3.1.4 \(\int x^3 (a+b \csc ^{-1}(c x)) \, dx\) [4]

Optimal. Leaf size=64 \[ \frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right ) \]

[Out]

1/4*x^4*(a+b*arccsc(c*x))+1/6*b*x*(1-1/c^2/x^2)^(1/2)/c^3+1/12*b*x^3*(1-1/c^2/x^2)^(1/2)/c

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Rubi [A]
time = 0.02, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5329, 277, 197} \begin {gather*} \frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x^3 \sqrt {1-\frac {1}{c^2 x^2}}}{12 c}+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{6 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*ArcCsc[c*x]),x]

[Out]

(b*Sqrt[1 - 1/(c^2*x^2)]*x)/(6*c^3) + (b*Sqrt[1 - 1/(c^2*x^2)]*x^3)/(12*c) + (x^4*(a + b*ArcCsc[c*x]))/4

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 277

Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^n)^(p + 1)/(a*(m + 1))), x]
- Dist[b*((m + n*(p + 1) + 1)/(a*(m + 1))), Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x]
&& ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]

Rule 5329

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

Rubi steps

\begin {align*} \int x^3 \left (a+b \csc ^{-1}(c x)\right ) \, dx &=\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {x^2}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{4 c}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )+\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}} \, dx}{6 c^3}\\ &=\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{6 c^3}+\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^3}{12 c}+\frac {1}{4} x^4 \left (a+b \csc ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 62, normalized size = 0.97 \begin {gather*} \frac {a x^4}{4}+b \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}} \left (\frac {x}{6 c^3}+\frac {x^3}{12 c}\right )+\frac {1}{4} b x^4 \csc ^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*ArcCsc[c*x]),x]

[Out]

(a*x^4)/4 + b*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)]*(x/(6*c^3) + x^3/(12*c)) + (b*x^4*ArcCsc[c*x])/4

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Maple [A]
time = 0.14, size = 74, normalized size = 1.16

method	result	size
derivativedivides	\(\frac {\frac {c^{4} x^{4} a}{4}+b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(74\)
default	\(\frac {\frac {c^{4} x^{4} a}{4}+b \left (\frac {c^{4} x^{4} \mathrm {arccsc}\left (c x \right )}{4}+\frac {\left (c^{2} x^{2}-1\right ) \left (c^{2} x^{2}+2\right )}{12 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{4}}\)	\(74\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a+b*arccsc(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^4*(1/4*c^4*x^4*a+b*(1/4*c^4*x^4*arccsc(c*x)+1/12*(c^2*x^2-1)*(c^2*x^2+2)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c/x))

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Maxima [A]
time = 0.27, size = 59, normalized size = 0.92 \begin {gather*} \frac {1}{4} \, a x^{4} + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arccsc}\left (c x\right ) + \frac {c^{2} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 3 \, x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{3}}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*x^4 + 1/12*(3*x^4*arccsc(c*x) + (c^2*x^3*(-1/(c^2*x^2) + 1)^(3/2) + 3*x*sqrt(-1/(c^2*x^2) + 1))/c^3)*b

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Fricas [A]
time = 0.36, size = 52, normalized size = 0.81 \begin {gather*} \frac {3 \, b c^{4} x^{4} \operatorname {arccsc}\left (c x\right ) + 3 \, a c^{4} x^{4} + {\left (b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} x^{2} - 1}}{12 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arccsc(c*x)),x, algorithm="fricas")

[Out]

1/12*(3*b*c^4*x^4*arccsc(c*x) + 3*a*c^4*x^4 + (b*c^2*x^2 + 2*b)*sqrt(c^2*x^2 - 1))/c^4

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Sympy [A]
time = 1.92, size = 107, normalized size = 1.67 \begin {gather*} \frac {a x^{4}}{4} + \frac {b x^{4} \operatorname {acsc}{\left (c x \right )}}{4} + \frac {b \left (\begin {cases} \frac {x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} + \frac {2 \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i x^{2} \sqrt {- c^{2} x^{2} + 1}}{3 c} + \frac {2 i \sqrt {- c^{2} x^{2} + 1}}{3 c^{3}} & \text {otherwise} \end {cases}\right )}{4 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*acsc(c*x)),x)

[Out]

a*x**4/4 + b*x**4*acsc(c*x)/4 + b*Piecewise((x**2*sqrt(c**2*x**2 - 1)/(3*c) + 2*sqrt(c**2*x**2 - 1)/(3*c**3),
Abs(c**2*x**2) > 1), (I*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + 2*I*sqrt(-c**2*x**2 + 1)/(3*c**3), True))/(4*c)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (54) = 108\).
time = 0.46, size = 352, normalized size = 5.50 \begin {gather*} \frac {1}{192} \, {\left (\frac {3 \, b x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {3 \, a x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}{c} + \frac {2 \, b x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}}{c^{2}} + \frac {12 \, b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {12 \, a x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c^{3}} + \frac {18 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{4}} + \frac {18 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{5}} + \frac {18 \, a}{c^{5}} - \frac {18 \, b}{c^{6} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {12 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {12 \, a}{c^{7} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} - \frac {2 \, b}{c^{8} x^{3} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{3}} + \frac {3 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}} + \frac {3 \, a}{c^{9} x^{4} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{4}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/192*(3*b*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4*arcsin(1/(c*x))/c + 3*a*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4/c + 2
*b*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3/c^2 + 12*b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c^3 + 12*a
*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c^3 + 18*b*x*(sqrt(-1/(c^2*x^2) + 1) + 1)/c^4 + 18*b*arcsin(1/(c*x))/c^5 +
18*a/c^5 - 18*b/(c^6*x*(sqrt(-1/(c^2*x^2) + 1) + 1)) + 12*b*arcsin(1/(c*x))/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1)
+ 1)^2) + 12*a/(c^7*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) - 2*b/(c^8*x^3*(sqrt(-1/(c^2*x^2) + 1) + 1)^3) + 3*b*a
rcsin(1/(c*x))/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4) + 3*a/(c^9*x^4*(sqrt(-1/(c^2*x^2) + 1) + 1)^4))*c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*asin(1/(c*x))),x)

[Out]

int(x^3*(a + b*asin(1/(c*x))), x)

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