3.27 \(\int x^3 \csc ^{-1}(a+b x)^2 \, dx\)

Optimal. Leaf size=366 \[ -\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {2 i a^3 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {i a \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {(a+b x)^2}{12 b^4}+\frac {\log (a+b x)}{3 b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}-\frac {2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {a x}{b^3}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2 \]

[Out]

-a*x/b^3+1/12*(b*x+a)^2/b^4-1/4*a^4*arccsc(b*x+a)^2/b^4+1/4*x^4*arccsc(b*x+a)^2-2*a*arccsc(b*x+a)*arctanh(I/(b
*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4-4*a^3*arccsc(b*x+a)*arctanh(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4+1/3*ln(b*x+a
)/b^4+3*a^2*ln(b*x+a)/b^4-2*I*a^3*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4+I*a*polylog(2,-I/(b*x+a)-(1-1
/(b*x+a)^2)^(1/2))/b^4+2*I*a^3*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^4-I*a*polylog(2,I/(b*x+a)+(1-1/(b
*x+a)^2)^(1/2))/b^4+1/3*(b*x+a)*arccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4+3*a^2*(b*x+a)*arccsc(b*x+a)*(1-1/(b*x+
a)^2)^(1/2)/b^4-a*(b*x+a)^2*arccsc(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4+1/6*(b*x+a)^3*arccsc(b*x+a)*(1-1/(b*x+a)^2
)^(1/2)/b^4

________________________________________________________________________________________

Rubi [A]  time = 0.31, antiderivative size = 366, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {5259, 4427, 4190, 4183, 2279, 2391, 4184, 3475, 4185} \[ \frac {2 i a^3 \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {i a \text {PolyLog}\left (2,-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \text {PolyLog}\left (2,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {3 a^2 \log (a+b x)}{b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {\log (a+b x)}{3 b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}-\frac {2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCsc[a + b*x]^2,x]

[Out]

-((a*x)/b^3) + (a + b*x)^2/(12*b^4) + ((a + b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/(3*b^4) + (3*a^2*(a
+ b*x)*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/b^4 - (a*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x
])/b^4 + ((a + b*x)^3*Sqrt[1 - (a + b*x)^(-2)]*ArcCsc[a + b*x])/(6*b^4) - (a^4*ArcCsc[a + b*x]^2)/(4*b^4) + (x
^4*ArcCsc[a + b*x]^2)/4 - (2*a*ArcCsc[a + b*x]*ArcTanh[E^(I*ArcCsc[a + b*x])])/b^4 - (4*a^3*ArcCsc[a + b*x]*Ar
cTanh[E^(I*ArcCsc[a + b*x])])/b^4 + Log[a + b*x]/(3*b^4) + (3*a^2*Log[a + b*x])/b^4 + (I*a*PolyLog[2, -E^(I*Ar
cCsc[a + b*x])])/b^4 + ((2*I)*a^3*PolyLog[2, -E^(I*ArcCsc[a + b*x])])/b^4 - (I*a*PolyLog[2, E^(I*ArcCsc[a + b*
x])])/b^4 - ((2*I)*a^3*PolyLog[2, E^(I*ArcCsc[a + b*x])])/b^4

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 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 3475

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4183

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

Rule 4184

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp[((c + d*x)^m*Cot[e + f*x])/f, x]
+ Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4185

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> -Simp[(b^2*(c + d*x)*Cot[e + f*x]*
(b*Csc[e + f*x])^(n - 2))/(f*(n - 1)), x] + (Dist[(b^2*(n - 2))/(n - 1), Int[(c + d*x)*(b*Csc[e + f*x])^(n - 2
), x], x] - Simp[(b^2*d*(b*Csc[e + f*x])^(n - 2))/(f^2*(n - 1)*(n - 2)), x]) /; FreeQ[{b, c, d, e, f}, x] && G
tQ[n, 1] && NeQ[n, 2]

Rule 4190

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Csc[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 4427

