3.87 \(\int \frac {(d+e x^2) (a+b \csc ^{-1}(c x))}{x^3} \, dx\)

Optimal. Leaf size=137 \[ -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )+\frac {1}{2} i b e \csc ^{-1}(c x)^2-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \]

[Out]

1/4*b*c^2*d*arccsc(c*x)+1/2*I*b*e*arccsc(c*x)^2-1/2*d*(a+b*arccsc(c*x))/x^2-b*e*arccsc(c*x)*ln(1-(I/c/x+(1-1/c
^2/x^2)^(1/2))^2)+b*e*arccsc(c*x)*ln(1/x)-e*(a+b*arccsc(c*x))*ln(1/x)+1/2*I*b*e*polylog(2,(I/c/x+(1-1/c^2/x^2)
^(1/2))^2)-1/4*b*c*d*(1-1/c^2/x^2)^(1/2)/x

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Rubi [A]  time = 0.30, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 13, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {5241, 14, 4731, 12, 6742, 321, 216, 2326, 4625, 3717, 2190, 2279, 2391} \[ \frac {1}{2} i b e \text {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \log \left (\frac {1}{x}\right ) \left (a+b \csc ^{-1}(c x)\right )-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \log \left (\frac {1}{x}\right ) \csc ^{-1}(c x) \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^3,x]

[Out]

-(b*c*d*Sqrt[1 - 1/(c^2*x^2)])/(4*x) + (b*c^2*d*ArcCsc[c*x])/4 + (I/2)*b*e*ArcCsc[c*x]^2 - (d*(a + b*ArcCsc[c*
x]))/(2*x^2) - b*e*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + b*e*ArcCsc[c*x]*Log[x^(-1)] - e*(a + b*ArcCsc[
c*x])*Log[x^(-1)] + (I/2)*b*e*PolyLog[2, E^((2*I)*ArcCsc[c*x])]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 321

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

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^3} \, dx &=-\operatorname {Subst}\left (\int \frac {\left (e+d x^2\right ) \left (a+b \sin ^{-1}\left (\frac {x}{c}\right )\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \frac {d x^2+2 e \log (x)}{2 \sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \frac {d x^2+2 e \log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {b \operatorname {Subst}\left (\int \left (\frac {d x^2}{\sqrt {1-\frac {x^2}{c^2}}}+\frac {2 e \log (x)}{\sqrt {1-\frac {x^2}{c^2}}}\right ) \, dx,x,\frac {1}{x}\right )}{2 c}\\ &=-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {(b d) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{2 c}+\frac {(b e) \operatorname {Subst}\left (\int \frac {\log (x)}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )}{c}\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{4} (b c d) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {x^2}{c^2}}} \, dx,x,\frac {1}{x}\right )-(b e) \operatorname {Subst}\left (\int \frac {\sin ^{-1}\left (\frac {x}{c}\right )}{x} \, dx,x,\frac {1}{x}\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-(b e) \operatorname {Subst}\left (\int x \cot (x) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(2 i b e) \operatorname {Subst}\left (\int \frac {e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+(b e) \operatorname {Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\csc ^{-1}(c x)\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )-\frac {1}{2} (i b e) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i \csc ^{-1}(c x)}\right )\\ &=-\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \csc ^{-1}(c x)+\frac {1}{2} i b e \csc ^{-1}(c x)^2-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{2 x^2}-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )+b e \csc ^{-1}(c x) \log \left (\frac {1}{x}\right )-e \left (a+b \csc ^{-1}(c x)\right ) \log \left (\frac {1}{x}\right )+\frac {1}{2} i b e \text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 125, normalized size = 0.91 \[ -\frac {a d}{2 x^2}+a e \log (x)-\frac {b c d \sqrt {\frac {c^2 x^2-1}{c^2 x^2}}}{4 x}+\frac {1}{4} b c^2 d \sin ^{-1}\left (\frac {1}{c x}\right )-\frac {b d \csc ^{-1}(c x)}{2 x^2}+\frac {1}{2} i b e \left (\csc ^{-1}(c x)^2+\text {Li}_2\left (e^{2 i \csc ^{-1}(c x)}\right )\right )-b e \csc ^{-1}(c x) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right ) \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d + e*x^2)*(a + b*ArcCsc[c*x]))/x^3,x]

