\(\int (a+b \arccos (c x))^{3/2} \, dx\) [180]

Optimal result

Integrand size = 12, antiderivative size = 159 \[ \int (a+b \arccos (c x))^{3/2} \, dx=-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c} \]

[Out]

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

Rubi [A] (verified)

Time = 0.18 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {4716, 4768, 4720, 3387, 3386, 3432, 3385, 3433} \[ \int (a+b \arccos (c x))^{3/2} \, dx=-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2} \]

[In]

Int[(a + b*ArcCos[c*x])^(3/2),x]

[Out]

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

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 4716

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

Rule 4720

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

Rule 4768

Int[((a_.) + ArcCos[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^(
p + 1)*((a + b*ArcCos[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p
], Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcCos[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]

Rubi steps \begin{align*} \text {integral}& = x (a+b \arccos (c x))^{3/2}+\frac {1}{2} (3 b c) \int \frac {x \sqrt {a+b \arccos (c x)}}{\sqrt {1-c^2 x^2}} \, dx \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}-\frac {1}{4} \left (3 b^2\right ) \int \frac {1}{\sqrt {a+b \arccos (c x)}} \, dx \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}-\frac {(3 b) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{4 c} \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{4 c}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arccos (c x)\right )}{4 c} \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {\left (3 b \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{2 c}-\frac {\left (3 b \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \arccos (c x)}\right )}{2 c} \\ & = -\frac {3 b \sqrt {1-c^2 x^2} \sqrt {a+b \arccos (c x)}}{2 c}+x (a+b \arccos (c x))^{3/2}+\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right )}{2 c}-\frac {3 b^{3/2} \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 c} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.96 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.82 \[ \int (a+b \arccos (c x))^{3/2} \, dx=\frac {\sqrt {b} \left (2 \sqrt {b} \sqrt {a+b \arccos (c x)} \left (-3 \sqrt {1-c^2 x^2}+2 c x \arccos (c x)\right )+\frac {2 i a \sqrt {b} e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arccos (c x))}{b}\right )-e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arccos (c x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arccos (c x))}{b}\right )\right )}{\sqrt {a+b \arccos (c x)}}+\sqrt {2 \pi } \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (3 b \cos \left (\frac {a}{b}\right )+2 a \sin \left (\frac {a}{b}\right )\right )+\sqrt {2 \pi } \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arccos (c x)}}{\sqrt {b}}\right ) \left (2 a \cos \left (\frac {a}{b}\right )-3 b \sin \left (\frac {a}{b}\right )\right )\right )}{4 c} \]

[In]

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

[Out]

(Sqrt[b]*(2*Sqrt[b]*Sqrt[a + b*ArcCos[c*x]]*(-3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcCos[c*x]) + ((2*I)*a*Sqrt[b]*(Sqr
t[((-I)*(a + b*ArcCos[c*x]))/b]*Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b] - E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcC
os[c*x]))/b]*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcCos[c*x]]) + Sqrt[2*Pi]*Fresne
lS[(Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) + Sqrt[2*Pi]*FresnelC[(Sqrt[2/P
i]*Sqrt[a + b*ArcCos[c*x]])/Sqrt[b]]*(2*a*Cos[a/b] - 3*b*Sin[a/b])))/(4*c)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(277\) vs. \(2(123)=246\).

Time = 2.07 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.75

method	result	size
default	\(\frac {-3 \sqrt {a +b \arccos \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}-3 \sqrt {a +b \arccos \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arccos \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{2}+4 \arccos \left (c x \right )^{2} \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+8 \arccos \left (c x \right ) \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b +6 \arccos \left (c x \right ) \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) b^{2}+4 \cos \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a^{2}+6 \sin \left (-\frac {a +b \arccos \left (c x \right )}{b}+\frac {a}{b}\right ) a b}{4 c \sqrt {a +b \arccos \left (c x \right )}}\)	\(278\)

[In]

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

[Out]

1/4/c/(a+b*arccos(c*x))^(1/2)*(-3*(a+b*arccos(c*x))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b
*arccos(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*b^2-3*(a+b*arccos(c*x))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/P
i^(1/2)/(-1/b)^(1/2)*(a+b*arccos(c*x))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*b^2+4*arccos(c*x)^2*cos(-(a+b*ar
ccos(c*x))/b+a/b)*b^2+8*arccos(c*x)*cos(-(a+b*arccos(c*x))/b+a/b)*a*b+6*arccos(c*x)*sin(-(a+b*arccos(c*x))/b+a
/b)*b^2+4*cos(-(a+b*arccos(c*x))/b+a/b)*a^2+6*sin(-(a+b*arccos(c*x))/b+a/b)*a*b)

Fricas [F(-2)]

Exception generated. \[ \int (a+b \arccos (c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((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)

Sympy [F]

\[ \int (a+b \arccos (c x))^{3/2} \, dx=\int \left (a + b \operatorname {acos}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

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

[Out]

Integral((a + b*acos(c*x))**(3/2), x)

Maxima [F]

\[ \int (a+b \arccos (c x))^{3/2} \, dx=\int { {\left (b \arccos \left (c x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.18 (sec) , antiderivative size = 993, normalized size of antiderivative = 6.25 \[ \int (a+b \arccos (c x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*a
rccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a
*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b)
)/b)*e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*I*sqrt(2)*sqrt(pi)*a^2*b^2*erf(1/2*I*sqrt(2)*
sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3
/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) + 1/2*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sq
rt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(a
bs(b)))*c) - 1/2*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*
sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 3/8*I*sqrt(2)*sq
rt(pi)*b^3*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(
abs(b))/b)*e^(I*a/b)/((I*b^2/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 1/2*sqrt(2)*sqrt(pi)*a*b^2*erf(1/2*I*sqrt(2)*
sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2
/sqrt(abs(b)) + b*sqrt(abs(b)))*c) + 3/8*I*sqrt(2)*sqrt(pi)*b^3*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt
(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b
)))*c) + I*sqrt(pi)*a^2*b*erf(-1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(
c*x) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) - I*sqrt(pi)*a^2
*b*erf(1/2*I*sqrt(2)*sqrt(b*arccos(c*x) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b
)*e^(-I*a/b)/((-I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*c) + 1/2*sqrt(b*arccos(c*x) + a)*b*arccos
(c*x)*e^(I*arccos(c*x))/c + 1/2*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(-I*arccos(c*x))/c + 1/2*sqrt(b*arccos
(c*x) + a)*a*e^(I*arccos(c*x))/c + 3/4*I*sqrt(b*arccos(c*x) + a)*b*e^(I*arccos(c*x))/c + 1/2*sqrt(b*arccos(c*x
) + a)*a*e^(-I*arccos(c*x))/c - 3/4*I*sqrt(b*arccos(c*x) + a)*b*e^(-I*arccos(c*x))/c

Mupad [F(-1)]

Timed out. \[ \int (a+b \arccos (c x))^{3/2} \, dx=\int {\left (a+b\,\mathrm {acos}\left (c\,x\right )\right )}^{3/2} \,d x \]

[In]

int((a + b*acos(c*x))^(3/2),x)

[Out]

int((a + b*acos(c*x))^(3/2), x)