Integrand size = 21, antiderivative size = 1234 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-1+c x} \sqrt {1+c x}}{16 \sqrt {-d} e \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {a+b \text {arccosh}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {a+b \text {arccosh}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {a+b \text {arccosh}(c x)}{16 d e^{3/2} \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (c \sqrt {-d}+\sqrt {e}\right )^{3/2} e^{3/2}}+\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\frac {b c^3 \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (c \sqrt {-d}+\sqrt {e}\right )^{3/2} e^{3/2}}-\frac {b c \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}+\sqrt {e}} \sqrt {1+c x}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {-1+c x}}\right )}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 (-d)^{3/2} e^{3/2}} \]
-1/16*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c *(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(3/2)/e^(3/2)+1/16*(a+b*arccosh(c*x))* ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1 /2)))/(-d)^(3/2)/e^(3/2)-1/16*(a+b*arccosh(c*x))*ln(1-(c*x+(c*x-1)^(1/2)*( c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(3/2)/e^(3/2)+ 1/16*(a+b*arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c* (-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(3/2)/e^(3/2)+1/16*b*polylog(2,-(c*x+(c *x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^( 3/2)/e^(3/2)-1/16*b*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c *(-d)^(1/2)-(-c^2*d-e)^(1/2)))/(-d)^(3/2)/e^(3/2)+1/16*b*polylog(2,-(c*x+( c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/(c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^ (3/2)/e^(3/2)-1/16*b*polylog(2,(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*e^(1/2)/( c*(-d)^(1/2)+(-c^2*d-e)^(1/2)))/(-d)^(3/2)/e^(3/2)+1/8*b*c^3*arctanh((c*x+ 1)^(1/2)*(c*(-d)^(1/2)-e^(1/2))^(1/2)/(c*x-1)^(1/2)/(c*(-d)^(1/2)+e^(1/2)) ^(1/2))/e^(3/2)/(c*(-d)^(1/2)-e^(1/2))^(3/2)/(c*(-d)^(1/2)+e^(1/2))^(3/2)- 1/8*b*c^3*arctanh((c*x+1)^(1/2)*(c*(-d)^(1/2)+e^(1/2))^(1/2)/(c*x-1)^(1/2) /(c*(-d)^(1/2)-e^(1/2))^(1/2))/e^(3/2)/(c*(-d)^(1/2)-e^(1/2))^(3/2)/(c*(-d )^(1/2)+e^(1/2))^(3/2)+1/16*(-a-b*arccosh(c*x))/e^(3/2)/(-d)^(1/2)/((-d)^( 1/2)-x*e^(1/2))^2+1/16*(-a-b*arccosh(c*x))/d/e^(3/2)/((-d)^(1/2)-x*e^(1/2) )+1/16*(a+b*arccosh(c*x))/e^(3/2)/(-d)^(1/2)/((-d)^(1/2)+x*e^(1/2))^2+1...
Result contains complex when optimal does not.
