Integrand size = 12, antiderivative size = 82 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=-\frac {1}{5} b c^5 \sqrt {1-\frac {1}{c^2 x^2}}+\frac {2}{15} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-\frac {1}{25} b c^5 \left (1-\frac {1}{c^2 x^2}\right )^{5/2}-\frac {a+b \csc ^{-1}(c x)}{5 x^5} \] Output:
-1/5*b*c^5*(1-1/c^2/x^2)^(1/2)+2/15*b*c^5*(1-1/c^2/x^2)^(3/2)-1/25*b*c^5*( 1-1/c^2/x^2)^(5/2)-1/5*(a+b*arccsc(c*x))/x^5
Time = 0.06 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.84 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=-\frac {a}{5 x^5}+b \left (-\frac {8 c^5}{75}-\frac {c}{25 x^4}-\frac {4 c^3}{75 x^2}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{5 x^5} \] Input:
Integrate[(a + b*ArcCsc[c*x])/x^6,x]
Output:
-1/5*a/x^5 + b*((-8*c^5)/75 - c/(25*x^4) - (4*c^3)/(75*x^2))*Sqrt[(-1 + c^ 2*x^2)/(c^2*x^2)] - (b*ArcCsc[c*x])/(5*x^5)
Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5744, 798, 53, 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.
\(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx\) |
\(\Big \downarrow \) 5744 |
\(\displaystyle -\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^7}dx}{5 c}-\frac {a+b \csc ^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^4}d\frac {1}{x^2}}{10 c}-\frac {a+b \csc ^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle \frac {b \int \left (\left (1-\frac {1}{c^2 x^2}\right )^{3/2} c^4-2 \sqrt {1-\frac {1}{c^2 x^2}} c^4+\frac {c^4}{\sqrt {1-\frac {1}{c^2 x^2}}}\right )d\frac {1}{x^2}}{10 c}-\frac {a+b \csc ^{-1}(c x)}{5 x^5}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (-\frac {2}{5} c^6 \left (1-\frac {1}{c^2 x^2}\right )^{5/2}+\frac {4}{3} c^6 \left (1-\frac {1}{c^2 x^2}\right )^{3/2}-2 c^6 \sqrt {1-\frac {1}{c^2 x^2}}\right )}{10 c}-\frac {a+b \csc ^{-1}(c x)}{5 x^5}\) |
Input:
Int[(a + b*ArcCsc[c*x])/x^6,x]
Output:
(b*(-2*c^6*Sqrt[1 - 1/(c^2*x^2)] + (4*c^6*(1 - 1/(c^2*x^2))^(3/2))/3 - (2* c^6*(1 - 1/(c^2*x^2))^(5/2))/5))/(10*c) - (a + b*ArcCsc[c*x])/(5*x^5)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Sim p[(d*x)^(m + 1)*((a + b*ArcCsc[c*x])/(d*(m + 1))), x] + Simp[b*(d/(c*(m + 1 ))) Int[(d*x)^(m - 1)/Sqrt[1 - 1/(c^2*x^2)], x], x] /; FreeQ[{a, b, c, d, m}, x] && NeQ[m, -1]
Time = 0.24 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.96
method | result | size |
parts | \(-\frac {a}{5 x^{5}}+b \,c^{5} \left (-\frac {\operatorname {arccsc}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (8 x^{4} c^{4}+4 c^{2} x^{2}+3\right )}{75 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )\) | \(79\) |
derivativedivides | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (8 x^{4} c^{4}+4 c^{2} x^{2}+3\right )}{75 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )\right )\) | \(83\) |
default | \(c^{5} \left (-\frac {a}{5 c^{5} x^{5}}+b \left (-\frac {\operatorname {arccsc}\left (c x \right )}{5 c^{5} x^{5}}-\frac {\left (c^{2} x^{2}-1\right ) \left (8 x^{4} c^{4}+4 c^{2} x^{2}+3\right )}{75 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{6} x^{6}}\right )\right )\) | \(83\) |
Input:
int((a+b*arccsc(c*x))/x^6,x,method=_RETURNVERBOSE)
Output:
-1/5*a/x^5+b*c^5*(-1/5/c^5/x^5*arccsc(c*x)-1/75*(c^2*x^2-1)*(8*c^4*x^4+4*c ^2*x^2+3)/((c^2*x^2-1)/c^2/x^2)^(1/2)/c^6/x^6)
