\(\int \frac {x^7 (a+b \csc ^{-1}(c x))}{\sqrt {1-c^4 x^4}} \, dx\) [175]

Optimal result

Integrand size = 26, antiderivative size = 268 \[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {b \sqrt {1-c^2 x^2} \sqrt {1+c^2 x^2}}{3 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}+\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{3/2}}{18 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {b \sqrt {1-c^2 x^2} \left (1+c^2 x^2\right )^{5/2}}{30 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}+\frac {b \sqrt {1-c^2 x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{3 c^9 \sqrt {1-\frac {1}{c^2 x^2}} x} \] Output:

-1/3*b*(-c^2*x^2+1)^(1/2)*(c^2*x^2+1)^(1/2)/c^9/(1-1/c^2/x^2)^(1/2)/x+1/18 
*b*(-c^2*x^2+1)^(1/2)*(c^2*x^2+1)^(3/2)/c^9/(1-1/c^2/x^2)^(1/2)/x-1/30*b*( 
-c^2*x^2+1)^(1/2)*(c^2*x^2+1)^(5/2)/c^9/(1-1/c^2/x^2)^(1/2)/x-1/2*(-c^4*x^ 
4+1)^(1/2)*(a+b*arccsc(c*x))/c^8+1/6*(-c^4*x^4+1)^(3/2)*(a+b*arccsc(c*x))/ 
c^8+1/3*b*(-c^2*x^2+1)^(1/2)*arctanh((c^2*x^2+1)^(1/2))/c^9/(1-1/c^2/x^2)^ 
(1/2)/x
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.59 \[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {15 a \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right )+\frac {b c \sqrt {1-\frac {1}{c^2 x^2}} x \sqrt {1-c^4 x^4} \left (28+c^2 x^2+3 c^4 x^4\right )}{-1+c^2 x^2}+15 b \sqrt {1-c^4 x^4} \left (2+c^4 x^4\right ) \csc ^{-1}(c x)+30 b \arctan \left (\frac {c \sqrt {1-\frac {1}{c^2 x^2}} x}{\sqrt {1-c^4 x^4}}\right )}{90 c^8} \] Input:

Integrate[(x^7*(a + b*ArcCsc[c*x]))/Sqrt[1 - c^4*x^4],x]
 

Output:

-1/90*(15*a*Sqrt[1 - c^4*x^4]*(2 + c^4*x^4) + (b*c*Sqrt[1 - 1/(c^2*x^2)]*x 
*Sqrt[1 - c^4*x^4]*(28 + c^2*x^2 + 3*c^4*x^4))/(-1 + c^2*x^2) + 15*b*Sqrt[ 
1 - c^4*x^4]*(2 + c^4*x^4)*ArcCsc[c*x] + 30*b*ArcTan[(c*Sqrt[1 - 1/(c^2*x^ 
2)]*x)/Sqrt[1 - c^4*x^4]])/c^8
 

Rubi [A] (warning: unable to verify)

Time = 1.46 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5770, 27, 7272, 1388, 1579, 517, 25, 1584, 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.

\(\displaystyle \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx\)

\(\Big \downarrow \) 5770

\(\displaystyle \frac {b \int -\frac {\sqrt {1-c^4 x^4} \left (c^4 x^4+2\right )}{6 c^8 \sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{c}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b \int \frac {\sqrt {1-c^4 x^4} \left (c^4 x^4+2\right )}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}dx}{6 c^9}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 7272

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {1-c^4 x^4} \left (c^4 x^4+2\right )}{x \sqrt {1-c^2 x^2}}dx}{6 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 1388

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {c^2 x^2+1} \left (c^4 x^4+2\right )}{x}dx}{6 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 1579

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int \frac {\sqrt {c^2 x^2+1} \left (c^4 x^4+2\right )}{x^2}dx^2}{12 c^9 x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 517

\(\displaystyle -\frac {b \sqrt {1-c^2 x^2} \int -\frac {x^4 \left (c^4 x^8-2 c^4 x^4+3 c^4\right )}{1-x^4}d\sqrt {c^2 x^2+1}}{6 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {b \sqrt {1-c^2 x^2} \int \frac {x^4 \left (c^4 x^8-2 c^4 x^4+3 c^4\right )}{1-x^4}d\sqrt {c^2 x^2+1}}{6 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 1584

