Integrand size = 21, antiderivative size = 727 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\frac {b c d \sqrt {1-\frac {1}{c^2 x^2}}}{8 e^2 \left (c^2 d+e\right ) \left (e+\frac {d}{x^2}\right ) x}-\frac {a+b \csc ^{-1}(c x)}{4 e \left (e+\frac {d}{x^2}\right )^2}-\frac {a+b \csc ^{-1}(c x)}{2 e^2 \left (e+\frac {d}{x^2}\right )}+\frac {b \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {c^2 d+e}}+\frac {b \left (c^2 d+2 e\right ) \arctan \left (\frac {\sqrt {c^2 d+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (c^2 d+e\right )^{3/2}}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1+\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {\left (a+b \csc ^{-1}(c x)\right ) \log \left (1-e^{2 i \csc ^{-1}(c x)}\right )}{e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}-\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,-\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}-\frac {i b \operatorname {PolyLog}\left (2,\frac {i c \sqrt {-d} e^{i \csc ^{-1}(c x)}}{\sqrt {e}+\sqrt {c^2 d+e}}\right )}{2 e^3}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \csc ^{-1}(c x)}\right )}{2 e^3} \]
1/4*(-a-b*arccsc(c*x))/e/(e+d/x^2)^2+1/2*(-a-b*arccsc(c*x))/e^2/(e+d/x^2)+ 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^(5/2)/(c^2*d+e)^(3/2)-(a+b*arccsc(c*x))*ln(1-(I/c/x+(1-1/c^2/x^2)^(1/2)) ^2)/e^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)))/e^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)))/e^3+1/2*(a+b*arc csc(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)))/e^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)))/e^3+1/2*I*b*polylog(2,(I/c/x+(1-1/c^ 2/x^2)^(1/2))^2)/e^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)))/e^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)))/e^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)))/e ^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)))/e^3+1/2*b*arctan((c^2*d+e)^(1/2)/c/x/e^(1/2)/(1-1/c^2/x^2 )^(1/2))/e^(5/2)/(c^2*d+e)^(1/2)+1/8*b*c*d*(1-1/c^2/x^2)^(1/2)/e^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. \(2053\) vs. \(2(727)=1454\).
Time = 7.39 (sec) , antiderivative size = 2053, normalized size of antiderivative = 2.82 \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Result too large to show} \]
-1/4*(a*d^2)/(e^3*(d + e*x^2)^2) + (a*d)/(e^3*(d + e*x^2)) + (a*Log[d + e* x^2])/(2*e^3) + b*((((7*I)/16)*Sqrt[d]*(-(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]))/e^( 5/2) - (((7*I)/16)*Sqrt[d]*(-(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*Sq rt[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]))/e^(5/2) - (d*((I* c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*((-I)*Sqrt[d] + Sq rt[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*e^(5/2) ) - (d*(((-I)*c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)/(Sqrt[d]*(c^2*d + e)*(I*S qrt[d] + Sqrt[e]*x)) - ArcCsc[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - A rcSin[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))))/(1 6*e^(5/2)) + ((I/16)*(Pi^2 - 4*Pi*ArcCsc[c*x] + 8*ArcCsc[c*x]^2 - 32*Ar...
Time = 1.93 (sec) , antiderivative size = 795, 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.
