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

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

[Out]

1/5*a^5*arccsc(b*x+a)/b^5+1/5*x^5*arccsc(b*x+a)+1/40*(40*a^4+40*a^2+3)*arctanh((1-1/(b*x+a)^2)^(1/2))/b^5-1/30
*a*(53*a^2+20)*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^5-11/60*a*x^2*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^3+1/20*x^3*(b*x+a
)*(1-1/(b*x+a)^2)^(1/2)/b^2+1/120*(58*a^2+9)*(b*x+a)^2*(1-1/(b*x+a)^2)^(1/2)/b^5

________________________________________________________________________________________

Rubi [A]  time = 0.23, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, 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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 3770

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

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 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 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^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 = 0.22, size = 174, normalized size = 0.88 \[ \frac {24 a^5 \sin ^{-1}\left (\frac {1}{a+b x}\right )-\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}} \left (-9 \left (4 a^2+1\right ) b^2 x^2+2 a \left (48 a^2+31\right ) b x+a^2 \left (154 a^2+71\right )+16 a b^3 x^3-6 b^4 x^4\right )+3 \left (40 a^4+40 a^2+3\right ) \log \left ((a+b x) \left (\sqrt {\frac {a^2+2 a b x+b^2 x^2-1}{(a+b x)^2}}+1\right )\right )+24 b^5 x^5 \csc ^{-1}(a+b x)}{120 b^5} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 151, normalized size = 0.77 \[ \frac {24 \, b^{5} x^{5} \operatorname {arccsc}\left (b x + a\right ) - 48 \, a^{5} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) - 3 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + {\left (6 \, b^{3} x^{3} - 22 \, a b^{2} x^{2} - 154 \, a^{3} + {\left (58 \, a^{2} + 9\right )} b x - 71 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}}{120 \, b^{5}} \]

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

________________________________________________________________________________________

giac [B]  time = 0.20, size = 408, normalized size = 2.07 \[ -\frac {1}{960} \, b {\left (\frac {192 \, {\left (b x + a\right )}^{5} {\left (\frac {5 \, a}{b x + a} - \frac {10 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {10 \, a^{3}}{{\left (b x + a\right )}^{3}} - \frac {5 \, a^{4}}{{\left (b x + a\right )}^{4}} - 1\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{6}} + \frac {3 \, {\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4} + 40 \, {\left (b x + a\right )}^{3} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 240 \, {\left (b x + a\right )}^{2} a^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 960 \, {\left (b x + a\right )} a^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 360 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 24 \, {\left (40 \, a^{4} + 40 \, a^{2} + 3\right )} \log \left (-{\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} {\left | b x + a \right |}\right ) - \frac {120 \, {\left (8 \, a^{3} + 3 \, a\right )} {\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 24 \, {\left (10 \, a^{2} + 1\right )} {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 40 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 3}{{\left (b x + a\right )}^{4} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{4}}}{b^{6}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/960*b*(192*(b*x + a)^5*(5*a/(b*x + a) - 10*a^2/(b*x + a)^2 + 10*a^3/(b*x + a)^3 - 5*a^4/(b*x + a)^4 - 1)*ar
csin(-1/((b*x + a)*(a/(b*x + a) - 1) - a))/b^6 + (3*(b*x + a)^4*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4 + 40*(b*x + a
)^3*a*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 240*(b*x + a)^2*a^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 960*(b*x + a)*
a^3*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 24*(b*x + a)^2*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 360*(b*x + a)*a*(sqrt(-
1/(b*x + a)^2 + 1) - 1) + 24*(40*a^4 + 40*a^2 + 3)*log(-(sqrt(-1/(b*x + a)^2 + 1) - 1)*abs(b*x + a)) - (120*(8
*a^3 + 3*a)*(b*x + a)^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 24*(10*a^2 + 1)*(b*x + a)^2*(sqrt(-1/(b*x + a)^2 +
1) - 1)^2 + 40*(b*x + a)*a*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 3)/((b*x + a)^4*(sqrt(-1/(b*x + a)^2 + 1) - 1)^4))
/b^6)

________________________________________________________________________________________

maple [B]  time = 0.07, size = 507, normalized size = 2.57 \[ -\frac {77 \left (-1+\left (b x +a \right )^{2}\right ) a^{3}}{60 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {71 \left (-1+\left (b x +a \right )^{2}\right ) a}{120 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {3 \sqrt {-1+\left (b x +a \right )^{2}}\, \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{40 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}-\frac {11 \left (-1+\left (b x +a \right )^{2}\right ) x^{2} a}{60 b^{3} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {29 \left (-1+\left (b x +a \right )^{2}\right ) x \,a^{2}}{60 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\left (-1+\left (b x +a \right )^{2}\right ) x^{3}}{20 b^{2} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{5} \arctan \left (\frac {1}{\sqrt {-1+\left (b x +a \right )^{2}}}\right )}{5 b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{4} \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {3 \left (-1+\left (b x +a \right )^{2}\right ) x}{40 b^{4} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {\sqrt {-1+\left (b x +a \right )^{2}}\, a^{2} \ln \left (b x +a +\sqrt {-1+\left (b x +a \right )^{2}}\right )}{b^{5} \sqrt {\frac {-1+\left (b x +a \right )^{2}}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}+\frac {x^{5} \mathrm {arccsc}\left (b x +a \right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-77/60/b^5*(-1+(b*x+a)^2)/((-1+(b*x+a)^2)/(b*x+a)^2)^(1/2)/(b*x+a)*a^3-71/120/b^5*(-1+(b*x+a)^2)/((-1+(b*x+a)^
2)/(b*x+a)^2)^(1/2)/(b*x+a)*a+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))-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+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+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+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/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
))+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^2*ln(b*x+a+(-1+(b*x+a)^2)^(1/2))+1/5*x^5*arccsc(b*x+a)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int x^4\,\mathrm {asin}\left (\frac {1}{a+b\,x}\right ) \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*asin(1/(a + b*x)),x)

[Out]

int(x^4*asin(1/(a + b*x)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{4} \operatorname {acsc}{\left (a + b x \right )}\, dx \]

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)

________________________________________________________________________________________