\(\displaystyle \frac {b \sqrt {1-c^2 x^2} \int \left (-c^4 x^8+c^4 x^4-2 c^4+\frac {2 c^4}{1-x^4}\right )d\sqrt {c^2 x^2+1}}{6 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}+\frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\left (1-c^4 x^4\right )^{3/2} \left (a+b \csc ^{-1}(c x)\right )}{6 c^8}-\frac {\sqrt {1-c^4 x^4} \left (a+b \csc ^{-1}(c x)\right )}{2 c^8}-\frac {b \sqrt {1-c^2 x^2} \left (-2 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {c^4 x^{10}}{5}-\frac {c^4 x^6}{3}+2 c^4 \sqrt {c^2 x^2+1}\right )}{6 c^{13} x \sqrt {1-\frac {1}{c^2 x^2}}}\)

Input:

Int[(x^7*(a + b*ArcCsc[c*x]))/Sqrt[1 - c^4*x^4],x]
 

Output:

-1/2*(Sqrt[1 - c^4*x^4]*(a + b*ArcCsc[c*x]))/c^8 + ((1 - c^4*x^4)^(3/2)*(a 
 + b*ArcCsc[c*x]))/(6*c^8) - (b*Sqrt[1 - c^2*x^2]*(-1/3*(c^4*x^6) + (c^4*x 
^10)/5 + 2*c^4*Sqrt[1 + c^2*x^2] - 2*c^4*ArcTanh[Sqrt[1 + c^2*x^2]]))/(6*c 
^13*Sqrt[1 - 1/(c^2*x^2)]*x)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 517

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> Simp[2*(e^m/d^(m + 2*p + 1))   Subst[Int[x^(2*n + 1)*(-c + x^ 
2)^m*(b*c^2 + a*d^2 - 2*b*c*x^2 + b*x^4)^p, x], x, Sqrt[c + d*x]], x] /; Fr 
eeQ[{a, b, c, d, e}, x] && IGtQ[p, 0] && ILtQ[m, 0] && IntegerQ[n + 1/2]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

rule 1579

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + c*x^2)^p, x], 
 x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x] && IntegerQ[(m + 1)/2]
 

rule 1584

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* 
(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ 
b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5770

Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*(u_), x_Symbol] :> With[{v = IntHide 
[u, x]}, Simp[(a + b*ArcCsc[c*x])   v, x] + Simp[b/c   Int[SimplifyIntegran 
d[v/(x^2*Sqrt[1 - 1/(c^2*x^2)]), x], x], x] /; InverseFunctionFreeQ[v, x]] 
/; FreeQ[{a, b, c}, x]
 

rule 7272

Int[(u_.)*((a_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[b^IntPart[p]*(( 
a + b*x^n)^FracPart[p]/(x^(n*FracPart[p])*(1 + a*(1/(x^n*b)))^FracPart[p])) 
   Int[u*x^(n*p)*(1 + a*(1/(x^n*b)))^p, x], x] /; FreeQ[{a, b, p}, x] &&  ! 
IntegerQ[p] && ILtQ[n, 0] &&  !RationalFunctionQ[u, x] && IntegerQ[p + 1/2]
 

Maple [F]

\[\int \frac {x^{7} \left (a +b \,\operatorname {arccsc}\left (c x \right )\right )}{\sqrt {-x^{4} c^{4}+1}}d x\]

Input:

int(x^7*(a+b*arccsc(c*x))/(-c^4*x^4+1)^(1/2),x)
 