| \(\displaystyle \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5764 |
| \(\displaystyle -\int \frac {x \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{\left (\frac {d}{x^2}+e\right )^3}d\frac {1}{x}\) |
| \(\Big \downarrow \) 5232 |
| \(\displaystyle -\int \left (\frac {x \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^3}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^3 \left (\frac {d}{x^2}+e\right ) x}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e^2 \left (\frac {d}{x^2}+e\right )^2 x}-\frac {d \left (a+b \arcsin \left (\frac {1}{c x}\right )\right )}{e \left (\frac {d}{x^2}+e\right )^3 x}\right )d\frac {1}{x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b c \sqrt {1-\frac {1}{c^2 x^2}} d}{8 e^2 \left (d c^2+e\right ) \left (\frac {d}{x^2}+e\right ) x}-\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{2 e^2 \left (\frac {d}{x^2}+e\right )}-\frac {a+b \arcsin \left (\frac {1}{c x}\right )}{4 e \left (\frac {d}{x^2}+e\right )^2}+\frac {b \left (d c^2+2 e\right ) \arctan \left (\frac {\sqrt {d c^2+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{8 e^{5/2} \left (d c^2+e\right )^{3/2}}+\frac {b \arctan \left (\frac {\sqrt {d c^2+e}}{c \sqrt {e} \sqrt {1-\frac {1}{c^2 x^2}} x}\right )}{2 e^{5/2} \sqrt {d c^2+e}}+\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 {d c^2+e}}\right )}{2 e^3}+\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (\frac {i \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )} c}{\sqrt {e}-\sqrt {d c^2+e}}+1\right )}{2 e^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 {d c^2+e}}\right )}{2 e^3}+\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (\frac {i \sqrt {-d} e^{i \arcsin \left (\frac {1}{c x}\right )} c}{\sqrt {e}+\sqrt {d c^2+e}}+1\right )}{2 e^3}-\frac {\left (a+b \arcsin \left (\frac {1}{c x}\right )\right ) \log \left (1-e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{e^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 e^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 e^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 e^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 e^3}+\frac {i b \operatorname {PolyLog}\left (2,e^{2 i \arcsin \left (\frac {1}{c x}\right )}\right )}{2 e^3}\) |
(b*c*d*Sqrt[1 - 1/(c^2*x^2)])/(8*e^2*(c^2*d + e)*(e + d/x^2)*x) - (a + b*A rcSin[1/(c*x)])/(4*e*(e + d/x^2)^2) - (a + b*ArcSin[1/(c*x)])/(2*e^2*(e + d/x^2)) + (b*ArcTan[Sqrt[c^2*d + e]/(c*Sqrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/ (2*e^(5/2)*Sqrt[c^2*d + e]) + (b*(c^2*d + 2*e)*ArcTan[Sqrt[c^2*d + e]/(c*S qrt[e]*Sqrt[1 - 1/(c^2*x^2)]*x)])/(8*e^(5/2)*(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] - S qrt[c^2*d + e])])/(2*e^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*e^3) + ((a + b*Ar cSin[1/(c*x)])*Log[1 - (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqr t[c^2*d + e])])/(2*e^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*e^3) - ((a + b*ArcS in[1/(c*x)])*Log[1 - E^((2*I)*ArcSin[1/(c*x)])])/e^3 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqrt[c^2*d + e])])/e^3 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[-d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] - Sqr t[c^2*d + e])])/e^3 - ((I/2)*b*PolyLog[2, ((-I)*c*Sqrt[-d]*E^(I*ArcSin[1/( c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 - ((I/2)*b*PolyLog[2, (I*c*Sqrt[ -d]*E^(I*ArcSin[1/(c*x)]))/(Sqrt[e] + Sqrt[c^2*d + e])])/e^3 + ((I/2)*b*Po lyLog[2, E^((2*I)*ArcSin[1/(c*x)])])/e^3
3.2.11.3.1 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 = 4.67 (sec) , antiderivative size = 1269, normalized size of antiderivative = 1.75
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1269\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1285\) |
| default | \(\text {Expression too large to display}\) | \(1285\) |
a*(-1/4*d^2/e^3/(e*x^2+d)^2+1/2/e^3*ln(e*x^2+d)+d/e^3/(e*x^2+d))+b/c^6*(-1 /8*c^6*(4*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+4*c^4*d*e*arccsc(c*x)*x^2+6*arccsc(c*x)*e^2* c^4*x^4)/e^2/(c^2*e*x^2+c^2*d)^2/(c^2*d+e)-1/4*I/(c^2*d+e)/e^2*c^8*d*sum(( _R1^2-1)/(_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))+I/(c^2*d+e)/e^2*c^6*dilog(1+I/c/x+(1-1/c ^2/x^2)^(1/2))-I/(c^2*d+e)/e^2*c^6*dilog(I/c/x+(1-1/c^2/x^2)^(1/2))-1/4*I/ (c^2*d+e)/e^2*c^6*sum((_R1^2*c^2*d-c^2*d-4*e)/(_R1^2*c^2*d-c^2*d-2*e)*(I*a rccsc(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))-1/( c^2*d+e)/e^2*c^6*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1/(c^2*d+e)/e ^3*c^8*d*arccsc(c*x)*ln(1+I/c/x+(1-1/c^2/x^2)^(1/2))-1/4*I/(c^2*d+e)/e^3*c ^8*d*sum((_R1^2*c^2*d-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))-1/4*I/(c^2*d+e)/ e^3*c^10*d^2*sum((_R1^2-1)/(_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))-5/8*I*(e*(c^2*d+e))...
\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arccsc}\left (c x\right ) + a\right )} x^{5}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
1/4*a*((4*d*e*x^2 + 3*d^2)/(e^5*x^4 + 2*d*e^4*x^2 + d^2*e^3) + 2*log(e*x^2 + d)/e^3) + b*integrate(x^5*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))/(e^3* x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3), x)
Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {x^5 \left (a+b \csc ^{-1}(c x)\right )}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]