3.2.79 \(\int x (a+b \text {ArcCos}(c x))^{3/2} \, dx\) [179]

Optimal. Leaf size=172 \[ -\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcCos}(c x)}}{8 c}-\frac {(a+b \text {ArcCos}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {ArcCos}(c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 b^{3/2} \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^2} \]

[Out]

-1/4*(a+b*arccos(c*x))^(3/2)/c^2+1/2*x^2*(a+b*arccos(c*x))^(3/2)+3/32*b^(3/2)*cos(2*a/b)*FresnelS(2*(a+b*arcco
s(c*x))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/c^2-3/32*b^(3/2)*FresnelC(2*(a+b*arccos(c*x))^(1/2)/b^(1/2)/Pi^(1/2))
*sin(2*a/b)*Pi^(1/2)/c^2-3/8*b*x*(-c^2*x^2+1)^(1/2)*(a+b*arccos(c*x))^(1/2)/c

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Rubi [A]
time = 0.28, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {4726, 4796, 4738, 4732, 4491, 12, 3387, 3386, 3432, 3385, 3433} \begin {gather*} -\frac {3 \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {\pi } \sqrt {b}}\right )}{32 c^2}+\frac {3 \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \text {ArcCos}(c x)}}{8 c}-\frac {(a+b \text {ArcCos}(c x))^{3/2}}{4 c^2}+\frac {1}{2} x^2 (a+b \text {ArcCos}(c x))^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCos[c*x])^(3/2),x]

[Out]

(-3*b*x*Sqrt[1 - c^2*x^2]*Sqrt[a + b*ArcCos[c*x]])/(8*c) - (a + b*ArcCos[c*x])^(3/2)/(4*c^2) + (x^2*(a + b*Arc
Cos[c*x])^(3/2))/2 + (3*b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]
)/(32*c^2) - (3*b^(3/2)*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcCos[c*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(32*c^
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 3385

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 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[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 3387

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

Rule 3432

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

Rule 3433

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

Rule 4491

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]

Rule 4726

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcCos[c*x])^n/(m
+ 1)), x] + Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcCos[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 4732

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

Rule 4738

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

Rule 4796

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[f
*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCos[c*x])^n/(e*(m + 2*p + 1))), x] + (Dist[f^2*((m - 1)/(c^2*(m
+ 2*p + 1))), Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcCos[c*x])^n, x], x] - Dist[b*f*(n/(c*(m + 2*p + 1)))*S
imp[(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*ArcCos[c*x])^(n - 1), x],
x]) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && IGtQ[m, 1] && NeQ[m + 2*p + 1, 0]

Rubi steps

\begin {align*} \int x \left (a+b \cos ^{-1}(c x)\right )^{3/2} \, dx &=\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {1}{4} (3 b c) \int \frac {x^2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}-\frac {1}{16} \left (3 b^2\right ) \int \frac {x}{\sqrt {a+b \cos ^{-1}(c x)}} \, dx+\frac {(3 b) \int \frac {\sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {1-c^2 x^2}} \, dx}{8 c}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{16 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{32 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b^2 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{32 c^2}-\frac {\left (3 b^2 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\cos ^{-1}(c x)\right )}{32 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {\left (3 b \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{16 c^2}-\frac {\left (3 b \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \cos ^{-1}(c x)}\right )}{16 c^2}\\ &=-\frac {3 b x \sqrt {1-c^2 x^2} \sqrt {a+b \cos ^{-1}(c x)}}{8 c}-\frac {\left (a+b \cos ^{-1}(c x)\right )^{3/2}}{4 c^2}+\frac {1}{2} x^2 \left (a+b \cos ^{-1}(c x)\right )^{3/2}+\frac {3 b^{3/2} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right )}{32 c^2}-\frac {3 b^{3/2} \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \cos ^{-1}(c x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{32 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.61, size = 155, normalized size = 0.90 \begin {gather*} \frac {3 b \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {\pi }}\right )-3 b \sqrt {\pi } \text {FresnelC}\left (\frac {2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}(c x)}}{\sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )+2 \sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}(c x)} (4 a \cos (2 \text {ArcCos}(c x))+4 b \text {ArcCos}(c x) \cos (2 \text {ArcCos}(c x))-3 b \sin (2 \text {ArcCos}(c x)))}{32 \sqrt {\frac {1}{b}} c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCos[c*x])^(3/2),x]

