Integrand size = 18, antiderivative size = 1092 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b c \sqrt {1-c^2 x^2}}{16 (-d)^{3/2} \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \arcsin (c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {3 (a+b \arcsin (c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \arcsin (c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {3 (a+b \arcsin (c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d \sqrt {e} \left (c^2 d+e\right )^{3/2}}+\frac {3 b c \text {arctanh}\left (\frac {\sqrt {e}+c^2 \sqrt {-d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{16 d^2 \sqrt {e} \sqrt {c^2 d+e}}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \arcsin (c x)) \log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}+\sqrt {c^2 d+e}}\right )}{16 (-d)^{5/2} \sqrt {e}} \]
1/16*b*c^3*arctanh((-c^2*x*(-d)^(1/2)+e^(1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1 )^(1/2))/d/(c^2*d+e)^(3/2)/e^(1/2)+1/16*b*c^3*arctanh((c^2*x*(-d)^(1/2)+e^ (1/2))/(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d/(c^2*d+e)^(3/2)/e^(1/2)+3/16* (a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)- (c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)-3/16*(a+b*arcsin(c*x))*ln(1+(I*c*x+(- c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^( 1/2)+3/16*(a+b*arcsin(c*x))*ln(1-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*( -d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)-3/16*(a+b*arcsin(c*x))*ln(1 +(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d) ^(5/2)/e^(1/2)-3/16*I*b*polylog(2,(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c* (-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)-3/16*I*b*polylog(2,(I*c*x+ (-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)/e ^(1/2)+3/16*I*b*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1 /2)-(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2)+3/16*I*b*polylog(2,-(I*c*x+(-c^2* x^2+1)^(1/2))*e^(1/2)/(I*c*(-d)^(1/2)+(c^2*d+e)^(1/2)))/(-d)^(5/2)/e^(1/2) +1/16*(-a-b*arcsin(c*x))/(-d)^(3/2)/e^(1/2)/((-d)^(1/2)-x*e^(1/2))^2-3/16* (a+b*arcsin(c*x))/d^2/e^(1/2)/((-d)^(1/2)-x*e^(1/2))+1/16*(a+b*arcsin(c*x) )/(-d)^(3/2)/e^(1/2)/((-d)^(1/2)+x*e^(1/2))^2+3/16*(a+b*arcsin(c*x))/d^2/e ^(1/2)/((-d)^(1/2)+x*e^(1/2))+3/16*b*c*arctanh((-c^2*x*(-d)^(1/2)+e^(1/2)) /(c^2*d+e)^(1/2)/(-c^2*x^2+1)^(1/2))/d^2/e^(1/2)/(c^2*d+e)^(1/2)+3/16*b...
Time = 6.06 (sec) , antiderivative size = 1055, normalized size of antiderivative = 0.97 \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\frac {a x}{4 d \left (d+e x^2\right )^2}+\frac {3 a x}{8 d^2 \left (d+e x^2\right )}+\frac {3 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{5/2} \sqrt {e}}+b \left (\frac {3 i \left (\frac {\arcsin (c x)}{\sqrt {d}+i \sqrt {e} x}-\frac {c \arctan \left (\frac {i \sqrt {e}+c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )}{16 d^2 \sqrt {e}}-\frac {3 \left (-\frac {\arcsin (c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \text {arctanh}\left (\frac {\sqrt {e}+i c^2 \sqrt {d} x}{\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d+e}}\right )}{16 d^2 \sqrt {e}}+\frac {i \left (-\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\arcsin (c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {i c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d+i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )}{16 d^{3/2}}-\frac {i \left (-\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\arcsin (c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {i c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (\sqrt {e}+i c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {1-c^2 x^2}\right )}{c^3 \left (d-i \sqrt {d} \sqrt {e} x\right )}\right )\right )}{\sqrt {e} \left (c^2 d+e\right )^{3/2}}\right )}{16 d^{3/2}}-\frac {3 \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{32 d^{5/2} \sqrt {e}}+\frac {3 \left (\arcsin (c x) \left (\arcsin (c x)+2 i \left (\log \left (1+\frac {\sqrt {e} e^{i \arcsin (c x)}}{-c \sqrt {d}+\sqrt {c^2 d+e}}\right )+\log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}-\sqrt {c^2 d+e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{c \sqrt {d}+\sqrt {c^2 d+e}}\right )\right )}{32 d^{5/2} \sqrt {e}}\right ) \]
