Integrand size = 21, antiderivative size = 704 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}+\frac {e^2 \left (a+b \csc ^{-1}(c x)\right )}{4 d^3 \left (e+\frac {d}{x^2}\right )^2}-\frac {e \left (a+b \csc ^{-1}(c x)\right )}{d^3 \left (e+\frac {d}{x^2}\right )}+\frac {i \left (a+b \csc ^{-1}(c x)\right )^2}{2 b d^3}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 d^3} \]
1/4*e^2*(a+b*arccsc(c*x))/d^3/(e+d/x^2)^2-e*(a+b*arccsc(c*x))/d^3/(e+d/x^2 )+1/2*I*(a+b*arccsc(c*x))^2/b/d^3-1/2*(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1 -1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e)^(1/2)))/d^3-1/2*(a+b*arcc sc(c*x))*ln(1+I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e )^(1/2)))/d^3-1/2*(a+b*arccsc(c*x))*ln(1-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*( -d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3-1/2*(a+b*arccsc(c*x))*ln(1+I*c*(I /c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3+1/2*I* b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)-(c^2*d+e) ^(1/2)))/d^3+1/2*I*b*polylog(2,I*c*(I/c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/ (e^(1/2)-(c^2*d+e)^(1/2)))/d^3+1/2*I*b*polylog(2,-I*c*(I/c/x+(1-1/c^2/x^2) ^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3+1/2*I*b*polylog(2,I*c*(I /c/x+(1-1/c^2/x^2)^(1/2))*(-d)^(1/2)/(e^(1/2)+(c^2*d+e)^(1/2)))/d^3-1/8*b* (c^2*d+2*e)*arctan((c^2*d+e)^(1/2)/c/x/e^(1/2)/(1-1/c^2/x^2)^(1/2))*e^(1/2 )/d^3/(c^2*d+e)^(3/2)+b*arctan((c^2*d+e)^(1/2)/c/x/e^(1/2)/(1-1/c^2/x^2)^( 1/2))*e^(1/2)/d^3/(c^2*d+e)^(1/2)-1/8*b*c*e*(1-1/c^2/x^2)^(1/2)/d^2/(c^2*d +e)/(e+d/x^2)/x
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2114\) vs. \(2(704)=1408\).
Time = 6.06 (sec) , antiderivative size = 2114, normalized size of antiderivative = 3.00 \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]
a/(4*d*(d + e*x^2)^2) + a/(2*d^2*(d + e*x^2)) + (a*Log[x])/d^3 - (a*Log[d + e*x^2])/(2*d^3) + b*((((5*I)/16)*Sqrt[e]*(-(ArcCsc[c*x]/((-I)*Sqrt[d]*Sq rt[e] + e*x)) + (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(Sqrt [e] + c*((-I)*c*Sqrt[d] - Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(S qrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d])) /d^(5/2) - (((5*I)/16)*Sqrt[e]*(-(ArcCsc[c*x]/(I*Sqrt[d]*Sqrt[e] + e*x)) - (I*(ArcSin[1/(c*x)]/Sqrt[e] - Log[(2*Sqrt[d]*Sqrt[e]*(-Sqrt[e] + c*((-I)* c*Sqrt[d] + Sqrt[-(c^2*d) - e]*Sqrt[1 - 1/(c^2*x^2)])*x))/(Sqrt[-(c^2*d) - e]*(Sqrt[d] - I*Sqrt[e]*x))]/Sqrt[-(c^2*d) - e]))/Sqrt[d]))/d^(5/2) + (Sq rt[e]*((I*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sq rt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(4*d*Sqrt[e]*Sqrt[c^2* d + e]*(I*Sqrt[e] + c*(c*Sqrt[d] - Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])* x))/((2*c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e)^(3/2))))/( 16*d^2) + (Sqrt[e]*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2 *d + e)*(I*Sqrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[ e]*x)^2) - ArcSin[1/(c*x)]/(d*Sqrt[e]) + (I*(2*c^2*d + e)*Log[(-4*d*Sqrt[e ]*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c*(c*Sqrt[d] + Sqrt[c^2*d + e]*Sqrt[1 - 1/(c^2*x^2)])*x))/((2*c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))])/(d*(c^2*d + e) ^(3/2))))/(16*d^2) - ((I/16)*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2...
Time = 1.83 (sec) , antiderivative size = 764, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5764, 5232, 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.
