3.86 \(\int \frac {(d+e x^2) (a+b \csc ^{-1}(c x))}{x} \, dx\)

Optimal. Leaf size=124 \[ -d \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}+\frac {1}{2} i b d \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b d \csc ^{-1}(c x)^2-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \]

[Out]

1/2*I*b*d*arccsc(c*x)^2+1/2*e*x^2*(a+b*arccsc(c*x))-b*d*arccsc(c*x)*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2))^2)+b*d*ar
ccsc(c*x)*ln(1/x)-d*(a+b*arccsc(c*x))*ln(1/x)+1/2*I*b*d*polylog(2,(I/c/x+(1-1/c^2/x^2)^(1/2))^2)+1/2*b*e*x*(1-
1/c^2/x^2)^(1/2)/c

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Rubi [A]  time = 0.29, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.579, Rules used = {5241, 14, 4731, 6742, 264, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac {1}{2} i b d \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-d \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )+\frac {1}{2} e x^2 \left (a+b \csc ^{-1}(c x)\right )+\frac {b e x \sqrt {1-\frac {1}{c^2 x^2}}}{2 c}+\frac {1}{2} i b d \csc ^{-1}(c x)^2-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b d \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 264

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

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2326

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(ArcSin[(Rt[-e, 2]*x)/S
qrt[d]]*(a + b*Log[c*x^n]))/Rt[-e, 2], x] - Dist[(b*n)/Rt[-e, 2], Int[ArcSin[(Rt[-e, 2]*x)/Sqrt[d]]/x, x], x]
/; FreeQ[{a, b, c, d, e, n}, x] && GtQ[d, 0] && NegQ[e]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3717

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 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 0]))

Rule 5241

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> -Subst[Int[
((e + d*x^2)^p*(a + b*ArcSin[x/c])^n)/x^(m + 2*(p + 1)), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n
, 0] && IntegerQ[m] && IntegerQ[p]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 0.09, size = 108, normalized size = 0.87 \[ a d \log (x)+\frac {1}{2} a e x^2+\frac {b e x \sqrt {\frac {c^2 x^2-1}{c^2 x^2}}}{2 c}+\frac {1}{2} i b d \left (\csc ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\right )-b d \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} b e x^2 \csc ^{-1}(c x) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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fricas [F]  time = 1.01, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arccsc}\left (c x\right )}{x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/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(x)]Undef/Un
signed Inf encountered in limitEvaluation time: 0.55Limit: Max order reached or unable to make series expansio
n Error: Bad Argument Value

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maple [A]  time = 3.30, size = 198, normalized size = 1.60 \[ \frac {a \,x^{2} e}{2}+d a \ln \left (c x \right )+\frac {i b d \mathrm {arccsc}\left (c x \right )^{2}}{2}+\frac {b \,\mathrm {arccsc}\left (c x \right ) x^{2} e}{2}+\frac {b \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x e}{2 c}-\frac {i b e}{2 c^{2}}-b d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b d \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b d \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b d \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccsc(c*x))/x,x)

[Out]

1/2*a*x^2*e+d*a*ln(c*x)+1/2*I*b*d*arccsc(c*x)^2+1/2*b*arccsc(c*x)*x^2*e+1/2*b/c*((c^2*x^2-1)/c^2/x^2)^(1/2)*x*
e-1/2*I*b/c^2*e-b*d*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-b*d*arccsc(c*x)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2)
)+I*b*d*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))+I*b*d*polylog(2,I/c/x+(1-1/c^2/x^2)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{2} \, a e x^{2} + a d \log \relax (x) + \frac {2 i \, b c^{2} d \log \left (-c x + 1\right ) \log \relax (x) + 2 i \, b c^{2} d \log \relax (x)^{2} + 2 i \, b c^{2} d {\rm Li}_2\left (c x\right ) + 2 i \, b c^{2} d {\rm Li}_2\left (-c x\right ) + {\left (2 \, b c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 2 i \, b c^{2} \log \relax (c)\right )} e x^{2} - i \, {\left (2 \, {\left ({\left (\log \left (c x + 1\right ) + \log \left (c x - 1\right ) - 2 \, \log \relax (x)\right )} \log \relax (x) - \log \left (c x - 1\right ) \log \relax (x) + \log \left (-c x + 1\right ) \log \relax (x) + \log \relax (x)^{2} + {\rm Li}_2\left (c x\right ) + {\rm Li}_2\left (-c x\right )\right )} b d + b e {\left (\frac {\log \left (c x + 1\right )}{c^{2}} + \frac {\log \left (c x - 1\right )}{c^{2}}\right )}\right )} c^{2} + 2 \, {\left (2 \, b d \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} \log \relax (x)}{c^{2} x^{3} - x}\,{d x} + \frac {\sqrt {c x + 1} \sqrt {c x - 1} b e}{c^{2}}\right )} c^{2} + i \, b e \log \left (c x - 1\right ) + {\left (-i \, b c^{2} e x^{2} - 2 i \, b c^{2} d \log \relax (x)\right )} \log \left (c^{2} x^{2}\right ) + {\left (2 i \, b c^{2} d \log \relax (x) + i \, b e\right )} \log \left (c x + 1\right ) + {\left (2 i \, b c^{2} e x^{2} + {\left (4 \, b c^{2} \arctan \left (1, \sqrt {c x + 1} \sqrt {c x - 1}\right ) + 4 i \, b c^{2} \log \relax (c)\right )} d\right )} \log \relax (x)}{4 \, c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a*e*x^2 + a*d*log(x) + 1/4*(2*I*b*c^2*d*log(-c*x + 1)*log(x) + 2*I*b*c^2*d*log(x)^2 + 2*I*b*c^2*d*dilog(c*
x) + 2*I*b*c^2*d*dilog(-c*x) + (2*b*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 2*I*b*c^2*log(c))*e*x^2 - I*
(b*e*(log(c*x + 1)/c^2 + log(c*x - 1)/c^2) + 8*b*d*integrate(1/2*log(x)/(c^2*x^3 - x), x))*c^2 + 4*c^2*integra
te(1/2*(b*e*x^2 + 2*b*d*log(x))*sqrt(c*x + 1)*sqrt(c*x - 1)/(c^2*x^3 - x), x) + I*b*e*log(c*x - 1) + (-I*b*c^2
*e*x^2 - 2*I*b*c^2*d*log(x))*log(c^2*x^2) + (2*I*b*c^2*d*log(x) + I*b*e)*log(c*x + 1) + (2*I*b*c^2*e*x^2 + (4*
b*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 4*I*b*c^2*log(c))*d)*log(x))/c^2

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mupad [B]  time = 0.92, size = 111, normalized size = 0.90 \[ \frac {a\,e\,x^2}{2}-a\,d\,\ln \left (\frac {1}{x}\right )-b\,d\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right )+\frac {b\,e\,x\,\left (\sqrt {1-\frac {1}{c^2\,x^2}}+c\,x\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{2\,c}+\frac {b\,d\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,d\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)*(a + b*asin(1/(c*x))))/x,x)

[Out]

(b*d*polylog(2, exp(asin(1/(c*x))*2i))*1i)/2 - a*d*log(1/x) + (b*d*asin(1/(c*x))^2*1i)/2 + (a*e*x^2)/2 - b*d*l
og(1 - exp(asin(1/(c*x))*2i))*asin(1/(c*x)) + (b*e*x*((1 - 1/(c^2*x^2))^(1/2) + c*x*asin(1/(c*x))))/(2*c)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*acsc(c*x))*(d + e*x**2)/x, x)

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