Integrand size = 10, antiderivative size = 155 \[ \int x^3 \csc ^{-1}(a+b x) \, dx=\frac {\left (2+17 a^2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^4}+\frac {x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}}{12 b^2}-\frac {a (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}}}{3 b^4}-\frac {a^4 \csc ^{-1}(a+b x)}{4 b^4}+\frac {1}{4} x^4 \csc ^{-1}(a+b x)-\frac {a \left (1+2 a^2\right ) \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )}{2 b^4} \] Output:
1/12*(17*a^2+2)*(b*x+a)*(1-1/(b*x+a)^2)^(1/2)/b^4+1/12*x^2*(b*x+a)*(1-1/(b *x+a)^2)^(1/2)/b^2-1/3*a*(b*x+a)^2*(1-1/(b*x+a)^2)^(1/2)/b^4-1/4*a^4*arccs c(b*x+a)/b^4+1/4*x^4*arccsc(b*x+a)-1/2*a*(2*a^2+1)*arctanh((1-1/(b*x+a)^2) ^(1/2))/b^4
Time = 0.18 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int x^3 \csc ^{-1}(a+b x) \, dx=\frac {\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}} \left (2 a+13 a^3+2 b x+9 a^2 b x-3 a b^2 x^2+b^3 x^3\right )+3 b^4 x^4 \csc ^{-1}(a+b x)-3 a^4 \arcsin \left (\frac {1}{a+b x}\right )-6 a \left (1+2 a^2\right ) \log \left ((a+b x) \left (1+\sqrt {\frac {-1+a^2+2 a b x+b^2 x^2}{(a+b x)^2}}\right )\right )}{12 b^4} \] Input:
Integrate[x^3*ArcCsc[a + b*x],x]
Output:
(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*Arc Sin[(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)
Time = 0.54 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.02, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {5782, 25, 4927, 3042, 4269, 3042, 4536, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^3 \csc ^{-1}(a+b x) \, dx\) |
| \(\Big \downarrow \) 5782 |
| \(\displaystyle -\frac {\int b^3 x^3 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int -b^3 x^3 (a+b x)^2 \sqrt {1-\frac {1}{(a+b x)^2}} \csc ^{-1}(a+b x)d\csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4927 |
| \(\displaystyle -\frac {\frac {1}{4} \int b^4 x^4d\csc ^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{4} \int \left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right )^4d\csc ^{-1}(a+b x)-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4269 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \int -b x \left (3 a^3+8 (a+b x)^2 a-\left (9 a^2+2\right ) (a+b x)\right )d\csc ^{-1}(a+b x)-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \int \left (a-\csc \left (\csc ^{-1}(a+b x)\right )\right ) \left (3 a^3+8 \csc \left (\csc ^{-1}(a+b x)\right )^2 a-\left (9 a^2+2\right ) \csc \left (\csc ^{-1}(a+b x)\right )\right )d\csc ^{-1}(a+b x)-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 4536 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \int \left (6 a^4-12 \left (2 a^2+1\right ) (a+b x) a+2 \left (17 a^2+2\right ) (a+b x)^2\right )d\csc ^{-1}(a+b x)+4 a \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)^2\right )-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)}{b^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{4} \left (\frac {1}{3} \left (\frac {1}{2} \left (6 a^4 \csc ^{-1}(a+b x)+12 \left (2 a^2+1\right ) a \text {arctanh}\left (\sqrt {1-\frac {1}{(a+b x)^2}}\right )-2 \left (17 a^2+2\right ) (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )+4 a \sqrt {1-\frac {1}{(a+b x)^2}} (a+b x)^2\right )-\frac {1}{3} b^2 x^2 (a+b x) \sqrt {1-\frac {1}{(a+b x)^2}}\right )-\frac {1}{4} b^4 x^4 \csc ^{-1}(a+b x)}{b^4}\) |
Input:
Int[x^3*ArcCsc[a + b*x],x]
Output:
-((-1/4*(b^4*x^4*ArcCsc[a + b*x]) + (-1/3*(b^2*x^2*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)]) + (4*a*(a + b*x)^2*Sqrt[1 - (a + b*x)^(-2)] + (-2*(2 + 17*a^2 )*(a + b*x)*Sqrt[1 - (a + b*x)^(-2)] + 6*a^4*ArcCsc[a + b*x] + 12*a*(1 + 2 *a^2)*ArcTanh[Sqrt[1 - (a + b*x)^(-2)]])/2)/3)/4)/b^4)
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Simp[(-b^2)*C ot[c + d*x]*((a + b*Csc[c + d*x])^(n - 2)/(d*(n - 1))), x] + Simp[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]
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] + Simp[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]
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] + Simp[f*(m/(b*d*( n + 1))) Int[(e + f*x)^(m - 1)*(a + b*Csc[c + d*x])^(n + 1), x], x] /; Fr eeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && NeQ[n, -1]
Int[((a_.) + ArcCsc[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m _.), x_Symbol] :> Simp[-(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]