(a*x)/(4*d*(d + e*x^2)^2) + (3*a*x)/(8*d^2*(d + e*x^2)) + (3*a*ArcTan[(Sqr t[e]*x)/Sqrt[d]])/(8*d^(5/2)*Sqrt[e]) + b*((((3*I)/16)*(ArcSin[c*x]/(Sqrt[ d] + I*Sqrt[e]*x) - (c*ArcTan[(I*Sqrt[e] + c^2*Sqrt[d]*x)/(Sqrt[c^2*d + e] *Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(d^2*Sqrt[e]) - (3*(-(ArcSin[c*x]/ (I*Sqrt[d] + Sqrt[e]*x)) - (c*ArcTanh[(Sqrt[e] + I*c^2*Sqrt[d]*x)/(Sqrt[c^ 2*d + e]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d + e]))/(16*d^2*Sqrt[e]) + ((I/16) *(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x))) - ArcS in[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) - (I*c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2 ))))/d^(3/2) - ((I/16)*(-((c*Sqrt[1 - c^2*x^2])/((c^2*d + e)*(I*Sqrt[d] + Sqrt[e]*x))) - ArcSin[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) + (I*c^3*Sq rt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*(Sqrt[e] + I*c^2*Sqrt[d]*x + Sqrt[c ^2*d + e]*Sqrt[1 - c^2*x^2]))/(c^3*(d - I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]* (c^2*d + e)^(3/2))))/d^(3/2) - (3*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] - Sqrt[c^2*d + e])] + Log[1 + (S qrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e])])) + 2*PolyLog[2, (Sqrt[e]*E^(I*ArcSin[c*x]))/(-(c*Sqrt[d]) + Sqrt[c^2*d + e])] + 2*PolyLog[ 2, -((Sqrt[e]*E^(I*ArcSin[c*x]))/(c*Sqrt[d] + Sqrt[c^2*d + e]))]))/(32*d^( 5/2)*Sqrt[e]) + (3*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(Log[1 + (Sqrt[e]*...
Time = 1.78 (sec) , antiderivative size = 1092, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {5172, 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.
| \(\displaystyle \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 5172 |
| \(\displaystyle \int \left (-\frac {3 e (a+b \arcsin (c x))}{8 d^2 \left (-d e-e^2 x^2\right )}-\frac {3 e (a+b \arcsin (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}-e x\right )^2}-\frac {3 e (a+b \arcsin (c x))}{16 d^2 \left (\sqrt {-d} \sqrt {e}+e x\right )^2}-\frac {e^{3/2} (a+b \arcsin (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}-e x\right )^3}-\frac {e^{3/2} (a+b \arcsin (c x))}{8 (-d)^{3/2} \left (\sqrt {-d} \sqrt {e}+e x\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c^3}{16 d \sqrt {e} \left (d c^2+e\right )^{3/2}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {e}-c^2 \sqrt {-d} x}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}+\frac {3 b \text {arctanh}\left (\frac {\sqrt {-d} x c^2+\sqrt {e}}{\sqrt {d c^2+e} \sqrt {1-c^2 x^2}}\right ) c}{16 d^2 \sqrt {e} \sqrt {d c^2+e}}+\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {b \sqrt {1-c^2 x^2} c}{16 (-d)^{3/2} \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {3 (a+b \arcsin (c x))}{16 d^2 \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {3 (a+b \arcsin (c x))}{16 d^2 \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \arcsin (c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \arcsin (c x)}{16 (-d)^{3/2} \sqrt {e} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \arcsin (c x)) \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i c \sqrt {-d}-\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 (a+b \arcsin (c x)) \log \left (1-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 (a+b \arcsin (c x)) \log \left (\frac {e^{i \arcsin (c x)} \sqrt {e}}{i \sqrt {-d} c+\sqrt {d c^2+e}}+1\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i c \sqrt {-d}-\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}+\frac {3 i b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}-\frac {3 i b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{i \arcsin (c x)}}{i \sqrt {-d} c+\sqrt {d c^2+e}}\right )}{16 (-d)^{5/2} \sqrt {e}}\) |
(b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) + (b*c*Sqrt[1 - c^2*x^2])/(16*(-d)^(3/2)*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]* x)) - (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)^2) - (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcSin[c*x])/(16*(-d)^(3/2)*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)^2) + (3*(a + b*ArcSin[c*x]))/(16*d^2*Sqrt[e]*(Sqrt[-d] + Sqrt[e]*x)) + (b*c^3*ArcTanh[( Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d*Sqrt [e]*(c^2*d + e)^(3/2)) + (3*b*c*ArcTanh[(Sqrt[e] - c^2*Sqrt[-d]*x)/(Sqrt[c ^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (b*c^3*A rcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x)/(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(1 6*d*Sqrt[e]*(c^2*d + e)^(3/2)) + (3*b*c*ArcTanh[(Sqrt[e] + c^2*Sqrt[-d]*x) /(Sqrt[c^2*d + e]*Sqrt[1 - c^2*x^2])])/(16*d^2*Sqrt[e]*Sqrt[c^2*d + e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b*ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e])])/(16*(-d) ^(5/2)*Sqrt[e]) + (3*(a + b*ArcSin[c*x])*Log[1 - (Sqrt[e]*E^(I*ArcSin[c*x] ))/(I*c*Sqrt[-d] + Sqrt[c^2*d + e])])/(16*(-d)^(5/2)*Sqrt[e]) - (3*(a + b* ArcSin[c*x])*Log[1 + (Sqrt[e]*E^(I*ArcSin[c*x]))/(I*c*Sqrt[-d] + Sqrt[c^2* d + e])])/(16*(-d)^(5/2)*Sqrt[e]) + (((3*I)/16)*b*PolyLog[2, -((Sqrt[e]*E^ (I*ArcSin[c*x]))/(I*c*Sqrt[-d] - Sqrt[c^2*d + e]))])/((-d)^(5/2)*Sqrt[e...
