Integrand size = 12, antiderivative size = 76 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=-\frac {b c \sqrt {1-\frac {1}{c^2 x^2}}}{16 x^3}-\frac {3 b c^3 \sqrt {1-\frac {1}{c^2 x^2}}}{32 x}+\frac {3}{32} b c^4 \csc ^{-1}(c x)-\frac {a+b \csc ^{-1}(c x)}{4 x^4} \]
3/32*b*c^4*arccsc(c*x)+1/4*(-a-b*arccsc(c*x))/x^4-1/16*b*c*(1-1/c^2/x^2)^( 1/2)/x^3-3/32*b*c^3*(1-1/c^2/x^2)^(1/2)/x
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.03 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=-\frac {a}{4 x^4}+b \left (-\frac {c}{16 x^3}-\frac {3 c^3}{32 x}\right ) \sqrt {\frac {-1+c^2 x^2}{c^2 x^2}}-\frac {b \csc ^{-1}(c x)}{4 x^4}+\frac {3}{32} b c^4 \arcsin \left (\frac {1}{c x}\right ) \]
-1/4*a/x^4 + b*(-1/16*c/x^3 - (3*c^3)/(32*x))*Sqrt[(-1 + c^2*x^2)/(c^2*x^2 )] - (b*ArcCsc[c*x])/(4*x^4) + (3*b*c^4*ArcSin[1/(c*x)])/32
Time = 0.25 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {5744, 858, 262, 262, 223}
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.
\(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx\) |
\(\Big \downarrow \) 5744 |
\(\displaystyle -\frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^6}dx}{4 c}-\frac {a+b \csc ^{-1}(c x)}{4 x^4}\) |
\(\Big \downarrow \) 858 |
\(\displaystyle \frac {b \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^4}d\frac {1}{x}}{4 c}-\frac {a+b \csc ^{-1}(c x)}{4 x^4}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {b \left (\frac {3}{4} c^2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}} x^2}d\frac {1}{x}-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{4 x^3}\right )}{4 c}-\frac {a+b \csc ^{-1}(c x)}{4 x^4}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {b \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^2 \int \frac {1}{\sqrt {1-\frac {1}{c^2 x^2}}}d\frac {1}{x}-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 x}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{4 x^3}\right )}{4 c}-\frac {a+b \csc ^{-1}(c x)}{4 x^4}\) |
\(\Big \downarrow \) 223 |
\(\displaystyle \frac {b \left (\frac {3}{4} c^2 \left (\frac {1}{2} c^3 \arcsin \left (\frac {1}{c x}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{2 x}\right )-\frac {c^2 \sqrt {1-\frac {1}{c^2 x^2}}}{4 x^3}\right )}{4 c}-\frac {a+b \csc ^{-1}(c x)}{4 x^4}\) |
-1/4*(a + b*ArcCsc[c*x])/x^4 + (b*(-1/4*(c^2*Sqrt[1 - 1/(c^2*x^2)])/x^3 + (3*c^2*(-1/2*(c^2*Sqrt[1 - 1/(c^2*x^2)])/x + (c^3*ArcSin[1/(c*x)])/2))/4)) /(4*c)
3.1.12.3.1 Defintions of rubi rules used
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt [a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^(m + 2), x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] && Int egerQ[m]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(137\) vs. \(2(67)=134\).
