3.1.19 \(\int \frac {(a+b \csc ^{-1}(c x))^2}{x} \, dx\) [19]

Optimal. Leaf size=91 \[ \frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \text {PolyLog}\left (3,e^{2 i \csc ^{-1}(c x)}\right ) \]

[Out]

1/3*I*(a+b*arccsc(c*x))^3/b-(a+b*arccsc(c*x))^2*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)+I*b*(a+b*arccsc(c*x))*poly
log(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)-1/2*b^2*polylog(3,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)

________________________________________________________________________________________

Rubi [A]
time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5331, 3798, 2221, 2611, 2320, 6724} \begin {gather*} i b \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \left (a+b \csc ^{-1}(c x)\right )^2-\frac {1}{2} b^2 \text {Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcCsc[c*x])^2/x,x]

[Out]

((I/3)*(a + b*ArcCsc[c*x])^3)/b - (a + b*ArcCsc[c*x])^2*Log[1 - E^((2*I)*ArcCsc[c*x])] + I*b*(a + b*ArcCsc[c*x
])*PolyLog[2, E^((2*I)*ArcCsc[c*x])] - (b^2*PolyLog[3, E^((2*I)*ArcCsc[c*x])])/2

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 3798

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(
m + 1))), x] - Dist[2*I, Int[(c + d*x)^m*E^(2*I*k*Pi)*(E^(2*I*(e + f*x))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x))))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5331

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[-(c^(m + 1))^(-1), Subst[Int[(a + b*
x)^n*Csc[x]^(m + 1)*Cot[x], x], x, ArcCsc[c*x]], x] /; FreeQ[{a, b, c}, x] && IntegerQ[n] && IntegerQ[m] && (G
tQ[n, 0] || LtQ[m, -1])

Rule 6724

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

Rubi steps

\begin {align*} \int \frac {\left (a+b \csc ^{-1}(c x)\right )^2}{x} \, dx &=-\text {Subst}\left (\int (a+b x)^2 \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}+2 i \text {Subst}\left (\int \frac {e^{2 i x} (a+b x)^2}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+(2 b) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\left (i b^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=\frac {i \left (a+b \csc ^{-1}(c x)\right )^3}{3 b}-\left (a+b \csc ^{-1}(c x)\right )^2 \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+i b \left (a+b \csc ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )-\frac {1}{2} b^2 \text {Li}_3\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.09, size = 137, normalized size = 1.51 \begin {gather*} -2 a b \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+a^2 \log (c x)+i a b \left (\csc ^{-1}(c x)^2+\text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )\right )+\frac {1}{24} i b^2 \left (\pi ^3-8 \csc ^{-1}(c x)^3+24 i \csc ^{-1}(c x)^2 \log \left (1-e^{-2 i \csc ^{-1}(c x)}\right )-24 \csc ^{-1}(c x) \text {PolyLog}\left (2,e^{-2 i \csc ^{-1}(c x)}\right )+12 i \text {PolyLog}\left (3,e^{-2 i \csc ^{-1}(c x)}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcCsc[c*x])^2/x,x]

[Out]

-2*a*b*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a^2*Log[c*x] + I*a*b*(ArcCsc[c*x]^2 + PolyLog[2, E^((2*I)*
ArcCsc[c*x])]) + (I/24)*b^2*(Pi^3 - 8*ArcCsc[c*x]^3 + (24*I)*ArcCsc[c*x]^2*Log[1 - E^((-2*I)*ArcCsc[c*x])] - 2
4*ArcCsc[c*x]*PolyLog[2, E^((-2*I)*ArcCsc[c*x])] + (12*I)*PolyLog[3, E^((-2*I)*ArcCsc[c*x])])