| \(\displaystyle \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5764 |
| \(\displaystyle -\int \frac {a+b \arcsin \left (\frac {1}{c x}\right )}{\left (\frac {d}{x^2}+e\right )^3 x^5}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle -\int \left (\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) e^2}{d^2 \left (\frac {d}{x^2}+e\right )^3 x}-\frac {2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) e}{d^2 \left (\frac {d}{x^2}+e\right )^2 x}+\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{d^2 \left (\frac {d}{x^2}+e\right ) x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 d^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {c^2 d+e}+\sqrt {e}}\right )}{2 d^3}+\frac {e^2 \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{4 d^3 \left (\frac {d}{x^2}+e\right )^2}-\frac {e \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{d^3 \left (\frac {d}{x^2}+e\right )}+\frac {i \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )^2}{2 b d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}-\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}+\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )}}{\sqrt {e}+\sqrt {d c^2+e}}\right )}{2 d^3}-\frac {b \sqrt {e} \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{8 d^3 \left (c^2 d+e\right )^{3/2}}+\frac {b \sqrt {e} \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} x \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{d^3 \sqrt {c^2 d+e}}-\frac {b c e \sqrt {1-\frac {1}{c^2 x^2}}}{8 d^2 x \left (c^2 d+e\right ) \left (\frac {d}{x^2}+e\right )}\) |
-1/8*(b*c*e*Sqrt[1 - 1/(c^2*x^2)])/(d^2*(c^2*d + e)*(e + d/x^2)*x) + (e^2* (a + b*ArcSin[1/(c*x)]))/(4*d^3*(e + d/x^2)^2) - (e*(a + b*ArcSin[1/(c*x)] ))/(d^3*(e + d/x^2)) + ((I/2)*(a + b*ArcSin[1/(c*x)])^2)/(b*d^3) + (b*Sqrt [e]*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(d^3*Sqrt [c^2*d + e]) - (b*Sqrt[e]*(c^2*d + 2*e)*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]* Sqrt[1 - 1/(c^2*x^2)]*x)])/(8*d^3*(c^2*d + e)^(3/2)) - ((a + b*ArcSin[1/(c *x)])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSi n[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/(2*d^3) - ((a + b*ArcSin[1/(c*x )])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e ])])/(2*d^3) - ((a + b*ArcSin[1/(c*x)])*Log[1 + (I*c*Sqrt[-d]*E^(I*ArcSin[ 1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/(2*d^3) + ((I/2)*b*PolyLog[2, ((- I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/d^3 + ( (I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^ 2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x) ]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^3 + ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]* E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/d^3
3.2.14.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, ( f*x)^m*(d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[p] && IntegerQ[m]
Int[((a_.) + ArcCsc[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_.) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> -Subst[Int[(e + d*x^2)^p*((a + b*ArcSin[x/c])^n/x^( m + 2*(p + 1))), x], x, 1/x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[n, 0] && IntegerQ[m] && IntegerQ[p]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 3.31 (sec) , antiderivative size = 3479, normalized size of antiderivative = 4.94
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(3479\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(3553\) |
| default | \(\text {Expression too large to display}\) | \(3553\) |
a/d^3*ln(x)+1/2*a/d^2/(e*x^2+d)-1/2*a/d^3*ln(e*x^2+d)+1/4*a/d/(e*x^2+d)^2+ b*(-5/4*I*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+2*e ^2)*arccsc(c*x)^2/d^3/(c^4*d^2+2*c^2*d*e+e^2)+I/(c^2*d+e)/d^3*e*arccsc(c*x )^2+1/2*I/(c^2*d+e)/d^2*c^2*sum((_R1^2*c^2*d-2*c^2*d-4*e)/(_R1^2*c^2*d-c^2 *d-2*e)*(I*arccsc(c*x)*ln((_R1-I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)+dilog((_R1- I/c/x-(1-1/c^2/x^2)^(1/2))/_R1)),_R1=RootOf(c^2*d*_Z^4+(-2*c^2*d-4*e)*_Z^2 +c^2*d))+5/4*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2)*e+ 2*e^2)*ln(1-d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d-2*(e*(c^2*d+e))^(1/ 2)+2*e))*arccsc(c*x)/d^3/(c^4*d^2+2*c^2*d*e+e^2)+1/2*I*arccsc(c*x)^2*(c^2* d+2*(e*(c^2*d+e))^(1/2)+2*e)/(c^2*d+e)/d^3-1/8*e*(8*c^6*d^2*arccsc(c*x)*x^ 2+6*c^6*d*e*arccsc(c*x)*x^4+((c^2*x^2-1)/c^2/x^2)^(1/2)*c^5*d^2*x+((c^2*x^ 2-1)/c^2/x^2)^(1/2)*c^5*d*e*x^3-I*c^4*d^2-2*I*c^4*d*e*x^2-I*e^2*c^4*x^4+8* c^4*d*e*arccsc(c*x)*x^2+6*arccsc(c*x)*e^2*c^4*x^4)/d^3/(c^2*d+e)/(c^2*e*x^ 2+c^2*d)^2-5/8*I*((e*(c^2*d+e))^(1/2)*c^2*d+2*c^2*d*e+2*(e*(c^2*d+e))^(1/2 )*e+2*e^2)*polylog(2,d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d-2*(e*(c^2* d+e))^(1/2)+2*e))/d^3/(c^4*d^2+2*c^2*d*e+e^2)+1/4*I*polylog(2,d*c^2*(I/c/x +(1-1/c^2/x^2)^(1/2))^2/(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e))*(c^2*d+2*(e*(c^ 2*d+e))^(1/2)+2*e)/(c^2*d+e)/d^3-1/2*(c^2*d+2*(e*(c^2*d+e))^(1/2)+2*e)*ln( 1-d*c^2*(I/c/x+(1-1/c^2/x^2)^(1/2))^2/(c^2*d-2*(e*(c^2*d+e))^(1/2)+2*e))*a rccsc(c*x)/(c^2*d+e)/d^3+1/2*I/(c^2*d+e)/d^3*e*sum((_R1^2*c^2*d-2*c^2*d...
\[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int { \frac {b \operatorname {arccsc}\left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3} x} \,d x } \]
1/4*a*((2*e*x^2 + 3*d)/(d^2*e^2*x^4 + 2*d^3*e*x^2 + d^4) - 2*log(e*x^2 + d )/d^3 + 4*log(x)/d^3) + b*integrate(arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1) )/(e^3*x^7 + 3*d*e^2*x^5 + 3*d^2*e*x^3 + d^3*x), x)
Exception generated. \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {a+b \csc ^{-1}(c x)}{x \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )}{x\,{\left (e\,x^2+d\right )}^3} \,d x \]