a x)\right )}{100 a^5}+\frac{9 \operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}-\frac{4 \operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{25 a^5}\\ &=\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{3 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{400 a^5}-\frac{2 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{25 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}-\frac{3 \sqrt{\frac{\pi }{10}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}-\frac{\operatorname{Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{50 a^5}+\frac{\operatorname{Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt{\sin ^{-1}(a x)}\right )}{50 a^5}\\ &=\frac{4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^5}+\frac{2 x^2 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{25 a^3}+\frac{3 x^4 \sqrt{1-a^2 x^2} \sqrt{\sin ^{-1}(a x)}}{50 a}+\frac{1}{5} x^5 \sin ^{-1}(a x)^{3/2}-\frac{11 \sqrt{\frac{\pi }{2}} C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{400 a^5}-\frac{2 \sqrt{2 \pi } C\left (\sqrt{\frac{2}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{25 a^5}+\frac{\sqrt{\frac{\pi }{6}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{50 a^5}+\frac{3 \sqrt{\frac{3 \pi }{2}} C\left (\sqrt{\frac{6}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}-\frac{3 \sqrt{\frac{\pi }{10}} C\left (\sqrt{\frac{10}{\pi }} \sqrt{\sin ^{-1}(a x)}\right )}{800 a^5}\\ \end{align*}

Mathematica [C]  time = 0.0647914, size = 202, normalized size = 0.94 \[ \frac{\sqrt{\sin ^{-1}(a x)} \left (2250 \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-i \sin ^{-1}(a x)\right )+2250 \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},i \sin ^{-1}(a x)\right )-125 \sqrt{3} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-3 i \sin ^{-1}(a x)\right )-125 \sqrt{3} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},3 i \sin ^{-1}(a x)\right )+9 \sqrt{5} \sqrt{i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},-5 i \sin ^{-1}(a x)\right )+9 \sqrt{5} \sqrt{-i \sin ^{-1}(a x)} \text{Gamma}\left (\frac{5}{2},5 i \sin ^{-1}(a x)\right )\right )}{36000 a^5 \sqrt{\sin ^{-1}(a x)^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^4*ArcSin[a*x]^(3/2),x]

[Out]

(Sqrt[ArcSin[a*x]]*(2250*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-I)*ArcSin[a*x]] + 2250*Sqrt[(-I)*ArcSin[a*x]]*Gamma[
5/2, I*ArcSin[a*x]] - 125*Sqrt[3]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-3*I)*ArcSin[a*x]] - 125*Sqrt[3]*Sqrt[(-I)*A
rcSin[a*x]]*Gamma[5/2, (3*I)*ArcSin[a*x]] + 9*Sqrt[5]*Sqrt[I*ArcSin[a*x]]*Gamma[5/2, (-5*I)*ArcSin[a*x]] + 9*S
qrt[5]*Sqrt[(-I)*ArcSin[a*x]]*Gamma[5/2, (5*I)*ArcSin[a*x]]))/(36000*a^5*Sqrt[ArcSin[a*x]^2])

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Maple [A]  time = 0.079, size = 193, normalized size = 0.9 \begin{align*} -{\frac{1}{24000\,{a}^{5}} \left ( -3000\,ax \left ( \arcsin \left ( ax \right ) \right ) ^{2}+9\,\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{5}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -125\,\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{3}\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) +1500\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sin \left ( 3\,\arcsin \left ( ax \right ) \right ) -300\, \left ( \arcsin \left ( ax \right ) \right ) ^{2}\sin \left ( 5\,\arcsin \left ( ax \right ) \right ) +2250\,\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }\sqrt{\pi }{\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{\arcsin \left ( ax \right ) }}{\sqrt{\pi }}} \right ) -4500\,\arcsin \left ( ax \right ) \sqrt{-{a}^{2}{x}^{2}+1}+750\,\arcsin \left ( ax \right ) \cos \left ( 3\,\arcsin \left ( ax \right ) \right ) -90\,\arcsin \left ( ax \right ) \cos \left ( 5\,\arcsin \left ( ax \right ) \right ) \right ){\frac{1}{\sqrt{\arcsin \left ( ax \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arcsin(a*x)^(3/2),x)

[Out]

-1/24000/a^5/arcsin(a*x)^(1/2)*(-3000*a*x*arcsin(a*x)^2+9*5^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(
2^(1/2)/Pi^(1/2)*5^(1/2)*arcsin(a*x)^(1/2))-125*3^(1/2)*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi
^(1/2)*3^(1/2)*arcsin(a*x)^(1/2))+1500*arcsin(a*x)^2*sin(3*arcsin(a*x))-300*arcsin(a*x)^2*sin(5*arcsin(a*x))+2
250*2^(1/2)*arcsin(a*x)^(1/2)*Pi^(1/2)*FresnelC(2^(1/2)/Pi^(1/2)*arcsin(a*x)^(1/2))-4500*arcsin(a*x)*(-a^2*x^2
+1)^(1/2)+750*arcsin(a*x)*cos(3*arcsin(a*x))-90*arcsin(a*x)*cos(5*arcsin(a*x)))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^(3/2),x, algorithm="fricas")

[Out]

Exception raised: UnboundLocalError

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*asin(a*x)**(3/2),x)

[Out]

Timed out

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Giac [C]  time = 1.40054, size = 479, normalized size = 2.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arcsin(a*x)^(3/2),x, algorithm="giac")

[Out]

-1/160*I*arcsin(a*x)^(3/2)*e^(5*I*arcsin(a*x))/a^5 + 1/32*I*arcsin(a*x)^(3/2)*e^(3*I*arcsin(a*x))/a^5 - 1/16*I
*arcsin(a*x)^(3/2)*e^(I*arcsin(a*x))/a^5 + 1/16*I*arcsin(a*x)^(3/2)*e^(-I*arcsin(a*x))/a^5 - 1/32*I*arcsin(a*x
)^(3/2)*e^(-3*I*arcsin(a*x))/a^5 + 1/160*I*arcsin(a*x)^(3/2)*e^(-5*I*arcsin(a*x))/a^5 + (3/32000*I + 3/32000)*
sqrt(10)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (3/32000*I - 3/32000)*sqrt(10)*sqrt(pi)*
erf(-(1/2*I + 1/2)*sqrt(10)*sqrt(arcsin(a*x)))/a^5 - (1/768*I + 1/768)*sqrt(6)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt
(6)*sqrt(arcsin(a*x)))/a^5 + (1/768*I - 1/768)*sqrt(6)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(6)*sqrt(arcsin(a*x)))/
a^5 + (3/128*I + 3/128)*sqrt(2)*sqrt(pi)*erf((1/2*I - 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 - (3/128*I - 3/128)*
sqrt(2)*sqrt(pi)*erf(-(1/2*I + 1/2)*sqrt(2)*sqrt(arcsin(a*x)))/a^5 + 3/1600*sqrt(arcsin(a*x))*e^(5*I*arcsin(a*
x))/a^5 - 1/64*sqrt(arcsin(a*x))*e^(3*I*arcsin(a*x))/a^5 + 3/32*sqrt(arcsin(a*x))*e^(I*arcsin(a*x))/a^5 + 3/32
*sqrt(arcsin(a*x))*e^(-I*arcsin(a*x))/a^5 - 1/64*sqrt(arcsin(a*x))*e^(-3*I*arcsin(a*x))/a^5 + 3/1600*sqrt(arcs
in(a*x))*e^(-5*I*arcsin(a*x))/a^5