3.3.7 \(\int \frac {a+b \text {ArcCos}(c x)}{(d x)^{3/2}} \, dx\) [207]

Optimal. Leaf size=55 \[ -\frac {2 (a+b \text {ArcCos}(c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}} \]

[Out]

-4*b*EllipticF(c^(1/2)*(d*x)^(1/2)/d^(1/2),I)*c^(1/2)/d^(3/2)-2*(a+b*arccos(c*x))/d/(d*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 55, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4724, 335, 227} \begin {gather*} -\frac {2 (a+b \text {ArcCos}(c x))}{d \sqrt {d x}}-\frac {4 b \sqrt {c} F\left (\left .\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*ArcCos[c*x]))/(d*Sqrt[d*x]) - (4*b*Sqrt[c]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[d*x])/Sqrt[d]], -1])/d^(3
/2)

Rule 227

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[Rt[-b, 4]*(x/Rt[a, 4])], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 4724

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

Rubi steps

\begin {align*} \int \frac {a+b \cos ^{-1}(c x)}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt {d x}}-\frac {(2 b c) \int \frac {1}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d}\\ &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt {d x}}-\frac {(4 b c) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {c^2 x^4}{d^2}}} \, dx,x,\sqrt {d x}\right )}{d^2}\\ &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )}{d \sqrt {d x}}-\frac {4 b \sqrt {c} F\left (\left .\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {d x}}{\sqrt {d}}\right )\right |-1\right )}{d^{3/2}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.10, size = 93, normalized size = 1.69 \begin {gather*} \frac {2 x \left (-a-b \text {ArcCos}(c x)+\frac {2 i b \sqrt {-\frac {1}{c}} c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^{3/2} F\left (\left .i \sinh ^{-1}\left (\frac {\sqrt {-\frac {1}{c}}}{\sqrt {x}}\right )\right |-1\right )}{\sqrt {1-c^2 x^2}}\right )}{(d x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-a - b*ArcCos[c*x] + ((2*I)*b*Sqrt[-c^(-1)]*c^2*Sqrt[1 - 1/(c^2*x^2)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[-
c^(-1)]/Sqrt[x]], -1])/Sqrt[1 - c^2*x^2]))/(d*x)^(3/2)

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 85, normalized size = 1.55

method	result	size
derivativedivides	\(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(85\)
default	\(\frac {-\frac {2 a}{\sqrt {d x}}+2 b \left (-\frac {\arccos \left (c x \right )}{\sqrt {d x}}-\frac {2 c \sqrt {-c x +1}\, \sqrt {c x +1}\, \EllipticF \left (\sqrt {d x}\, \sqrt {\frac {c}{d}}, i\right )}{d \sqrt {\frac {c}{d}}\, \sqrt {-c^{2} x^{2}+1}}\right )}{d}\)	\(85\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d*(-a/(d*x)^(1/2)+b*(-1/(d*x)^(1/2)*arccos(c*x)-2*c/d/(c/d)^(1/2)*(-c*x+1)^(1/2)*(c*x+1)^(1/2)/(-c^2*x^2+1)^
(1/2)*EllipticF((d*x)^(1/2)*(c/d)^(1/2),I)))

________________________________________________________________________________________

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

[Out]

-(2*b*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x) - (2*b*c*d^2*integrate(sqrt(c*x + 1)*sqrt(-c*x + 1)*sqrt(x)/(
c^2*d^2*x^3 - d^2*x), x) - (2*b*arctan(1/(sqrt(c)*sqrt(x))) - b*log(-(c*x - 1)/(c*x + 2*sqrt(c)*sqrt(x) + 1)))
*sqrt(c))*sqrt(x))/(d^(3/2)*sqrt(x))

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.55, size = 50, normalized size = 0.91 \begin {gather*} \frac {2 \, {\left (2 \, \sqrt {-c^{2} d} b x {\rm weierstrassPInverse}\left (\frac {4}{c^{2}}, 0, x\right ) - {\left (b c \arccos \left (c x\right ) + a c\right )} \sqrt {d x}\right )}}{c d^{2} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(2*sqrt(-c^2*d)*b*x*weierstrassPInverse(4/c^2, 0, x) - (b*c*arccos(c*x) + a*c)*sqrt(d*x))/(c*d^2*x)

________________________________________________________________________________________

Sympy [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((a+b*acos(c*x))/(d*x)**(3/2),x)

[Out]

Exception raised: TypeError >> Invalid comparison of non-real zoo

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\mathrm {acos}\left (c\,x\right )}{{\left (d\,x\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*acos(c*x))/(d*x)^(3/2),x)

[Out]

int((a + b*acos(c*x))/(d*x)^(3/2), x)

________________________________________________________________________________________