[Out]

-1/2*(a*d)/x^2 - (b*c*d*Sqrt[(-1 + c^2*x^2)/(c^2*x^2)])/(4*x) - (b*d*ArcCsc[c*x])/(2*x^2) + (b*c^2*d*ArcSin[1/
(c*x)])/4 - b*e*ArcCsc[c*x]*Log[1 - E^((2*I)*ArcCsc[c*x])] + a*e*Log[x] + (I/2)*b*e*(ArcCsc[c*x]^2 + PolyLog[2
, E^((2*I)*ArcCsc[c*x])])

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fricas [F]  time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arccsc}\left (c x\right )}{x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^3,x, algorithm="fricas")

[Out]

integral((a*e*x^2 + a*d + (b*e*x^2 + b*d)*arccsc(c*x))/x^3, x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )} {\left (b \operatorname {arccsc}\left (c x\right ) + a\right )}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^3,x, algorithm="giac")

[Out]

integrate((e*x^2 + d)*(b*arccsc(c*x) + a)/x^3, x)

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maple [A]  time = 0.78, size = 185, normalized size = 1.35 \[ a e \ln \left (c x \right )-\frac {d a}{2 x^{2}}+\frac {i b e \mathrm {arccsc}\left (c x \right )^{2}}{2}-b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (1+\frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b e \,\mathrm {arccsc}\left (c x \right ) \ln \left (1-\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b e \polylog \left (2, -\frac {i}{c x}-\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+i b e \polylog \left (2, \frac {i}{c x}+\sqrt {1-\frac {1}{c^{2} x^{2}}}\right )+\frac {c^{2} b d \,\mathrm {arccsc}\left (c x \right ) \cos \left (2 \,\mathrm {arccsc}\left (c x \right )\right )}{4}-\frac {c^{2} b \sin \left (2 \,\mathrm {arccsc}\left (c x \right )\right ) d}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)*(a+b*arccsc(c*x))/x^3,x)

[Out]

a*e*ln(c*x)-1/2*d*a/x^2+1/2*I*b*e*arccsc(c*x)^2-b*e*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-b*e*arccsc(c*x
)*ln(1-I/c/x-(1-1/c^2/x^2)^(1/2))+I*b*e*polylog(2,-I/c/x-(1-1/c^2/x^2)^(1/2))+I*b*e*polylog(2,I/c/x+(1-1/c^2/x
^2)^(1/2))+1/4*c^2*b*d*arccsc(c*x)*cos(2*arccsc(c*x))-1/8*c^2*b*sin(2*arccsc(c*x))*d

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)*(a+b*arccsc(c*x))/x^3,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.99, size = 124, normalized size = 0.91 \[ -a\,e\,\ln \left (\frac {1}{x}\right )-\frac {a\,d}{2\,x^2}-b\,e\,\ln \left (1-{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,\mathrm {asin}\left (\frac {1}{c\,x}\right )-\frac {b\,c\,d\,\sqrt {1-\frac {1}{c^2\,x^2}}}{4\,x}-\frac {b\,c^2\,d\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\,\left (\frac {2}{c^2\,x^2}-1\right )}{4}+\frac {b\,e\,\mathrm {polylog}\left (2,{\mathrm {e}}^{\mathrm {asin}\left (\frac {1}{c\,x}\right )\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}}{2}+\frac {b\,e\,{\mathrm {asin}\left (\frac {1}{c\,x}\right )}^2\,1{}\mathrm {i}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x^2)*(a + b*asin(1/(c*x))))/x^3,x)

[Out]

(b*e*polylog(2, exp(asin(1/(c*x))*2i))*1i)/2 - a*e*log(1/x) + (b*e*asin(1/(c*x))^2*1i)/2 - (a*d)/(2*x^2) - b*e
*log(1 - exp(asin(1/(c*x))*2i))*asin(1/(c*x)) - (b*c*d*(1 - 1/(c^2*x^2))^(1/2))/(4*x) - (b*c^2*d*asin(1/(c*x))
*(2/(c^2*x^2) - 1))/4

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsc}{\left (c x \right )}\right ) \left (d + e x^{2}\right )}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)*(a+b*acsc(c*x))/x**3,x)

[Out]

Integral((a + b*acsc(c*x))*(d + e*x**2)/x**3, x)

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