Time = 0.22 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.61
| method | result | size |
| derivativedivides | \(\frac {\frac {\operatorname {arccsc}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccsc}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccsc}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (3 a^{4} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+12 a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-\left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-2 \sqrt {\left (b x +a \right )^{2}-1}\right )}{12 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}}\) | \(249\) |
| default | \(\frac {\frac {\operatorname {arccsc}\left (b x +a \right ) a^{4}}{4}-\operatorname {arccsc}\left (b x +a \right ) a^{3} \left (b x +a \right )+\frac {3 \,\operatorname {arccsc}\left (b x +a \right ) a^{2} \left (b x +a \right )^{2}}{2}-\operatorname {arccsc}\left (b x +a \right ) a \left (b x +a \right )^{3}+\frac {\operatorname {arccsc}\left (b x +a \right ) \left (b x +a \right )^{4}}{4}-\frac {\sqrt {\left (b x +a \right )^{2}-1}\, \left (3 a^{4} \arctan \left (\frac {1}{\sqrt {\left (b x +a \right )^{2}-1}}\right )+12 a^{3} \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-18 a^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 a \left (b x +a \right ) \sqrt {\left (b x +a \right )^{2}-1}-\left (b x +a \right )^{2} \sqrt {\left (b x +a \right )^{2}-1}+6 a \ln \left (b x +a +\sqrt {\left (b x +a \right )^{2}-1}\right )-2 \sqrt {\left (b x +a \right )^{2}-1}\right )}{12 \sqrt {\frac {\left (b x +a \right )^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right )}}{b^{4}}\) | \(249\) |
| parts | \(\frac {x^{4} \operatorname {arccsc}\left (b x +a \right )}{4}-\frac {\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (3 a^{4} \arctan \left (\frac {1}{\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}\right ) \sqrt {b^{2}}-x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, b^{2} \sqrt {b^{2}}+12 \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) a^{3} b +4 \sqrt {b^{2}}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a b x -13 \sqrt {b^{2}}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}+6 a \ln \left (\frac {b^{2} x +\sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}+a b}{\sqrt {b^{2}}}\right ) b -2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \sqrt {b^{2}}\right )}{12 b^{4} \sqrt {\frac {b^{2} x^{2}+2 a b x +a^{2}-1}{\left (b x +a \right )^{2}}}\, \left (b x +a \right ) \sqrt {b^{2}}}\) | \(315\) |
Input:
int(x^3*arccsc(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b^4*(1/4*arccsc(b*x+a)*a^4-arccsc(b*x+a)*a^3*(b*x+a)+3/2*arccsc(b*x+a)*a ^2*(b*x+a)^2-arccsc(b*x+a)*a*(b*x+a)^3+1/4*arccsc(b*x+a)*(b*x+a)^4-1/12*(( b*x+a)^2-1)^(1/2)*(3*a^4*arctan(1/((b*x+a)^2-1)^(1/2))+12*a^3*ln(b*x+a+((b *x+a)^2-1)^(1/2))-18*a^2*((b*x+a)^2-1)^(1/2)+6*a*(b*x+a)*((b*x+a)^2-1)^(1/ 2)-(b*x+a)^2*((b*x+a)^2-1)^(1/2)+6*a*ln(b*x+a+((b*x+a)^2-1)^(1/2))-2*((b*x +a)^2-1)^(1/2))/(((b*x+a)^2-1)/(b*x+a)^2)^(1/2)/(b*x+a))
Time = 0.13 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.83 \[ \int x^3 \csc ^{-1}(a+b x) \, dx=\frac {3 \, b^{4} x^{4} \operatorname {arccsc}\left (b x + a\right ) + 6 \, a^{4} \arctan \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + 6 \, {\left (2 \, a^{3} + a\right )} \log \left (-b x - a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right ) + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} - 1} {\left (b^{2} x^{2} - 4 \, a b x + 13 \, a^{2} + 2\right )}}{12 \, b^{4}} \] Input:
integrate(x^3*arccsc(b*x+a),x, algorithm="fricas")
Output:
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)*log(-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
\[ \int x^3 \csc ^{-1}(a+b x) \, dx=\int x^{3} \operatorname {acsc}{\left (a + b x \right )}\, dx \] Input:
integrate(x**3*acsc(b*x+a),x)
Output:
Integral(x**3*acsc(a + b*x), x)
\[ \int x^3 \csc ^{-1}(a+b x) \, dx=\int { x^{3} \operatorname {arccsc}\left (b x + a\right ) \,d x } \] Input:
integrate(x^3*arccsc(b*x+a),x, algorithm="maxima")
Output:
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) + log (b*x + a - 1)) - 1), x)
Leaf count of result is larger than twice the leaf count of optimal. 303 vs. \(2 (135) = 270\).