3.7.48.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(a + b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x ] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (G tQ[p, 0] || IGtQ[n, 0])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.07 (sec) , antiderivative size = 1772, normalized size of antiderivative = 1.62
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1772\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1793\) |
| default | \(\text {Expression too large to display}\) | \(1793\) |
1/4*a*x/d/(e*x^2+d)^2+3/8*a/d^2*x/(e*x^2+d)+3/8*a/d^2/(d*e)^(1/2)*arctan(e *x/(d*e)^(1/2))+b/c*(1/8*c^2*(5*arcsin(c*x)*d^2*c^5*x+3*arcsin(c*x)*d*c^5* e*x^3+d^2*c^4*(-c^2*x^2+1)^(1/2)+(-c^2*x^2+1)^(1/2)*c^4*d*e*x^2+5*arcsin(c *x)*c^3*d*e*x+3*arcsin(c*x)*e^2*c^3*x^3)/d^2/(c^2*d+e)/(c^2*e*x^2+c^2*d)^2 +1/2*((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(2*c^2*d-2*(d*c^2*(c^ 2*d+e))^(1/2)+e)*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(d*c^2*( c^2*d+e))^(1/2)+e)*e)^(1/2))*c^4/(c^2*d+e)/d/e^3-1/2*((2*c^2*d+2*(d*c^2*(c ^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*(d*c^2*(c^2*d+e))^(1/2)*d*c^2+2*d^2*c^4+2*c ^2*e*d-(d*c^2*(c^2*d+e))^(1/2)*e)*c^4*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2)) /((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2))/(c^2*d+e)^2/d/e^3+3/16/( c^2*d+e)/d^2*c^2*e*sum(1/_R1/(_R1^2*e-2*c^2*d-e)*(I*arcsin(c*x)*ln((_R1-I* c*x-(-c^2*x^2+1)^(1/2))/_R1)+dilog((_R1-I*c*x-(-c^2*x^2+1)^(1/2))/_R1)),_R 1=RootOf(e*_Z^4+(-4*c^2*d-2*e)*_Z^2+e))+1/2*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e ))^(1/2)+e))^(1/2)*(2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*arctan(e*(I*c*x+( -c^2*x^2+1)^(1/2))/((-2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)-e)*e)^(1/2))*c^4/( c^2*d+e)/d/e^3-3/8*((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/2)+e)*e)^(1/2)*(-2*(d* c^2*(c^2*d+e))^(1/2)*d*c^2+2*d^2*c^4+2*c^2*e*d-(d*c^2*(c^2*d+e))^(1/2)*e)* c^2*arctanh(e*(I*c*x+(-c^2*x^2+1)^(1/2))/((2*c^2*d+2*(d*c^2*(c^2*d+e))^(1/ 2)+e)*e)^(1/2))/(c^2*d+e)^2/d^2/e^2-1/2*(-e*(2*c^2*d-2*(d*c^2*(c^2*d+e))^( 1/2)+e))^(1/2)*(2*d^2*c^4+2*(d*c^2*(c^2*d+e))^(1/2)*d*c^2+2*c^2*e*d+(d*...
\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a + b \operatorname {asin}{\left (c x \right )}}{\left (d + e x^{2}\right )^{3}}\, dx \]
Exception generated. \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\int { \frac {b \arcsin \left (c x\right ) + a}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {a+b \arcsin (c x)}{\left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asin}\left (c\,x\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]