Time = 0.11 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.61 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=-\frac {15 \, b \operatorname {arccsc}\left (c x\right ) + {\left (8 \, b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 3 \, b\right )} \sqrt {c^{2} x^{2} - 1} + 15 \, a}{75 \, x^{5}} \] Input:
integrate((a+b*arccsc(c*x))/x^6,x, algorithm="fricas")
Output:
-1/75*(15*b*arccsc(c*x) + (8*b*c^4*x^4 + 4*b*c^2*x^2 + 3*b)*sqrt(c^2*x^2 - 1) + 15*a)/x^5
Time = 4.49 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.93 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=- \frac {a}{5 x^{5}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{5 x^{5}} - \frac {b \left (\begin {cases} \frac {8 c^{5} \sqrt {c^{2} x^{2} - 1}}{15 x} + \frac {4 c^{3} \sqrt {c^{2} x^{2} - 1}}{15 x^{3}} + \frac {c \sqrt {c^{2} x^{2} - 1}}{5 x^{5}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {8 i c^{5} \sqrt {- c^{2} x^{2} + 1}}{15 x} + \frac {4 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{15 x^{3}} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{5 x^{5}} & \text {otherwise} \end {cases}\right )}{5 c} \] Input:
integrate((a+b*acsc(c*x))/x**6,x)
Output:
-a/(5*x**5) - b*acsc(c*x)/(5*x**5) - b*Piecewise((8*c**5*sqrt(c**2*x**2 - 1)/(15*x) + 4*c**3*sqrt(c**2*x**2 - 1)/(15*x**3) + c*sqrt(c**2*x**2 - 1)/( 5*x**5), Abs(c**2*x**2) > 1), (8*I*c**5*sqrt(-c**2*x**2 + 1)/(15*x) + 4*I* c**3*sqrt(-c**2*x**2 + 1)/(15*x**3) + I*c*sqrt(-c**2*x**2 + 1)/(5*x**5), T rue))/(5*c)
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.93 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=-\frac {1}{75} \, b {\left (\frac {3 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {5}{2}} - 10 \, c^{6} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} + \frac {15 \, \operatorname {arccsc}\left (c x\right )}{x^{5}}\right )} - \frac {a}{5 \, x^{5}} \] Input:
integrate((a+b*arccsc(c*x))/x^6,x, algorithm="maxima")
Output:
-1/75*b*((3*c^6*(-1/(c^2*x^2) + 1)^(5/2) - 10*c^6*(-1/(c^2*x^2) + 1)^(3/2) + 15*c^6*sqrt(-1/(c^2*x^2) + 1))/c + 15*arccsc(c*x)/x^5) - 1/5*a/x^5
Leaf count of result is larger than twice the leaf count of optimal. 149 vs. \(2 (68) = 136\).
Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.82 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=-\frac {1}{75} \, {\left (3 \, b c^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 10 \, b c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + \frac {15 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right )}{x} + 15 \, b c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {30 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {15 \, b c^{3} \arcsin \left (\frac {1}{c x}\right )}{x} + \frac {15 \, a}{c x^{5}}\right )} c \] Input:
integrate((a+b*arccsc(c*x))/x^6,x, algorithm="giac")
Output:
-1/75*(3*b*c^4*(1/(c^2*x^2) - 1)^2*sqrt(-1/(c^2*x^2) + 1) - 10*b*c^4*(-1/( c^2*x^2) + 1)^(3/2) + 15*b*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x))/x + 15* b*c^4*sqrt(-1/(c^2*x^2) + 1) + 30*b*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/ x + 15*b*c^3*arcsin(1/(c*x))/x + 15*a/(c*x^5))*c
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^6} \,d x \] Input:
int((a + b*asin(1/(c*x)))/x^6,x)
Output:
int((a + b*asin(1/(c*x)))/x^6, x)
\[ \int \frac {a+b \csc ^{-1}(c x)}{x^6} \, dx=\frac {5 \left (\int \frac {\mathit {acsc} \left (c x \right )}{x^{6}}d x \right ) b \,x^{5}-a}{5 x^{5}} \] Input:
int((a+b*acsc(c*x))/x^6,x)
Output:
(5*int(acsc(c*x)/x**6,x)*b*x**5 - a)/(5*x**5)