3.1.6 \(\int x (a+b \csc ^{-1}(c x)) \, dx\) [6]

Optimal. Leaf size=39 \[ \frac {b \sqrt {1-\frac {1}{c^2 x^2}} x}{2 c}+\frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 39, 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 = {5329, 197} \begin {gather*} \frac {1}{2} x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcCsc[c*x]),x]

[Out]

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

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

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Mathematica [A]
time = 0.02, size = 50, normalized size = 1.28 \begin {gather*} \frac {a x^2}{2}+\frac {b x \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}}{2 c}+\frac {1}{2} b x^2 \csc ^{-1}(c x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcCsc[c*x]),x]

[Out]

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

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Maple [A]
time = 0.15, size = 65, normalized size = 1.67

method	result	size
derivativedivides	\(\frac {\frac {a \,c^{2} x^{2}}{2}+b \left (\frac {c^{2} x^{2} \mathrm {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(65\)
default	\(\frac {\frac {a \,c^{2} x^{2}}{2}+b \left (\frac {c^{2} x^{2} \mathrm {arccsc}\left (c x \right )}{2}+\frac {c^{2} x^{2}-1}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}\)	\(65\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 36, normalized size = 0.92 \begin {gather*} \frac {1}{2} \, a x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arccsc}\left (c x\right ) + \frac {x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c}\right )} b \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.34, size = 39, normalized size = 1.00 \begin {gather*} \frac {b c^{2} x^{2} \operatorname {arccsc}\left (c x\right ) + a c^{2} x^{2} + \sqrt {c^{2} x^{2} - 1} b}{2 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 1.41, size = 58, normalized size = 1.49 \begin {gather*} \frac {a x^{2}}{2} + \frac {b x^{2} \operatorname {acsc}{\left (c x \right )}}{2} + \frac {b \left (\begin {cases} \frac {\sqrt {c^{2} x^{2} - 1}}{c} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {i \sqrt {- c^{2} x^{2} + 1}}{c} & \text {otherwise} \end {cases}\right )}{2 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (33) = 66\).
time = 0.45, size = 182, normalized size = 4.67 \begin {gather*} \frac {1}{8} \, {\left (\frac {b x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{c} + \frac {a x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}{c} + \frac {2 \, b x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}}{c^{2}} + \frac {2 \, b \arcsin \left (\frac {1}{c x}\right )}{c^{3}} + \frac {2 \, a}{c^{3}} - \frac {2 \, b}{c^{4} x {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}} + \frac {b \arcsin \left (\frac {1}{c x}\right )}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}} + \frac {a}{c^{5} x^{2} {\left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}^{2}}\right )} c \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(b*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2*arcsin(1/(c*x))/c + a*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2/c + 2*b*x*(
sqrt(-1/(c^2*x^2) + 1) + 1)/c^2 + 2*b*arcsin(1/(c*x))/c^3 + 2*a/c^3 - 2*b/(c^4*x*(sqrt(-1/(c^2*x^2) + 1) + 1))
+ b*arcsin(1/(c*x))/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2) + a/(c^5*x^2*(sqrt(-1/(c^2*x^2) + 1) + 1)^2))*c

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Mupad [B]
time = 0.65, size = 40, normalized size = 1.03 \begin {gather*} \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{2}+\frac {b\,x\,\sqrt {1-\frac {1}{c^2\,x^2}}}{2\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(1/(c*x))),x)

[Out]

(a*x^2)/2 + (b*x^2*asin(1/(c*x)))/2 + (b*x*(1 - 1/(c^2*x^2))^(1/2))/(2*c)

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