Integrand size = 19, antiderivative size = 105 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-\frac {b c \left (2 c^2 d+9 e\right ) \sqrt {-1+c^2 x^2}}{9 \sqrt {c^2 x^2}}-\frac {b c d \sqrt {-1+c^2 x^2}}{9 x^2 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{x} \]
-1/3*d*(a+b*arccsc(c*x))/x^3-e*(a+b*arccsc(c*x))/x-1/9*b*c*(2*c^2*d+9*e)*( c^2*x^2-1)^(1/2)/(c^2*x^2)^(1/2)-1/9*b*c*d*(c^2*x^2-1)^(1/2)/x^2/(c^2*x^2) ^(1/2)
Time = 0.09 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.66 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-\frac {3 a \left (d+3 e x^2\right )+b c \sqrt {1-\frac {1}{c^2 x^2}} x \left (d+2 c^2 d x^2+9 e x^2\right )+3 b \left (d+3 e x^2\right ) \csc ^{-1}(c x)}{9 x^3} \]
-1/9*(3*a*(d + 3*e*x^2) + b*c*Sqrt[1 - 1/(c^2*x^2)]*x*(d + 2*c^2*d*x^2 + 9 *e*x^2) + 3*b*(d + 3*e*x^2)*ArcCsc[c*x])/x^3
Time = 0.29 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.96, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {5762, 27, 359, 242}
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.
\(\displaystyle \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx\) |
\(\Big \downarrow \) 5762 |
\(\displaystyle \frac {b c x \int -\frac {3 e x^2+d}{3 x^4 \sqrt {c^2 x^2-1}}dx}{\sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {b c x \int \frac {3 e x^2+d}{x^4 \sqrt {c^2 x^2-1}}dx}{3 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
\(\Big \downarrow \) 359 |
\(\displaystyle -\frac {b c x \left (\frac {1}{3} \left (2 c^2 d+9 e\right ) \int \frac {1}{x^2 \sqrt {c^2 x^2-1}}dx+\frac {d \sqrt {c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {c^2 x^2}}-\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{x}\) |
\(\Big \downarrow \) 242 |
\(\displaystyle -\frac {d \left (a+b \csc ^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{x}-\frac {b c x \left (\frac {\sqrt {c^2 x^2-1} \left (2 c^2 d+9 e\right )}{3 x}+\frac {d \sqrt {c^2 x^2-1}}{3 x^3}\right )}{3 \sqrt {c^2 x^2}}\) |
-1/3*(b*c*x*((d*Sqrt[-1 + c^2*x^2])/(3*x^3) + ((2*c^2*d + 9*e)*Sqrt[-1 + c ^2*x^2])/(3*x)))/Sqrt[c^2*x^2] - (d*(a + b*ArcCsc[c*x]))/(3*x^3) - (e*(a + b*ArcCsc[c*x]))/x
3.1.80.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^ (m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, p}, x ] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(a*e^2*(m + 1)) Int[(e*x)^(m + 2)* (a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && !ILtQ[p, -1]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x _)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Sim p[(a + b*ArcCsc[c*x]) u, x] + Simp[b*c*(x/Sqrt[c^2*x^2]) Int[SimplifyIn tegrand[u/(x*Sqrt[c^2*x^2 - 1]), x], x], x]] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && ((IGtQ[p, 0] && !(ILtQ[(m - 1)/2, 0] && GtQ[m + 2*p + 3, 0])) | | (IGtQ[(m + 1)/2, 0] && !(ILtQ[p, 0] && GtQ[m + 2*p + 3, 0])) || (ILtQ[(m + 2*p + 1)/2, 0] && !ILtQ[(m - 1)/2, 0]))
Time = 0.32 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03
method | result | size |
parts | \(a \left (-\frac {e}{x}-\frac {d}{3 x^{3}}\right )+b \,c^{3} \left (-\frac {\operatorname {arccsc}\left (c x \right ) e}{c^{3} x}-\frac {\operatorname {arccsc}\left (c x \right ) d}{3 x^{3} c^{3}}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{4} d \,x^{2}+9 c^{2} e \,x^{2}+c^{2} d \right )}{9 c^{6} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x^{4}}\right )\) | \(108\) |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{c x}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{4} d \,x^{2}+9 c^{2} e \,x^{2}+c^{2} d \right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{2}}\right )\) | \(121\) |
default | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\operatorname {arccsc}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\operatorname {arccsc}\left (c x \right ) e}{c x}-\frac {\left (c^{2} x^{2}-1\right ) \left (2 c^{4} d \,x^{2}+9 c^{2} e \,x^{2}+c^{2} d \right )}{9 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{2}}\right )\) | \(121\) |
a*(-e/x-1/3*d/x^3)+b*c^3*(-1/c^3*arccsc(c*x)*e/x-1/3*arccsc(c*x)*d/x^3/c^3 -1/9/c^6*(c^2*x^2-1)*(2*c^4*d*x^2+9*c^2*e*x^2+c^2*d)/((c^2*x^2-1)/c^2/x^2) ^(1/2)/x^4)
Time = 0.28 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.63 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-\frac {9 \, a e x^{2} + 3 \, a d + 3 \, {\left (3 \, b e x^{2} + b d\right )} \operatorname {arccsc}\left (c x\right ) + \sqrt {c^{2} x^{2} - 1} {\left ({\left (2 \, b c^{2} d + 9 \, b e\right )} x^{2} + b d\right )}}{9 \, x^{3}} \]
-1/9*(9*a*e*x^2 + 3*a*d + 3*(3*b*e*x^2 + b*d)*arccsc(c*x) + sqrt(c^2*x^2 - 1)*((2*b*c^2*d + 9*b*e)*x^2 + b*d))/x^3
Time = 2.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.44 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=- \frac {a d}{3 x^{3}} - \frac {a e}{x} - b c e \sqrt {1 - \frac {1}{c^{2} x^{2}}} - \frac {b d \operatorname {acsc}{\left (c x \right )}}{3 x^{3}} - \frac {b e \operatorname {acsc}{\left (c x \right )}}{x} - \frac {b d \left (\begin {cases} \frac {2 c^{3} \sqrt {c^{2} x^{2} - 1}}{3 x} + \frac {c \sqrt {c^{2} x^{2} - 1}}{3 x^{3}} & \text {for}\: \left |{c^{2} x^{2}}\right | > 1 \\\frac {2 i c^{3} \sqrt {- c^{2} x^{2} + 1}}{3 x} + \frac {i c \sqrt {- c^{2} x^{2} + 1}}{3 x^{3}} & \text {otherwise} \end {cases}\right )}{3 c} \]
-a*d/(3*x**3) - a*e/x - b*c*e*sqrt(1 - 1/(c**2*x**2)) - b*d*acsc(c*x)/(3*x **3) - b*e*acsc(c*x)/x - b*d*Piecewise((2*c**3*sqrt(c**2*x**2 - 1)/(3*x) + c*sqrt(c**2*x**2 - 1)/(3*x**3), Abs(c**2*x**2) > 1), (2*I*c**3*sqrt(-c**2 *x**2 + 1)/(3*x) + I*c*sqrt(-c**2*x**2 + 1)/(3*x**3), True))/(3*c)
Time = 0.21 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.90 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=-{\left (c \sqrt {-\frac {1}{c^{2} x^{2}} + 1} + \frac {\operatorname {arccsc}\left (c x\right )}{x}\right )} b e + \frac {1}{9} \, b d {\left (\frac {c^{4} {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arccsc}\left (c x\right )}{x^{3}}\right )} - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \]
-(c*sqrt(-1/(c^2*x^2) + 1) + arccsc(c*x)/x)*b*e + 1/9*b*d*((c^4*(-1/(c^2*x ^2) + 1)^(3/2) - 3*c^4*sqrt(-1/(c^2*x^2) + 1))/c - 3*arccsc(c*x)/x^3) - a* e/x - 1/3*a*d/x^3
Time = 0.30 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.30 \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=\frac {1}{9} \, {\left (b c^{2} d {\left (-\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, b c^{2} d \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {3 \, b c d {\left (\frac {1}{c^{2} x^{2}} - 1\right )} \arcsin \left (\frac {1}{c x}\right )}{x} - \frac {3 \, b c d \arcsin \left (\frac {1}{c x}\right )}{x} - 9 \, b e \sqrt {-\frac {1}{c^{2} x^{2}} + 1} - \frac {9 \, b e \arcsin \left (\frac {1}{c x}\right )}{c x} - \frac {9 \, a e}{c x} - \frac {3 \, a d}{c x^{3}}\right )} c \]
1/9*(b*c^2*d*(-1/(c^2*x^2) + 1)^(3/2) - 3*b*c^2*d*sqrt(-1/(c^2*x^2) + 1) - 3*b*c*d*(1/(c^2*x^2) - 1)*arcsin(1/(c*x))/x - 3*b*c*d*arcsin(1/(c*x))/x - 9*b*e*sqrt(-1/(c^2*x^2) + 1) - 9*b*e*arcsin(1/(c*x))/(c*x) - 9*a*e/(c*x) - 3*a*d/(c*x^3))*c
Timed out. \[ \int \frac {\left (d+e x^2\right ) \left (a+b \csc ^{-1}(c x)\right )}{x^4} \, dx=\int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]