3.2.22 \(\int (b x)^m \text {ArcCos}(a x) \, dx\) [122]

Optimal. Leaf size=68 \[ \frac {(b x)^{1+m} \text {ArcCos}(a x)}{b (1+m)}+\frac {a (b x)^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{b^2 (1+m) (2+m)} \]

[Out]

(b*x)^(1+m)*arccos(a*x)/b/(1+m)+a*(b*x)^(2+m)*hypergeom([1/2, 1+1/2*m],[2+1/2*m],a^2*x^2)/b^2/(1+m)/(2+m)

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Rubi [A]
time = 0.02, antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4724, 371} \begin {gather*} \frac {a (b x)^{m+2} \, _2F_1\left (\frac {1}{2},\frac {m+2}{2};\frac {m+4}{2};a^2 x^2\right )}{b^2 (m+1) (m+2)}+\frac {\text {ArcCos}(a x) (b x)^{m+1}}{b (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(b*x)^m*ArcCos[a*x],x]

[Out]

((b*x)^(1 + m)*ArcCos[a*x])/(b*(1 + m)) + (a*(b*x)^(2 + m)*Hypergeometric2F1[1/2, (2 + m)/2, (4 + m)/2, a^2*x^
2])/(b^2*(1 + m)*(2 + m))

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

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 (b x)^m \cos ^{-1}(a x) \, dx &=\frac {(b x)^{1+m} \cos ^{-1}(a x)}{b (1+m)}+\frac {a \int \frac {(b x)^{1+m}}{\sqrt {1-a^2 x^2}} \, dx}{b (1+m)}\\ &=\frac {(b x)^{1+m} \cos ^{-1}(a x)}{b (1+m)}+\frac {a (b x)^{2+m} \, _2F_1\left (\frac {1}{2},\frac {2+m}{2};\frac {4+m}{2};a^2 x^2\right )}{b^2 (1+m) (2+m)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 54, normalized size = 0.79 \begin {gather*} \frac {x (b x)^m \left ((2+m) \text {ArcCos}(a x)+a x \, _2F_1\left (\frac {1}{2},1+\frac {m}{2};2+\frac {m}{2};a^2 x^2\right )\right )}{(1+m) (2+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(b*x)^m*ArcCos[a*x],x]

[Out]

(x*(b*x)^m*((2 + m)*ArcCos[a*x] + a*x*Hypergeometric2F1[1/2, 1 + m/2, 2 + m/2, a^2*x^2]))/((1 + m)*(2 + m))

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Maple [F]
time = 1.82, size = 0, normalized size = 0.00 \[\int \left (b x \right )^{m} \arccos \left (a x \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x)^m*arccos(a*x),x)

[Out]

int((b*x)^m*arccos(a*x),x)

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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((b*x)^m*arccos(a*x),x, algorithm="maxima")

[Out]

(b^m*x*x^m*arctan2(sqrt(a*x + 1)*sqrt(-a*x + 1), a*x) - (a*b^m*m + a*b^m)*integrate(sqrt(a*x + 1)*sqrt(-a*x +
1)*x*x^m/((a^2*m + a^2)*x^2 - m - 1), x))/(m + 1)

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Fricas [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((b*x)^m*arccos(a*x),x, algorithm="fricas")

[Out]

integral((b*x)^m*arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x)**m*acos(a*x),x)

[Out]

Integral((b*x)**m*acos(a*x), x)

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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((b*x)^m*arccos(a*x),x, algorithm="giac")

[Out]

integrate((b*x)^m*arccos(a*x), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acos(a*x)*(b*x)^m,x)

[Out]

int(acos(a*x)*(b*x)^m, x)

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