________________________________________________________________________________________

Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (127 ) = 254\).
time = 0.36, size = 361, normalized size = 3.97

method	result	size
derivativedivides	\(a^{2} \ln \left (c x \right )+\frac {i b^{2} \mathrm {arccsc}\left (c x \right )^{3}}{3}-b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 b^{2} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 b^{2} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i a b \mathrm {arccsc}\left (c x \right )^{2}-2 a b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 a b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i a b \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i a b \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\)	\(361\)
default	\(a^{2} \ln \left (c x \right )+\frac {i b^{2} \mathrm {arccsc}\left (c x \right )^{3}}{3}-b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 b^{2} \polylog \left (3, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{2} \mathrm {arccsc}\left (c x \right )^{2} \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i b^{2} \mathrm {arccsc}\left (c x \right ) \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 b^{2} \polylog \left (3, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i a b \mathrm {arccsc}\left (c x \right )^{2}-2 a b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-2 a b \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i a b \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+2 i a b \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\)	\(361\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*ln(c*x)+1/3*I*b^2*arccsc(c*x)^3-b^2*arccsc(c*x)^2*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+2*I*b^2*arccsc(c*x)*poly
log(2,I/c/x+(1-1/c^2/x^2)^(1/2))-2*b^2*polylog(3,I/c/x+(1-1/c^2/x^2)^(1/2))-b^2*arccsc(c*x)^2*ln(1+I/c/x+(1-1/
c^2/x^2)^(1/2))+2*I*b^2*arccsc(c*x)*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))-2*b^2*polylog(3,-I/c/x-(1-1/c^2/x^2)
^(1/2))+I*a*b*arccsc(c*x)^2-2*a*b*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-2*a*b*arccsc(c*x)*ln(1-I/c/x-(1-
1/c^2/x^2)^(1/2))+2*I*a*b*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))+2*I*a*b*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))

________________________________________________________________________________________

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*arccsc(c*x))^2/x,x, algorithm="maxima")

[Out]

-1/2*b^2*c^2*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2)*log(c)^2 + b^2*c^2*integrate(x^2*log(c^2*x^2)/(c^2*x^3 - x)
, x)*log(c) - 2*b^2*c^2*integrate(x^2*log(x)/(c^2*x^3 - x), x)*log(c) + 2*b^2*c^2*integrate(x^2*log(c^2*x^2)*l
og(x)/(c^2*x^3 - x), x) - b^2*c^2*integrate(x^2*log(x)^2/(c^2*x^3 - x), x) + 2*a*b*c^2*integrate(x^2*arctan(1/
(sqrt(c*x + 1)*sqrt(c*x - 1)))/(c^2*x^3 - x), x) + 1/2*b^2*(log(c*x + 1) + log(c*x - 1) - 2*log(x))*log(c)^2 +
b^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2*log(x) - 1/4*b^2*log(c^2*x^2)^2*log(x) - b^2*integrate(log(c^2*
x^2)/(c^2*x^3 - x), x)*log(c) + 2*b^2*integrate(log(x)/(c^2*x^3 - x), x)*log(c) + 2*b^2*integrate(sqrt(c*x + 1
)*sqrt(c*x - 1)*arctan(1/(sqrt(c*x + 1)*sqrt(c*x - 1)))*log(x)/(c^2*x^3 - x), x) - 2*b^2*integrate(log(c^2*x^2
)*log(x)/(c^2*x^3 - x), x) + b^2*integrate(log(x)^2/(c^2*x^3 - x), x) - 2*a*b*integrate(arctan(1/(sqrt(c*x + 1
)*sqrt(c*x - 1)))/(c^2*x^3 - x), x) + a^2*log(x)

________________________________________________________________________________________

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((a+b*arccsc(c*x))^2/x,x, algorithm="fricas")

[Out]

integral((b^2*arccsc(c*x)^2 + 2*a*b*arccsc(c*x) + a^2)/x, x)

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{2}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*acsc(c*x))**2/x,x)

[Out]

Integral((a + b*acsc(c*x))**2/x, x)

________________________________________________________________________________________

Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(sageVARx)]l
n of unsigned

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^2}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*asin(1/(c*x)))^2/x,x)

[Out]

int((a + b*asin(1/(c*x)))^2/x, x)

________________________________________________________________________________________