3.180 \(\int (a+b \cos ^{-1}(c x))^{3/2} \, dx\)

Optimal. Leaf size=159 \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{3 b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2} \]

[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)

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Rubi [A]  time = 0.237264, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4620, 4678, 4624, 3306, 3305, 3351, 3304, 3352} \[ -\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \sin \left (\frac{a}{b}\right ) \text{FresnelC}\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}+\frac{3 \sqrt{\frac{\pi }{2}} b^{3/2} \cos \left (\frac{a}{b}\right ) S\left (\frac{\sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}}{\sqrt{b}}\right )}{2 c}-\frac{3 b \sqrt{1-c^2 x^2} \sqrt{a+b \cos ^{-1}(c x)}}{2 c}+x \left (a+b \cos ^{-1}(c x)\right )^{3/2} \]

Antiderivative was successfully verified.

[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 4620

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 4678

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*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*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]

Rule 4624

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

Rule 3306

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 3305

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 3351

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

Rubi steps

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

Mathematica [C]  time = 2.37094, size = 295, normalized size = 1.86 \[ \frac{b \left (2 \left (2 c x \cos ^{-1}(c x)-3 \sqrt{1-c^2 x^2}\right ) \sqrt{a+b \cos ^{-1}(c x)}-\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (3 b \sin \left (\frac{a}{b}\right )-2 a \cos \left (\frac{a}{b}\right )\right ) \text{FresnelC}\left (\sqrt{\frac{2}{\pi }} \sqrt{\frac{1}{b}} \sqrt{a+b \cos ^{-1}(c x)}\right )+\sqrt{2 \pi } \sqrt{\frac{1}{b}} \left (2 a \sin \left (\frac{a}{b}\right )+3 b \cos \left (\frac{a}{b}\right )\right ) S\left (\sqrt{\frac{1}{b}} \sqrt{\frac{2}{\pi }} \sqrt{a+b \cos ^{-1}(c x)}\right )\right )-2 a e^{-\frac{i a}{b}} \sqrt{a+b \cos ^{-1}(c x)} \left (-\frac{\text{Gamma}\left (\frac{3}{2},-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{\sqrt{-\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}}}-\frac{e^{\frac{2 i a}{b}} \text{Gamma}\left (\frac{3}{2},\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}\right )}{\sqrt{\frac{i \left (a+b \cos ^{-1}(c x)\right )}{b}}}\right )}{4 c} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-2*a*Sqrt[a + b*ArcCos[c*x]]*(-(Gamma[3/2, ((-I)*(a + b*ArcCos[c*x]))/b]/Sqrt[((-I)*(a + b*ArcCos[c*x]))/b])
- (E^(((2*I)*a)/b)*Gamma[3/2, (I*(a + b*ArcCos[c*x]))/b])/Sqrt[(I*(a + b*ArcCos[c*x]))/b]))/E^((I*a)/b) + b*(
2*Sqrt[a + b*ArcCos[c*x]]*(-3*Sqrt[1 - c^2*x^2] + 2*c*x*ArcCos[c*x]) + Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelS[Sqrt[b
^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(3*b*Cos[a/b] + 2*a*Sin[a/b]) - Sqrt[b^(-1)]*Sqrt[2*Pi]*FresnelC[Sq
rt[b^(-1)]*Sqrt[2/Pi]*Sqrt[a + b*ArcCos[c*x]]]*(-2*a*Cos[a/b] + 3*b*Sin[a/b])))/(4*c)

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Maple [B]  time = 0.11, size = 270, normalized size = 1.7 \begin{align*}{\frac{1}{4\,c} \left ( 3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }\cos \left ({\frac{a}{b}} \right ){\it FresnelS} \left ({\frac{\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}-3\,\sqrt{{b}^{-1}}\sqrt{\pi }\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }\sin \left ({\frac{a}{b}} \right ){\it FresnelC} \left ({\frac{\sqrt{2}\sqrt{a+b\arccos \left ( cx \right ) }}{\sqrt{{b}^{-1}}\sqrt{\pi }b}} \right ){b}^{2}+4\, \left ( \arccos \left ( cx \right ) \right ) ^{2}\cos \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ){b}^{2}+8\,\arccos \left ( cx \right ) \cos \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) ab-6\,\arccos \left ( cx \right ) \sin \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ){b}^{2}+4\,\cos \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ){a}^{2}-6\,\sin \left ({\frac{a+b\arccos \left ( cx \right ) }{b}}-{\frac{a}{b}} \right ) ab \right ){\frac{1}{\sqrt{a+b\arccos \left ( cx \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \arccos \left (c x\right ) + a\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acos(c*x))**(3/2), x)

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Giac [B]  time = 2.23727, size = 905, normalized size = 5.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/8*sqrt(2)*sqrt(pi)*b^4*i*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcco
s(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs(b)) + b^2*sqrt(abs(b)))*c) - 3/8*sqrt(2)*sqrt(pi)*b^4*i
*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*
e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*c) - 1/4*sqrt(2)*sqrt(pi)*a*b^3*erf(-1/2*sqrt(2)*sqrt(b*ar
ccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^3*i/sqrt(abs
(b)) + b^2*sqrt(abs(b)))*c) + 1/4*sqrt(2)*sqrt(pi)*a*b^3*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)
) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*e^(-a*i/b)/((b^3*i/sqrt(abs(b)) - b^2*sqrt(abs(b)))*c)
+ 1/4*sqrt(2)*sqrt(pi)*a*b^2*erf(-1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arc
cos(c*x) + a)*sqrt(abs(b))/b)*e^(a*i/b)/((b^2*i/sqrt(abs(b)) + b*sqrt(abs(b)))*c) - 1/4*sqrt(2)*sqrt(pi)*a*b^2
*erf(1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*i/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arccos(c*x) + a)*sqrt(abs(b))/b)*
e^(-a*i/b)/((b^2*i/sqrt(abs(b)) - b*sqrt(abs(b)))*c) + 3/4*sqrt(b*arccos(c*x) + a)*b*i*e^(i*arccos(c*x))/c + 1
/2*sqrt(b*arccos(c*x) + a)*b*arccos(c*x)*e^(i*arccos(c*x))/c - 3/4*sqrt(b*arccos(c*x) + a)*b*i*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*ar
ccos(c*x))/c + 1/2*sqrt(b*arccos(c*x) + a)*a*e^(-i*arccos(c*x))/c