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

Rule 5259

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

Rubi steps

\begin {align*} \int x^3 \csc ^{-1}(a+b x)^2 \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \cot (x) \csc (x) (-a+\csc (x))^3 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}\\ &=\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int x (-a+\csc (x))^4 \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}\\ &=\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int \left (a^4 x-4 a^3 x \csc (x)+6 a^2 x \csc ^2(x)-4 a x \csc ^3(x)+x \csc ^4(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}\\ &=-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {\operatorname {Subst}\left (\int x \csc ^4(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}+\frac {(2 a) \operatorname {Subst}\left (\int x \csc ^3(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {\operatorname {Subst}\left (\int x \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^4}+\frac {a \operatorname {Subst}\left (\int x \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}-\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {\left (2 a^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {3 a^2 \log (a+b x)}{b^4}-\frac {\operatorname {Subst}\left (\int \cot (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{3 b^4}-\frac {a \operatorname {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {a \operatorname {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{b^4}+\frac {\left (2 i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {\left (2 i a^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {\log (a+b x)}{3 b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {2 i a^3 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {(i a) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {(i a) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \csc ^{-1}(a+b x)}\right )}{b^4}\\ &=-\frac {a x}{b^3}+\frac {(a+b x)^2}{12 b^4}+\frac {(a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{3 b^4}+\frac {3 a^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{b^4}+\frac {(a+b x)^3 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)}{6 b^4}-\frac {a^4 \csc ^{-1}(a+b x)^2}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)^2-\frac {2 a \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {4 a^3 \csc ^{-1}(a+b x) \tanh ^{-1}\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {\log (a+b x)}{3 b^4}+\frac {3 a^2 \log (a+b x)}{b^4}+\frac {i a \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}+\frac {2 i a^3 \text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {i a \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}-\frac {2 i a^3 \text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )}{b^4}\\ \end {align*}

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Mathematica [A]  time = 5.33, size = 420, normalized size = 1.15 \[ \frac {192 \left (2 a^3+a\right ) \left (i \left (\text {Li}_2\left (-e^{i \csc ^{-1}(a+b x)}\right )-\text {Li}_2\left (e^{i \csc ^{-1}(a+b x)}\right )\right )+\csc ^{-1}(a+b x) \left (\log \left (1-e^{i \csc ^{-1}(a+b x)}\right )-\log \left (1+e^{i \csc ^{-1}(a+b x)}\right )\right )\right )-64 \left (9 a^2+1\right ) \log \left (\frac {1}{a+b x}\right )+2 \left (\left (36 a^2+3\right ) \csc ^{-1}(a+b x)^2-24 a \csc ^{-1}(a+b x)+2\right ) \csc ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+2 \left (\left (36 a^2+3\right ) \csc ^{-1}(a+b x)^2+24 a \csc ^{-1}(a+b x)+2\right ) \sec ^2\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-16 \left (3 \left (2 a^3+a\right ) \csc ^{-1}(a+b x)^2-2 \left (9 a^2+1\right ) \csc ^{-1}(a+b x)+6 a\right ) \cot \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-16 \left (3 \left (2 a^3+a\right ) \csc ^{-1}(a+b x)^2+2 \left (9 a^2+1\right ) \csc ^{-1}(a+b x)+6 a\right ) \tan \left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )-\frac {2 \csc ^{-1}(a+b x) \left (6 a \csc ^{-1}(a+b x)-1\right ) \csc ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{a+b x}-32 (a+b x)^3 \csc ^{-1}(a+b x) \left (6 a \csc ^{-1}(a+b x)+1\right ) \sin ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )+3 \csc ^{-1}(a+b x)^2 \sec ^4\left (\frac {1}{2} \csc ^{-1}(a+b x)\right )}{192 b^4} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[x^3*ArcCsc[a + b*x]^2,x]

[Out]