Output:

int(x^7*(a+b*arccsc(c*x))/(-c^4*x^4+1)^(1/2),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.72 \[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=-\frac {{\left (3 \, b c^{4} x^{4} + b c^{2} x^{2} + 28 \, b\right )} \sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1} - 30 \, {\left (b c^{2} x^{2} - b\right )} \arctan \left (\frac {\sqrt {-c^{4} x^{4} + 1} \sqrt {c^{2} x^{2} - 1}}{c^{4} x^{4} - 1}\right ) + 15 \, {\left (a c^{6} x^{6} - a c^{4} x^{4} + 2 \, a c^{2} x^{2} + {\left (b c^{6} x^{6} - b c^{4} x^{4} + 2 \, b c^{2} x^{2} - 2 \, b\right )} \operatorname {arccsc}\left (c x\right ) - 2 \, a\right )} \sqrt {-c^{4} x^{4} + 1}}{90 \, {\left (c^{10} x^{2} - c^{8}\right )}} \] Input:

integrate(x^7*(a+b*arccsc(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="fricas")
 

Output:

-1/90*((3*b*c^4*x^4 + b*c^2*x^2 + 28*b)*sqrt(-c^4*x^4 + 1)*sqrt(c^2*x^2 - 
1) - 30*(b*c^2*x^2 - b)*arctan(sqrt(-c^4*x^4 + 1)*sqrt(c^2*x^2 - 1)/(c^4*x 
^4 - 1)) + 15*(a*c^6*x^6 - a*c^4*x^4 + 2*a*c^2*x^2 + (b*c^6*x^6 - b*c^4*x^ 
4 + 2*b*c^2*x^2 - 2*b)*arccsc(c*x) - 2*a)*sqrt(-c^4*x^4 + 1))/(c^10*x^2 - 
c^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Timed out} \] Input:

integrate(x**7*(a+b*acsc(c*x))/(-c**4*x**4+1)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{7}}{\sqrt {-c^{4} x^{4} + 1}} \,d x } \] Input:

integrate(x^7*(a+b*arccsc(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="maxima")
 

Output:

1/6*a*((-c^4*x^4 + 1)^(3/2)/c^8 - 3*sqrt(-c^4*x^4 + 1)/c^8) + 1/6*(c^8*x^8 
*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + 6*sqrt(c^2*x^2 + 1)*sqrt(c*x + 
1)*sqrt(-c*x + 1)*c^8*integrate(1/6*(c^6*x^7 + c^4*x^5 + 2*c^2*x^3 + 2*x)* 
e^(-1/2*log(c^2*x^2 + 1) + 1/2*log(c*x - 1))/(c^6*e^(log(c*x + 1) + log(c* 
x - 1) + 1/2*log(-c*x + 1)) + sqrt(-c*x + 1)*c^6), x) + c^4*x^4*arctan2(1, 
 sqrt(c*x + 1)*sqrt(c*x - 1)) - 2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))) 
*b/(sqrt(c^2*x^2 + 1)*sqrt(c*x + 1)*sqrt(-c*x + 1)*c^8)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^7*(a+b*arccsc(c*x))/(-c^4*x^4+1)^(1/2),x, algorithm="giac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\int \frac {x^7\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{\sqrt {1-c^4\,x^4}} \,d x \] Input:

int((x^7*(a + b*asin(1/(c*x))))/(1 - c^4*x^4)^(1/2),x)
 

Output:

int((x^7*(a + b*asin(1/(c*x))))/(1 - c^4*x^4)^(1/2), x)
 

Reduce [F]

\[ \int \frac {x^7 \left (a+b \csc ^{-1}(c x)\right )}{\sqrt {1-c^4 x^4}} \, dx=\frac {-\sqrt {-c^{4} x^{4}+1}\, a \,c^{4} x^{4}-2 \sqrt {-c^{4} x^{4}+1}\, a -6 \left (\int \frac {\sqrt {-c^{4} x^{4}+1}\, \mathit {acsc} \left (c x \right ) x^{7}}{c^{4} x^{4}-1}d x \right ) b \,c^{8}}{6 c^{8}} \] Input:

int(x^7*(a+b*acsc(c*x))/(-c^4*x^4+1)^(1/2),x)
 

Output:

( - sqrt( - c**4*x**4 + 1)*a*c**4*x**4 - 2*sqrt( - c**4*x**4 + 1)*a - 6*in 
t((sqrt( - c**4*x**4 + 1)*acsc(c*x)*x**7)/(c**4*x**4 - 1),x)*b*c**8)/(6*c* 
*8)