Time = 11.03 (sec) , antiderivative size = 1193, normalized size of antiderivative = 0.97 \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {a x}{4 e \left (d+e x^2\right )^2}+\frac {a x}{8 d e \left (d+e x^2\right )}+\frac {a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 d^{3/2} e^{3/2}}+b \left (\frac {\frac {\text {arccosh}(c x)}{-i \sqrt {d}+\sqrt {e} x}+\frac {c \log \left (\frac {2 e \left (i \sqrt {e}+c^2 \sqrt {d} x-i \sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (\sqrt {d}+i \sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}}{16 d e^{3/2}}-\frac {-\frac {\text {arccosh}(c x)}{i \sqrt {d}+\sqrt {e} x}-\frac {c \log \left (\frac {2 e \left (-\sqrt {e}-i c^2 \sqrt {d} x+\sqrt {-c^2 d-e} \sqrt {-1+c x} \sqrt {1+c x}\right )}{c \sqrt {-c^2 d-e} \left (i \sqrt {d}+\sqrt {e} x\right )}\right )}{\sqrt {-c^2 d-e}}}{16 d e^{3/2}}-\frac {i \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (-i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (-i \sqrt {d}+\sqrt {e} x\right )^2}+\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}-c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\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 \sqrt {d} e}+\frac {i \left (\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{\left (c^2 d+e\right ) \left (i \sqrt {d}+\sqrt {e} x\right )}-\frac {\text {arccosh}(c x)}{\sqrt {e} \left (i \sqrt {d}+\sqrt {e} x\right )^2}-\frac {c^3 \sqrt {d} \left (\log (4)+\log \left (\frac {e \sqrt {c^2 d+e} \left (-i \sqrt {e}+c^2 \sqrt {d} x+\sqrt {c^2 d+e} \sqrt {-1+c x} \sqrt {1+c x}\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 \sqrt {d} e}+\frac {i \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}-\sqrt {-c^2 d-e}}\right )+\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{32 d^{3/2} e^{3/2}}-\frac {i \left (\text {arccosh}(c x) \left (-\text {arccosh}(c x)+2 \left (\log \left (1+\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+\log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{-i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )+2 \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{i c \sqrt {d}+\sqrt {-c^2 d-e}}\right )\right )}{32 d^{3/2} e^{3/2}}\right ) \]
-1/4*(a*x)/(e*(d + e*x^2)^2) + (a*x)/(8*d*e*(d + e*x^2)) + (a*ArcTan[(Sqrt [e]*x)/Sqrt[d]])/(8*d^(3/2)*e^(3/2)) + b*((ArcCosh[c*x]/((-I)*Sqrt[d] + Sq rt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2*d) - e]*S qrt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(Sqrt[d] + I*Sqrt[e]*x ))])/Sqrt[-(c^2*d) - e])/(16*d*e^(3/2)) - (-(ArcCosh[c*x]/(I*Sqrt[d] + Sqr t[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2*d) - e]*Sq rt[-1 + c*x]*Sqrt[1 + c*x]))/(c*Sqrt[-(c^2*d) - e]*(I*Sqrt[d] + Sqrt[e]*x) )])/Sqrt[-(c^2*d) - e])/(16*d*e^(3/2)) - ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]* ((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] - c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d + I*Sqrt[d]*Sqrt[e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/ (Sqrt[d]*e) + ((I/16)*((c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*(I*Sq rt[d] + Sqrt[e]*x)) - ArcCosh[c*x]/(Sqrt[e]*(I*Sqrt[d] + Sqrt[e]*x)^2) - ( c^3*Sqrt[d]*(Log[4] + Log[(e*Sqrt[c^2*d + e]*((-I)*Sqrt[e] + c^2*Sqrt[d]*x + Sqrt[c^2*d + e]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]))/(c^3*(d - I*Sqrt[d]*Sqrt [e]*x))]))/(Sqrt[e]*(c^2*d + e)^(3/2))))/(Sqrt[d]*e) + ((I/32)*(ArcCosh[c* x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] - Sqr t[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sqrt[d] + Sqrt[- (c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/((-I)*c*Sqrt[d...
Time = 3.11 (sec) , antiderivative size = 1234, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6374, 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.
