3.17 \(\int x^4 \csc ^{-1}(a+b x) \, dx\)

Optimal. Leaf size=197 \[ -\frac{\left (53 a^2+20\right ) a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{30 b^5}+\frac{\left (58 a^2+9\right ) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{a^5 \csc ^{-1}(a+b x)}{5 b^5}+\frac{\left (40 a^4+40 a^2+3\right ) \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{40 b^5}-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{1}{5} x^5 \csc ^{-1}(a+b x) \]

[Out]

-(a*(20 + 53*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(30*b^5) - (11*a*x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])
/(60*b^3) + (x^3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(20*b^2) + ((9 + 58*a^2)*(a + b*x)^2*Sqrt[1 - (a + b*x)^(
-2)])/(120*b^5) + (a^5*ArcCsc[a + b*x])/(5*b^5) + (x^5*ArcCsc[a + b*x])/5 + ((3 + 40*a^2 + 40*a^4)*ArcTanh[Sqr
t[1 - (a + b*x)^(-2)]])/(40*b^5)

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Rubi [A]  time = 0.232028, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.8, Rules used = {5259, 4427, 3782, 4056, 4048, 3770, 3767, 8} \[ -\frac{\left (53 a^2+20\right ) a (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{30 b^5}+\frac{\left (58 a^2+9\right ) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{a^5 \csc ^{-1}(a+b x)}{5 b^5}+\frac{\left (40 a^4+40 a^2+3\right ) \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{40 b^5}-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{1}{5} x^5 \csc ^{-1}(a+b x) \]

Antiderivative was successfully verified.

[In]

Int[x^4*ArcCsc[a + b*x],x]

[Out]

-(a*(20 + 53*a^2)*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(30*b^5) - (11*a*x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])
/(60*b^3) + (x^3*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)])/(20*b^2) + ((9 + 58*a^2)*(a + b*x)^2*Sqrt[1 - (a + b*x)^(
-2)])/(120*b^5) + (a^5*ArcCsc[a + b*x])/(5*b^5) + (x^5*ArcCsc[a + b*x])/5 + ((3 + 40*a^2 + 40*a^4)*ArcTanh[Sqr
t[1 - (a + b*x)^(-2)]])/(40*b^5)

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 4056

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_.) +
(a_))^(m_.), x_Symbol] :> -Simp[(C*Cot[e + f*x]*(a + b*Csc[e + f*x])^m)/(f*(m + 1)), x] + Dist[1/(m + 1), Int
[(a + b*Csc[e + f*x])^(m - 1)*Simp[a*A*(m + 1) + ((A*b + a*B)*(m + 1) + b*C*m)*Csc[e + f*x] + (b*B*(m + 1) + a
*C*m)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 0]

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^4 \csc ^{-1}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x \cot (x) \csc (x) (-a+\csc (x))^4 \, dx,x,\csc ^{-1}(a+b x)\right )}{b^5}\\ &=\frac{1}{5} x^5 \csc ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\csc (x))^5 \, dx,x,\csc ^{-1}(a+b x)\right )}{5 b^5}\\ &=\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\csc (x))^2 \left (-4 a^3+3 \left (1+4 a^2\right ) \csc (x)-11 a \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{20 b^5}\\ &=-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\csc (x)) \left (12 a^4-a \left (31+48 a^2\right ) \csc (x)+\left (9+58 a^2\right ) \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{60 b^5}\\ &=-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{\left (9+58 a^2\right ) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \left (-24 a^5+3 \left (3+40 a^2+40 a^4\right ) \csc (x)-4 a \left (20+53 a^2\right ) \csc ^2(x)\right ) \, dx,x,\csc ^{-1}(a+b x)\right )}{120 b^5}\\ &=-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{\left (9+58 a^2\right ) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{a^5 \csc ^{-1}(a+b x)}{5 b^5}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)+\frac{\left (a \left (20+53 a^2\right )\right ) \operatorname{Subst}\left (\int \csc ^2(x) \, dx,x,\csc ^{-1}(a+b x)\right )}{30 b^5}-\frac{\left (3+40 a^2+40 a^4\right ) \operatorname{Subst}\left (\int \csc (x) \, dx,x,\csc ^{-1}(a+b x)\right )}{40 b^5}\\ &=-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{\left (9+58 a^2\right ) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{a^5 \csc ^{-1}(a+b x)}{5 b^5}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)+\frac{\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{40 b^5}-\frac{\left (a \left (20+53 a^2\right )\right ) \operatorname{Subst}\left (\int 1 \, dx,x,(a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}\right )}{30 b^5}\\ &=-\frac{a \left (20+53 a^2\right ) (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{30 b^5}-\frac{11 a x^2 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{60 b^3}+\frac{x^3 (a+b x) \sqrt{1-\frac{1}{(a+b x)^2}}}{20 b^2}+\frac{\left (9+58 a^2\right ) (a+b x)^2 \sqrt{1-\frac{1}{(a+b x)^2}}}{120 b^5}+\frac{a^5 \csc ^{-1}(a+b x)}{5 b^5}+\frac{1}{5} x^5 \csc ^{-1}(a+b x)+\frac{\left (3+40 a^2+40 a^4\right ) \tanh ^{-1}\left (\sqrt{1-\frac{1}{(a+b x)^2}}\right )}{40 b^5}\\ \end{align*}