Time = 0.36 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.82
method | result | size |
parts | \(-\frac {a}{4 x^{4}}-\frac {b \,\operatorname {arccsc}\left (c x \right )}{4 x^{4}}+\frac {3 b \,c^{3} \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x}-\frac {3 b c \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{3}}-\frac {b \left (c^{2} x^{2}-1\right )}{16 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{5}}\) | \(138\) |
derivativedivides | \(c^{4} \left (-\frac {a}{4 c^{4} x^{4}}-\frac {b \,\operatorname {arccsc}\left (c x \right )}{4 c^{4} x^{4}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {3 b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}-\frac {b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(150\) |
default | \(c^{4} \left (-\frac {a}{4 c^{4} x^{4}}-\frac {b \,\operatorname {arccsc}\left (c x \right )}{4 c^{4} x^{4}}+\frac {3 b \sqrt {c^{2} x^{2}-1}\, \arctan \left (\frac {1}{\sqrt {c^{2} x^{2}-1}}\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}-\frac {3 b \left (c^{2} x^{2}-1\right )}{32 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{3} x^{3}}-\frac {b \left (c^{2} x^{2}-1\right )}{16 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{5} x^{5}}\right )\) | \(150\) |
-1/4*a/x^4-1/4*b/x^4*arccsc(c*x)+3/32*b*c^3*(c^2*x^2-1)^(1/2)/((c^2*x^2-1) /c^2/x^2)^(1/2)/x*arctan(1/(c^2*x^2-1)^(1/2))-3/32*b*c*(c^2*x^2-1)/((c^2*x ^2-1)/c^2/x^2)^(1/2)/x^3-1/16*b/c*(c^2*x^2-1)/((c^2*x^2-1)/c^2/x^2)^(1/2)/ x^5
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.70 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=\frac {{\left (3 \, b c^{4} x^{4} - 8 \, b\right )} \operatorname {arccsc}\left (c x\right ) - {\left (3 \, b c^{2} x^{2} + 2 \, b\right )} \sqrt {c^{2} x^{2} - 1} - 8 \, a}{32 \, x^{4}} \]
Time = 3.46 (sec) , antiderivative size = 194, normalized size of antiderivative = 2.55 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=- \frac {a}{4 x^{4}} - \frac {b \operatorname {acsc}{\left (c x \right )}}{4 x^{4}} - \frac {b \left (\begin {cases} \frac {3 i c^{5} \operatorname {acosh}{\left (\frac {1}{c x} \right )}}{8} - \frac {3 i c^{4}}{8 x \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i c^{2}}{8 x^{3} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} + \frac {i}{4 x^{5} \sqrt {-1 + \frac {1}{c^{2} x^{2}}}} & \text {for}\: \frac {1}{\left |{c^{2} x^{2}}\right |} > 1 \\- \frac {3 c^{5} \operatorname {asin}{\left (\frac {1}{c x} \right )}}{8} + \frac {3 c^{4}}{8 x \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {c^{2}}{8 x^{3} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} - \frac {1}{4 x^{5} \sqrt {1 - \frac {1}{c^{2} x^{2}}}} & \text {otherwise} \end {cases}\right )}{4 c} \]
-a/(4*x**4) - b*acsc(c*x)/(4*x**4) - b*Piecewise((3*I*c**5*acosh(1/(c*x))/ 8 - 3*I*c**4/(8*x*sqrt(-1 + 1/(c**2*x**2))) + I*c**2/(8*x**3*sqrt(-1 + 1/( c**2*x**2))) + I/(4*x**5*sqrt(-1 + 1/(c**2*x**2))), 1/Abs(c**2*x**2) > 1), (-3*c**5*asin(1/(c*x))/8 + 3*c**4/(8*x*sqrt(1 - 1/(c**2*x**2))) - c**2/(8 *x**3*sqrt(1 - 1/(c**2*x**2))) - 1/(4*x**5*sqrt(1 - 1/(c**2*x**2))), True) )/(4*c)
Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.64 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=-\frac {1}{32} \, b {\left (\frac {3 \, c^{5} \arctan \left (c x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}\right ) + \frac {3 \, c^{8} x^{3} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} + 5 \, c^{6} x \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{4} x^{4} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} - 2 \, c^{2} x^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + 1}}{c} + \frac {8 \, \operatorname {arccsc}\left (c x\right )}{x^{4}}\right )} - \frac {a}{4 \, x^{4}} \]
-1/32*b*((3*c^5*arctan(c*x*sqrt(-1/(c^2*x^2) + 1)) + (3*c^8*x^3*(-1/(c^2*x ^2) + 1)^(3/2) + 5*c^6*x*sqrt(-1/(c^2*x^2) + 1))/(c^4*x^4*(1/(c^2*x^2) - 1 )^2 - 2*c^2*x^2*(1/(c^2*x^2) - 1) + 1))/c + 8*arccsc(c*x)/x^4) - 1/4*a/x^4
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=-\frac {1}{32} \, {\left (8 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} \arcsin \left (\frac {1}{c x}\right ) + 16 \, b c^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right ) + 5 \, b c^{3} \arcsin \left (\frac {1}{c x}\right ) - \frac {2 \, b c^{2} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}}}{x} + \frac {5 \, b c^{2} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{x} + \frac {8 \, a}{c x^{4}}\right )} c \]
-1/32*(8*b*c^3*(1/(c^2*x^2) - 1)^2*arcsin(1/(c*x)) + 16*b*c^3*(1/(c^2*x^2) - 1)*arcsin(1/(c*x)) + 5*b*c^3*arcsin(1/(c*x)) - 2*b*c^2*(-1/(c^2*x^2) + 1)^(3/2)/x + 5*b*c^2*sqrt(-1/(c^2*x^2) + 1)/x + 8*a/(c*x^4))*c
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x^5} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x^5} \,d x \]