(-16*(6*a - 2*(1 + 9*a^2)*ArcCsc[a + b*x] + 3*(a + 2*a^3)*ArcCsc[a + b*x]^2)*Cot[ArcCsc[a + b*x]/2] + 2*(2 - 2
4*a*ArcCsc[a + b*x] + (3 + 36*a^2)*ArcCsc[a + b*x]^2)*Csc[ArcCsc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Csc[ArcCs
c[a + b*x]/2]^4 - (2*ArcCsc[a + b*x]*(-1 + 6*a*ArcCsc[a + b*x])*Csc[ArcCsc[a + b*x]/2]^4)/(a + b*x) - 64*(1 +
9*a^2)*Log[(a + b*x)^(-1)] + 192*(a + 2*a^3)*(ArcCsc[a + b*x]*(Log[1 - E^(I*ArcCsc[a + b*x])] - Log[1 + E^(I*A
rcCsc[a + b*x])]) + I*(PolyLog[2, -E^(I*ArcCsc[a + b*x])] - PolyLog[2, E^(I*ArcCsc[a + b*x])])) + 2*(2 + 24*a*
ArcCsc[a + b*x] + (3 + 36*a^2)*ArcCsc[a + b*x]^2)*Sec[ArcCsc[a + b*x]/2]^2 + 3*ArcCsc[a + b*x]^2*Sec[ArcCsc[a
+ b*x]/2]^4 - 32*(a + b*x)^3*ArcCsc[a + b*x]*(1 + 6*a*ArcCsc[a + b*x])*Sin[ArcCsc[a + b*x]/2]^4 - 16*(6*a + 2*
(1 + 9*a^2)*ArcCsc[a + b*x] + 3*(a + 2*a^3)*ArcCsc[a + b*x]^2)*Tan[ArcCsc[a + b*x]/2])/(192*b^4)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^3*arccsc(b*x + a)^2, 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(x^3*arccsc(b*x+a)^2,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(a+b*x)]Simp
lification assuming t_nostep near 0Simplification assuming t_nostep near 0Evaluation time: 0.65sym2poly/r2sym(
const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 6.97, size = 769, normalized size = 2.10 \[ -\frac {5 a x}{6 b^{3}}-\frac {a^{4} \mathrm {arccsc}\left (b x +a \right )^{2}}{4 b^{4}}+\frac {3 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x \,a^{2}}{2 b^{3}}-\frac {\ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3 b^{4}}+\frac {2 \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{3 b^{4}}-\frac {\ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )}{3 b^{4}}-\frac {11 a^{2}}{12 b^{4}}+\frac {x^{2}}{12 b^{2}}+\frac {x^{4} \mathrm {arccsc}\left (b x +a \right )^{2}}{4}+\frac {13 \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) a^{3}}{6 b^{4}}+\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) a}{3 b^{4}}+\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x^{3}}{6 b}+\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x}{3 b^{3}}+\frac {2 a^{3} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {2 a^{3} \mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {3 i \mathrm {arccsc}\left (b x +a \right ) a^{2}}{b^{4}}+\frac {a \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1-\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {a \,\mathrm {arccsc}\left (b x +a \right ) \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {i a \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}+\frac {2 i a^{3} \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}+\frac {i a \polylog \left (2, -\frac {i}{b x +a}-\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {2 i a^{3} \polylog \left (2, \frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}+\frac {6 a^{2} \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {3 a^{2} \ln \left (\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}-1\right )}{b^{4}}-\frac {i \mathrm {arccsc}\left (b x +a \right )}{3 b^{4}}-\frac {3 a^{2} \ln \left (1+\frac {i}{b x +a}+\sqrt {1-\frac {1}{\left (b x +a \right )^{2}}}\right )}{b^{4}}-\frac {\sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \mathrm {arccsc}\left (b x +a \right ) x^{2} a}{2 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arccsc(b*x+a)^2,x)

[Out]

