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

Optimal. Leaf size=155 \[ \frac{\left (17 a^2+2\right ) (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^4}-\frac{a^4 \csc ^{-1}(a+b x)}{4 b^4}-\frac{\left (2 a^2+1\right ) a \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{2 b^4}+\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x) \]

[Out]

((2 + 17*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(12*b^4) + (x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(12*b^2)
- (a*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)])/(3*b^4) - (a^4*ArcCsc[a + b*x])/(4*b^4) + (x^4*ArcCsc[a + b*x])/4
- (a*(1 + 2*a^2)*ArcTanh[Sqrt[1 - (a + b*x)^(-2)]])/(2*b^4)

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Rubi [A]  time = 0.137291, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {5259, 4427, 3782, 4048, 3770, 3767, 8} \[ \frac{\left (17 a^2+2\right ) (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^4}-\frac{a^4 \csc ^{-1}(a+b x)}{4 b^4}-\frac{\left (2 a^2+1\right ) a \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{2 b^4}+\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x) \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcCsc[a + b*x],x]

[Out]

((2 + 17*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(12*b^4) + (x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(12*b^2)
- (a*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)])/(3*b^4) - (a^4*ArcCsc[a + b*x])/(4*b^4) + (x^4*ArcCsc[a + b*x])/4
- (a*(1 + 2*a^2)*ArcTanh[Sqrt[1 - (a + b*x)^(-2)]])/(2*b^4)

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]

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 3782

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(b^2*Cot[c + d*x]*(a + b*Csc[c + d*x])^(n
- 2))/(d*(n - 1)), x] + Dist[1/(n - 1), Int[(a + b*Csc[c + d*x])^(n - 3)*Simp[a^3*(n - 1) + (b*(b^2*(n - 2) +
3*a^2*(n - 1)))*Csc[c + d*x] + (a*b^2*(3*n - 4))*Csc[c + d*x]^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a
^2 - b^2, 0] && GtQ[n, 2] && IntegerQ[2*n]

Rule 4048

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_)), x_Symbol] :> -Simp[(b*C*Csc[e + f*x]*Cot[e + f*x])/(2*f), x] + Dist[1/2, Int[Simp[2*A*a + (2*B*a + b*(
2*A + C))*Csc[e + f*x] + 2*(a*C + B*b)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x]

Rule 3770

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

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int x^3 \csc ^{-1}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x \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)-\frac{\operatorname{Subst}\left (\int (-a+\csc (x))^4 \, dx,x,\csc ^{-1}(a+b x)\right )}{4 b^4}\\ &=\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\csc (x)) \left (-3 a^3+\left (2+9 a^2\right ) \csc (x)-8 a \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{12 b^4}\\ &=\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \left (6 a^4-12 a \left (1+2 a^2\right ) \csc (x)+2 \left (2+17 a^2\right ) \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{24 b^4}\\ &=\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}-\frac{a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)+\frac{\left (a \left (1+2 a^2\right )\right ) \operatorname{Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{2 b^4}-\frac{\left (2+17 a^2\right ) \operatorname{Subst}\left (\int \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{12 b^4}\\ &=\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}-\frac{a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)-\frac{a \left (1+2 a^2\right ) \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{2 b^4}+\frac{\left (2+17 a^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}\right )}{12 b^4}\\ &=\frac{\left (2+17 a^2\right ) (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^4}+\frac{x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{12 b^2}-\frac{a (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{3 b^4}-\frac{a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac{1}{4} x^4 \csc ^{-1}(a+b x)-\frac{a \left (1+2 a^2\right ) \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{2 b^4}\\ \end{align*}

Mathematica [A]  time = 0.273983, size = 149, normalized size = 0.96 \[ \frac{\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (9 a^2 b x+13 a^3-3 a b^2 x^2+2 a+b^3 x^3+2 b x\right )-6 \left (2 a^2+1\right ) a \log \left ((a+b x) \left (\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right )\right )-3 a^4 \sin ^{-1}\left (\frac{1}{a+b x}\right )+3 b^4 x^4 \csc ^{-1}(a+b x)}{12 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcCsc[a + b*x],x]