[Out]

(3*b*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]] - 3*b*Sqrt[Pi]*FresnelC
[(2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[Pi]]*Sin[(2*a)/b] + 2*Sqrt[b^(-1)]*Sqrt[a + b*ArcCos[c*x]]*(4*a
*Cos[2*ArcCos[c*x]] + 4*b*ArcCos[c*x]*Cos[2*ArcCos[c*x]] - 3*b*Sin[2*ArcCos[c*x]]))/(32*Sqrt[b^(-1)]*c^2)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(286\) vs. \(2(134)=268\).
time = 0.31, size = 287, normalized size = 1.67

method	result	size
default	\(\frac {-3 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}-3 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{2}+16 \arccos \left (c x \right )^{2} \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+32 \arccos \left (c x \right ) \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a b +12 \arccos \left (c x \right ) \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) b^{2}+16 \cos \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2}+12 \sin \left (-\frac {2 \left (a +b \arccos \left (c x \right )\right )}{b}+\frac {2 a}{b}\right ) a b}{64 c^{2} \sqrt {a +b \arccos \left (c x \right )}}\)	\(287\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arccos(c*x))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/64/c^2/(a+b*arccos(c*x))^(1/2)*(-3*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*x))^(1/2)*cos(2*a/b)*FresnelS
(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b^2-3*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arccos(c*
x))^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*b^2+16*arccos(c*x)^2*
cos(-2*(a+b*arccos(c*x))/b+2*a/b)*b^2+32*arccos(c*x)*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*a*b+12*arccos(c*x)*sin(
-2*(a+b*arccos(c*x))/b+2*a/b)*b^2+16*cos(-2*(a+b*arccos(c*x))/b+2*a/b)*a^2+12*sin(-2*(a+b*arccos(c*x))/b+2*a/b
)*a*b)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccos(c*x))^(3/2),x, algorithm="maxima")

[Out]

integrate((b*arccos(c*x) + a)^(3/2)*x, x)

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccos(c*x))^(3/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*acos(c*x))**(3/2),x)

[Out]

Integral(x*(a + b*acos(c*x))**(3/2), x)