Mathematica [A]  time = 0.214322, size = 174, normalized size = 0.88 \[ \frac{-\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (-9 \left (4 a^2+1\right ) b^2 x^2+2 a \left (48 a^2+31\right ) b x+a^2 \left (154 a^2+71\right )+16 a b^3 x^3-6 b^4 x^4\right )+3 \left (40 a^4+40 a^2+3\right ) \log \left ((a+b x) \left (\sqrt{\frac{a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right )\right )+24 a^5 \sin ^{-1}\left (\frac{1}{a+b x}\right )+24 b^5 x^5 \csc ^{-1}(a+b x)}{120 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*ArcCsc[a + b*x],x]

[Out]

(-(Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2]*(a^2*(71 + 154*a^2) + 2*a*(31 + 48*a^2)*b*x - 9*(1 + 4*a^2
)*b^2*x^2 + 16*a*b^3*x^3 - 6*b^4*x^4)) + 24*b^5*x^5*ArcCsc[a + b*x] + 24*a^5*ArcSin[(a + b*x)^(-1)] + 3*(3 + 4
0*a^2 + 40*a^4)*Log[(a + b*x)*(1 + Sqrt[(-1 + a^2 + 2*a*b*x + b^2*x^2)/(a + b*x)^2])])/(120*b^5)

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Maple [B]  time = 0.233, size = 507, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*arccsc(b*x+a),x)

[Out]

3/40/b^5*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*ln(b*x+a+(-1+(b*x+a)^2)^(1/2))-71/120/b
^5*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a-77/60/b^5*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^
2)^(1/2)/(b*x+a)*a^3+29/60/b^4*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*x*a^2-11/60/b^3*(-1+(b*
x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*x^2*a+1/5*x^5*arccsc(b*x+a)+1/b^5*(-1+(b*x+a)^2)^(1/2)/((-1+(
b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^2*ln(b*x+a+(-1+(b*x+a)^2)^(1/2))+3/40/b^4*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/
(b*x+a)^2)^(1/2)/(b*x+a)*x+1/b^5*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^4*ln(b*x+a+(-
1+(b*x+a)^2)^(1/2))+1/5/b^5*(-1+(b*x+a)^2)^(1/2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^5*arctan(1/(-1+(b*
x+a)^2)^(1/2))+1/20/b^2*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*x^3

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{5} \, x^{5} \arctan \left (1, \sqrt{b x + a + 1} \sqrt{b x + a - 1}\right ) + \int \frac{{\left (b^{2} x^{6} + a b x^{5}\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (b x + a - 1\right )\right )}}{5 \,{\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^4*arccsc(b*x+a),x, algorithm="maxima")

[Out]

1/5*x^5*arctan2(1, sqrt(b*x + a + 1)*sqrt(b*x + a - 1)) + integrate(1/5*(b^2*x^6 + a*b*x^5)*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 = 3.55555, size = 374, normalized size = 1.9 \begin{align*} \frac{24 \, b^{5} x^{5} \operatorname{arccsc}\left (b x + a\right ) - 48 \, a^{5} \arctan \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 3 \,{\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-b x - a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) +{\left (6 \, b^{3} x^{3} - 22 \, a b^{2} x^{2} - 154 \, a^{3} +{\left (58 \, a^{2} + 9\right )} b x - 71 \, a\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{120 \, b^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccsc(b*x+a),x, algorithm="fricas")

[Out]

1/120*(24*b^5*x^5*arccsc(b*x + a) - 48*a^5*arctan(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) - 3*(40*a^4 +
40*a^2 + 3)*log(-b*x - a + sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1)) + (6*b^3*x^3 - 22*a*b^2*x^2 - 154*a^3 + (58*a^2
+ 9)*b*x - 71*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 - 1))/b^5

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{acsc}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*acsc(b*x+a),x)

[Out]

Integral(x**4*acsc(a + b*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \operatorname{arccsc}\left (b x + a\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*arccsc(b*x+a),x, algorithm="giac")

[Out]

integrate(x^4*arccsc(b*x + a), x)