[Out]

(Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(2*a + 13*a^3 + 2*b*x + 9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3) +
3*b^4*x^4*ArcCsc[a + b*x] - 3*a^4*ArcSin[(a + b*x)^(-1)] - 6*a*(1 + 2*a^2)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2
+ 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(12*b^4)

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Maple [B]  time = 0.224, size = 360, normalized size = 2.3 \begin{align*}{\frac{{x}^{4}{\rm arccsc} \left (bx+a\right )}{4}}+{\frac{ \left ( -1+ \left ( bx+a \right ) ^{2} \right ){x}^{2}}{12\,{b}^{2} \left ( bx+a \right ) }{\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{{a}^{4}}{4\,{b}^{4} \left ( bx+a \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\arctan \left ({\frac{1}{\sqrt{-1+ \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{ \left ( -1+ \left ( bx+a \right ) ^{2} \right ) xa}{3\,{b}^{3} \left ( bx+a \right ) }{\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{{a}^{3}}{{b}^{4} \left ( bx+a \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\ln \left ( bx+a+\sqrt{-1+ \left ( bx+a \right ) ^{2}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}+{\frac{ \left ( -13+13\, \left ( bx+a \right ) ^{2} \right ){a}^{2}}{12\,{b}^{4} \left ( bx+a \right ) }{\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}-{\frac{a}{2\,{b}^{4} \left ( bx+a \right ) }\sqrt{-1+ \left ( bx+a \right ) ^{2}}\ln \left ( bx+a+\sqrt{-1+ \left ( bx+a \right ) ^{2}} \right ){\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}}+{\frac{-1+ \left ( bx+a \right ) ^{2}}{6\,{b}^{4} \left ( bx+a \right ) }{\frac{1}{\sqrt{{\frac{-1+ \left ( bx+a \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arccsc(b*x+a)+1/12/b^2*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*x^2-1/4/b^4*(-1+(b*x+a)
^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^4*arctan(1/(-1+(b*x+a)^2)^(1/2))-1/3/b^3*(-1+(b*x+a)^2)/(
(-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*x*a-1/b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a
)*a^3*ln(b*x+a+(-1+(b*x+a)^2)^(1/2))+13/12/b^4*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^2-1/2
/b^4*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a*ln(b*x+a+(-1+(b*x+a)^2)^(1/2))+1/6/b^4*(-
1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, x^{4} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) + \int \frac{{\left (b^{2} x^{5} + a b x^{4}\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (b x + a - 1\right )\right )}}{4 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} +{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\log \left (b x + a + 1\right ) + \log \left (b x + a - 1\right )\right )} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*x^4*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + integrate(1/4*(b^2*x^5 + a*b*x^4)*e^(1/2*log(b*x + a
+ 1) + 1/2*log(b*x + a - 1))/(b^2*x^2 + 2*a*b*x + a^2 + (b^2*x^2 + 2*a*b*x + a^2 - 1)*e^(log(b*x + a + 1) + l
og(b*x + a - 1)) - 1), x)

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Fricas [A]  time = 3.57049, size = 316, normalized size = 2.04 \begin{align*} \frac{3 \, b^{4} x^{4} \operatorname{arccsc}\left (b x + a\right ) + 6 \, a^{4} \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 6 \,{\left (2 \, a^{3} + a\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{\left (b^{2} x^{2} - 4 \, a b x + 13 \, a^{2} + 2\right )}}{12 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*x^4*arccsc(b*x + a) + 6*a^4*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + 6*(2*a^3 + a)*l
og(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)*(b^2*x^2 - 4*a*b*x + 13*a
^2 + 2))/b^4

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{acsc}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x**3*acsc(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{3} \operatorname{arccsc}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^3*arccsc(b*x + a), x)