-5/6*a*x/b^3-1/4*a^4*arccsc(b*x+a)^2/b^4-I*a*polylog(2,I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4+2*I*a^3*polylog(2,
-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^4+I*a*polylog(2,-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))/b^4-2*I*a^3*polylog(2,I/
(b*x+a)+(1-1/(b*x+a)^2)^(1/2))/b^4+3/2/b^3*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*arccsc(b*x+a)*x*a^2-11/12/b^4*a^2+
1/12/b^2*x^2-1/3/b^4*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+2/3/b^4*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-1/3/b^4
*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2)-1)+6/b^4*a^2*ln(I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-3/b^4*a^2*ln(I/(b*x+a)+(1
-1/(b*x+a)^2)^(1/2)-1)-1/3*I/b^4*arccsc(b*x+a)-3/b^4*a^2*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))+1/4*x^4*arccsc(
b*x+a)^2+2/b^4*a^3*arccsc(b*x+a)*ln(1-I/(b*x+a)-(1-1/(b*x+a)^2)^(1/2))-2/b^4*a^3*arccsc(b*x+a)*ln(1+I/(b*x+a)+
(1-1/(b*x+a)^2)^(1/2))+13/6/b^4*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*arccsc(b*x+a)*a^3+1/3/b^4*((-1+(b*x+a)^2)/(b*
x+a)^2)^(1/2)*arccsc(b*x+a)*a+1/6/b*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)*arccsc(b*x+a)*x^3+1/3/b^3*((-1+(b*x+a)^2)
/(b*x+a)^2)^(1/2)*arccsc(b*x+a)*x-3*I/b^4*arccsc(b*x+a)*a^2+1/b^4*a*arccsc(b*x+a)*ln(1-I/(b*x+a)-(1-1/(b*x+a)^
2)^(1/2))-1/b^4*a*arccsc(b*x+a)*ln(1+I/(b*x+a)+(1-1/(b*x+a)^2)^(1/2))-1/2/b^2*((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)
*arccsc(b*x+a)*x^2*a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, x^{4} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right )^{2} - \frac {1}{16} \, x^{4} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )^{2} + \int \frac {2 \, \sqrt {b x + a + 1} \sqrt {b x + a - 1} b x^{4} \arctan \left (1, \sqrt {b x + a + 1} \sqrt {b x + a - 1}\right ) - 4 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} + {\left (3 \, a^{2} - 1\right )} b x^{4} + {\left (a^{3} - a\right )} x^{3}\right )} \log \left (b x + a\right )^{2} + {\left (b^{3} x^{6} + 2 \, a b^{2} x^{5} + {\left (a^{2} - 1\right )} b x^{4} + 4 \, {\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} + {\left (3 \, a^{2} - 1\right )} b x^{4} + {\left (a^{3} - a\right )} x^{3}\right )} \log \left (b x + a\right )\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}{4 \, {\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} - 1\right )} b x - a\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1))^2 - 1/16*x^4*log(b^2*x^2 + 2*a*b*x + a^2)^2 + integrat
e(1/4*(2*sqrt(b*x + a + 1)*sqrt(b*x + a - 1)*b*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) - 4*(b^3*x^
6 + 3*a*b^2*x^5 + (3*a^2 - 1)*b*x^4 + (a^3 - a)*x^3)*log(b*x + a)^2 + (b^3*x^6 + 2*a*b^2*x^5 + (a^2 - 1)*b*x^4
+ 4*(b^3*x^6 + 3*a*b^2*x^5 + (3*a^2 - 1)*b*x^4 + (a^3 - a)*x^3)*log(b*x + a))*log(b^2*x^2 + 2*a*b*x + a^2))/(
b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2 - 1)*b*x - a), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^3\,{\mathrm {asin}\left (\frac {1}{a+b\,x}\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*asin(1/(a + b*x))^2,x)

[Out]

int(x^3*asin(1/(a + b*x))^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*acsc(b*x+a)**2,x)

[Out]

Integral(x**3*acsc(a + b*x)**2, x)

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