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Giac [C] Result contains complex when optimal does not.
time = 0.99, size = 845, normalized size = 4.91 \begin {gather*} -\frac {i \, \sqrt {\pi } a^{2} b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, {\left (b^{2} + \frac {i \, b^{3}}{{\left | b \right |}}\right )} c^{2}} + \frac {\sqrt {\pi } a b^{\frac {5}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{2} + \frac {i \, b^{3}}{{\left | b \right |}}\right )} c^{2}} + \frac {i \, \sqrt {\pi } a^{2} b^{\frac {3}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, {\left (b^{2} - \frac {i \, b^{3}}{{\left | b \right |}}\right )} c^{2}} + \frac {\sqrt {\pi } a b^{\frac {5}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{2} - \frac {i \, b^{3}}{{\left | b \right |}}\right )} c^{2}} - \frac {\sqrt {\pi } a b^{2} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{\frac {3}{2}} + \frac {i \, b^{\frac {5}{2}}}{{\left | b \right |}}\right )} c^{2}} - \frac {i \, \sqrt {\pi } a^{2} b \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{4 \, {\left (b^{\frac {3}{2}} - \frac {i \, b^{\frac {5}{2}}}{{\left | b \right |}}\right )} c^{2}} - \frac {\sqrt {\pi } a b^{2} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{8 \, {\left (b^{\frac {3}{2}} - \frac {i \, b^{\frac {5}{2}}}{{\left | b \right |}}\right )} c^{2}} + \frac {i \, \sqrt {\pi } a^{2} \sqrt {b} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{4 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} - \frac {3 i \, \sqrt {\pi } b^{\frac {5}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} - \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (\frac {2 i \, a}{b}\right )}}{64 \, {\left (b + \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} + \frac {3 i \, \sqrt {\pi } b^{\frac {5}{2}} \operatorname {erf}\left (-\frac {\sqrt {b \arccos \left (c x\right ) + a}}{\sqrt {b}} + \frac {i \, \sqrt {b \arccos \left (c x\right ) + a} \sqrt {b}}{{\left | b \right |}}\right ) e^{\left (-\frac {2 i \, a}{b}\right )}}{64 \, {\left (b - \frac {i \, b^{2}}{{\left | b \right |}}\right )} c^{2}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} b \arccos \left (c x\right ) e^{\left (2 i \, \arccos \left (c x\right )\right )}}{8 \, c^{2}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} b \arccos \left (c x\right ) e^{\left (-2 i \, \arccos \left (c x\right )\right )}}{8 \, c^{2}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} a e^{\left (2 i \, \arccos \left (c x\right )\right )}}{8 \, c^{2}} + \frac {3 i \, \sqrt {b \arccos \left (c x\right ) + a} b e^{\left (2 i \, \arccos \left (c x\right )\right )}}{32 \, c^{2}} + \frac {\sqrt {b \arccos \left (c x\right ) + a} a e^{\left (-2 i \, \arccos \left (c x\right )\right )}}{8 \, c^{2}} - \frac {3 i \, \sqrt {b \arccos \left (c x\right ) + a} b e^{\left (-2 i \, \arccos \left (c x\right )\right )}}{32 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arccos(c*x))^(3/2),x, algorithm="giac")

[Out]

-1/4*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e
^(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) + 1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b
*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*c^2) + 1/4*I*sqrt(pi)*a^2*b^(3/2)*erf(-sqr
t(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^
2) + 1/8*sqrt(pi)*a*b^(5/2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e
^(-2*I*a/b)/((b^2 - I*b^3/abs(b))*c^2) - 1/8*sqrt(pi)*a*b^2*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*ar
ccos(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^(3/2) + I*b^(5/2)/abs(b))*c^2) - 1/4*I*sqrt(pi)*a^2*b*erf(-sqrt
(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs
(b))*c^2) - 1/8*sqrt(pi)*a*b^2*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b)
)*e^(-2*I*a/b)/((b^(3/2) - I*b^(5/2)/abs(b))*c^2) + 1/4*I*sqrt(pi)*a^2*sqrt(b)*erf(-sqrt(b*arccos(c*x) + a)/sq
rt(b) - I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/abs(b))*c^2) - 3/64*I*sqrt(pi)*b^(5/
2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) - I*sqrt(b*arccos(c*x) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b + I*b^2/ab
s(b))*c^2) + 3/64*I*sqrt(pi)*b^(5/2)*erf(-sqrt(b*arccos(c*x) + a)/sqrt(b) + I*sqrt(b*arccos(c*x) + a)*sqrt(b)/
abs(b))*e^(-2*I*a/b)/((b - I*b^2/abs(b))*c^2) + 1/8*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(2*I*arccos(c*x))/
c^2 + 1/8*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(-2*I*arccos(c*x))/c^2 + 1/8*sqrt(b*arccos(c*x) + a)*a*e^(2*
I*arccos(c*x))/c^2 + 3/32*I*sqrt(b*arccos(c*x) + a)*b*e^(2*I*arccos(c*x))/c^2 + 1/8*sqrt(b*arccos(c*x) + a)*a*
e^(-2*I*arccos(c*x))/c^2 - 3/32*I*sqrt(b*arccos(c*x) + a)*b*e^(-2*I*arccos(c*x))/c^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*acos(c*x))^(3/2),x)

[Out]

int(x*(a + b*acos(c*x))^(3/2), x)

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