| \(\displaystyle \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (\frac {a+b \text {arccosh}(c x)}{e \left (d+e x^2\right )^2}-\frac {d (a+b \text {arccosh}(c x))}{e \left (d+e x^2\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) c^3}{8 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} e^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) c^3}{8 \left (c \sqrt {-d}-\sqrt {e}\right )^{3/2} \left (\sqrt {-d} c+\sqrt {e}\right )^{3/2} e^{3/2}}+\frac {b \text {arctanh}\left (\frac {\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x+1}}{\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x-1}}\right ) c}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{3/2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {\sqrt {-d} c+\sqrt {e}} \sqrt {c x+1}}{\sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c x-1}}\right ) c}{8 d \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{3/2}}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {c x-1} \sqrt {c x+1} c}{16 \sqrt {-d} e \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \text {arccosh}(c x)}{16 d e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {a+b \text {arccosh}(c x)}{16 d e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {a+b \text {arccosh}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {a+b \text {arccosh}(c x)}{16 \sqrt {-d} e^{3/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{c \sqrt {-d}-\sqrt {-d c^2-e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {(a+b \text {arccosh}(c x)) \log \left (1-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {e^{\text {arccosh}(c x)} \sqrt {e}}{\sqrt {-d} c+\sqrt {-d c^2-e}}+1\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}+\frac {b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}-\frac {b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 (-d)^{3/2} e^{3/2}}\) |
-1/16*(b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] - Sqrt[e]*x)) - (b*c*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*Sqrt[-d]*e*(c^2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (a + b*ArcCosh[c*x])/(16*Sqrt[-d]*e^(3/2)*( Sqrt[-d] - Sqrt[e]*x)^2) - (a + b*ArcCosh[c*x])/(16*d*e^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (a + b*ArcCosh[c*x])/(16*Sqrt[-d]*e^(3/2)*(Sqrt[-d] + Sqrt[e ]*x)^2) + (a + b*ArcCosh[c*x])/(16*d*e^(3/2)*(Sqrt[-d] + Sqrt[e]*x)) + (b* c^3*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + S qrt[e])^(3/2)*e^(3/2)) + (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(3/2)) - (b*c^3*ArcTanh[(Sqrt[c*Sq rt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x ])])/(8*(c*Sqrt[-d] - Sqrt[e])^(3/2)*(c*Sqrt[-d] + Sqrt[e])^(3/2)*e^(3/2)) - (b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d ] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*d*Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt [-d] + Sqrt[e]]*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCos h[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*(-d)^(3/2)*e^(3/2)) + ((a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^ 2*d) - e])])/(16*(-d)^(3/2)*e^(3/2)) - ((a + b*ArcCosh[c*x])*Log[1 - (Sqrt [e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])])/(16*(-d)^(3/2)*...
3.6.12.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e _.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcCosh[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]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 23.20 (sec) , antiderivative size = 1222, normalized size of antiderivative = 0.99
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1222\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1237\) |
| default | \(\text {Expression too large to display}\) | \(1237\) |
a*((1/8/d*x^3-1/8/e*x)/(e*x^2+d)^2+1/8/e/d/(d*e)^(1/2)*arctan(e*x/(d*e)^(1 /2)))+b/c^3*(1/8*c^4*((c*x+1)^(1/2)*(c*x-1)^(1/2)*c^4*d*e*x^2+(c*x-1)^(1/2 )*(c*x+1)^(1/2)*c^4*d^2+arccosh(c*x)*d*c^5*e*x^3-arccosh(c*x)*d^2*c^5*x+ar ccosh(c*x)*e^2*c^3*x^3-arccosh(c*x)*d*c^3*e*x)/e/(c^2*e*x^2+c^2*d)^2/d/(c^ 2*d+e)+1/16/(c^2*d+e)/d*c^4*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln(( _R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c* x+1)^(1/2))/_R1)),_R1=RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/16/(c^2*d+e)/ e*c^6*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*ln((_R1-c*x-(c*x-1)^(1/2)* (c*x+1)^(1/2))/_R1)+dilog((_R1-c*x-(c*x-1)^(1/2)*(c*x+1)^(1/2))/_R1)),_R1= RootOf(e*_Z^4+(4*c^2*d+2*e)*_Z^2+e))+1/8*(-(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)*c^2*d+2*c^4*d^2+2*c^2*d*e+(d*c^ 2*(c^2*d+e))^(1/2)*e)*c^4*arctanh(e*(c*x+(c*x-1)^(1/2)*(c*x+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^3-1/8*(-(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*(c*x+(c*x-1)^(1/2)*(c*x+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+1/8*((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)*c^2*d+2*c^4*d^2+2*c^2 *d*e-(d*c^2*(c^2*d+e))^(1/2)*e)*c^4*arctan(e*(c*x+(c*x-1)^(1/2)*(c*x+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^3-1/ 8*((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^...
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{2}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^2 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]