3.3.18 \(\int \frac {(a+b \text {ArcCos}(c x))^3}{(d x)^{3/2}} \, dx\) [218]

Optimal. Leaf size=65 \[ -\frac {2 (a+b \text {ArcCos}(c x))^3}{d \sqrt {d x}}-\frac {6 b c \text {Int}\left (\frac {(a+b \text {ArcCos}(c x))^2}{\sqrt {d x} \sqrt {1-c^2 x^2}},x\right )}{d} \]

[Out]

-2*(a+b*arccos(c*x))^3/d/(d*x)^(1/2)-6*b*c*Unintegrable((a+b*arccos(c*x))^2/(d*x)^(1/2)/(-c^2*x^2+1)^(1/2),x)/
d

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Rubi [A]
time = 0.10, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {(a+b \text {ArcCos}(c x))^3}{(d x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*(a + b*ArcCos[c*x])^3)/(d*Sqrt[d*x]) - (6*b*c*Defer[Int][(a + b*ArcCos[c*x])^2/(Sqrt[d*x]*Sqrt[1 - c^2*x^2
]), x])/d

Rubi steps

\begin {align*} \int \frac {\left (a+b \cos ^{-1}(c x)\right )^3}{(d x)^{3/2}} \, dx &=-\frac {2 \left (a+b \cos ^{-1}(c x)\right )^3}{d \sqrt {d x}}-\frac {(6 b c) \int \frac {\left (a+b \cos ^{-1}(c x)\right )^2}{\sqrt {d x} \sqrt {1-c^2 x^2}} \, dx}{d}\\ \end {align*}

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Mathematica [A]
time = 95.84, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(a+b \text {ArcCos}(c x))^3}{(d x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \arccos \left (c x \right )\right )^{3}}{\left (d x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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))^3/(d*x)^(3/2),x, algorithm="maxima")

[Out]

-1/2*(4*b^3*arctan2(sqrt(c*x + 1)*sqrt(-c*x + 1), c*x)^3 - (a^3*c^2*sqrt(d)*(2*arctan(sqrt(c)*sqrt(x))/(c^(3/2
)*d^2) + log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/(c^(3/2)*d^2)) + 6*a*b^2*c^2*sqrt(d)*integrate(x^(5/
2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*d^2*x^4 - d^2*x^2), x) + 6*a^2*b*c^2*sqrt(d)*integrate(x^
(5/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d^2*x^4 - d^2*x^2), x) + 12*b^3*c*sqrt(d)*integrate(sqrt
(c*x + 1)*sqrt(-c*x + 1)*x^(3/2)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*d^2*x^4 - d^2*x^2), x) - a^
3*sqrt(d)*(2*sqrt(c)*arctan(sqrt(c)*sqrt(x))/d^2 + sqrt(c)*log((c*sqrt(x) - sqrt(c))/(c*sqrt(x) + sqrt(c)))/d^
2 + 4/(d^2*sqrt(x))) - 6*a*b^2*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))^2/(c^2*d^2
*x^4 - d^2*x^2), x) - 6*a^2*b*sqrt(d)*integrate(sqrt(x)*arctan(sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*x))/(c^2*d^2*x^
4 - d^2*x^2), x))*d^(3/2)*sqrt(x))/(d^(3/2)*sqrt(x))

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Fricas [A]
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))^3/(d*x)^(3/2),x, algorithm="fricas")

[Out]

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

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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))**3/(d*x)**(3/2),x)

[Out]

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

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Giac [A]
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))^3/(d*x)^(3/2),x, algorithm="giac")

[Out]

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

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

[Out]

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

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