Time = 0.16 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.95 \[ \int x^3 \csc ^{-1}(a+b x) \, dx=-\frac {1}{96} \, b {\left (\frac {24 \, {\left (b x + a\right )}^{4} {\left (\frac {4 \, a}{b x + a} - \frac {6 \, a^{2}}{{\left (b x + a\right )}^{2}} + \frac {4 \, a^{3}}{{\left (b x + a\right )}^{3}} - 1\right )} \arcsin \left (-\frac {1}{{\left (b x + a\right )} {\left (\frac {a}{b x + a} - 1\right )} - a}\right )}{b^{5}} - \frac {{\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3} + 12 \, {\left (b x + a\right )}^{2} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 72 \, {\left (b x + a\right )} a^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 9 \, {\left (b x + a\right )} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 48 \, {\left (2 \, a^{3} + a\right )} \log \left (\frac {1}{2} \, {\left | b x + a \right |} {\left | -2 \, \sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} + 2 \right |}\right ) - \frac {9 \, {\left (8 \, a^{2} + 1\right )} {\left (b x + a\right )}^{2} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{2} + 12 \, {\left (b x + a\right )} a {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )} + 1}{{\left (b x + a\right )}^{3} {\left (\sqrt {-\frac {1}{{\left (b x + a\right )}^{2}} + 1} - 1\right )}^{3}}}{b^{5}}\right )} \] Input:
integrate(x^3*arccsc(b*x+a),x, algorithm="giac")
Output:
-1/96*b*(24*(b*x + a)^4*(4*a/(b*x + a) - 6*a^2/(b*x + a)^2 + 4*a^3/(b*x + a)^3 - 1)*arcsin(-1/((b*x + a)*(a/(b*x + a) - 1) - a))/b^5 - ((b*x + a)^3* (sqrt(-1/(b*x + a)^2 + 1) - 1)^3 + 12*(b*x + a)^2*a*(sqrt(-1/(b*x + a)^2 + 1) - 1)^2 + 72*(b*x + a)*a^2*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 9*(b*x + a) *(sqrt(-1/(b*x + a)^2 + 1) - 1) + 48*(2*a^3 + a)*log(1/2*abs(b*x + a)*abs( -2*sqrt(-1/(b*x + a)^2 + 1) + 2)) - (9*(8*a^2 + 1)*(b*x + a)^2*(sqrt(-1/(b *x + a)^2 + 1) - 1)^2 + 12*(b*x + a)*a*(sqrt(-1/(b*x + a)^2 + 1) - 1) + 1) /((b*x + a)^3*(sqrt(-1/(b*x + a)^2 + 1) - 1)^3))/b^5)
Timed out. \[ \int x^3 \csc ^{-1}(a+b x) \, dx=\int x^3\,\mathrm {asin}\left (\frac {1}{a+b\,x}\right ) \,d x \] Input:
int(x^3*asin(1/(a + b*x)),x)
Output:
int(x^3*asin(1/(a + b*x)), x)
\[ \int x^3 \csc ^{-1}(a+b x) \, dx=\int \mathit {acsc} \left (b x +a \right ) x^{3}d x \] Input:
int(x^3*acsc(b*x+a),x)
Output:
int(acsc(a + b*x)*x**3,x)