Integrand size = 21, antiderivative size = 1224 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=-\frac {b c \sqrt {-d} \sqrt {-1+c x} \sqrt {1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b c \sqrt {-d} \sqrt {-1+c x} \sqrt {1+c x}}{16 e^2 \left (c^2 d+e\right ) \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {5 (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {5 (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {b c^3 d \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^{5/2}}-\frac {5 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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {b c^3 d \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^{5/2}}+\frac {5 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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {c \sqrt {-d}+\sqrt {e}} e^{5/2}}+\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 (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 \sqrt {-d} e^{5/2}}+\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-c^2 d-e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}+\sqrt {-c^2 d-e}}\right )}{16 \sqrt {-d} e^{5/2}} \]
3/16*(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)))/e^(5/2)/(-d)^(1/2)-3/16*(a+b*arccosh(c*x))*l n(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)))/e^(5/2)/(-d)^(1/2)+3/16*(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)))/e^(5/2)/(-d)^(1/2)-3 /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)))/e^(5/2)/(-d)^(1/2)-3/16*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)))/e^(5/2) /(-d)^(1/2)+3/16*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)))/e^(5/2)/(-d)^(1/2)-3/16*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)))/e^(5/2 )/(-d)^(1/2)+3/16*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)))/e^(5/2)/(-d)^(1/2)-1/8*b*c^3*d*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^(5/2)/(c*(-d)^(1/2)-e^(1/2))^(3/2)/(c*(-d)^(1/2)+e^(1/2))^(3/2) +1/8*b*c^3*d*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^(5/2)/(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))*(-d)^(1/2)/e^(5/2)/((-d) ^(1/2)-x*e^(1/2))^2+5/16*(a+b*arccosh(c*x))/e^(5/2)/((-d)^(1/2)-x*e^(1/2)) +1/16*(a+b*arccosh(c*x))*(-d)^(1/2)/e^(5/2)/((-d)^(1/2)+x*e^(1/2))^2-5/...
Result contains complex when optimal does not.
Time = 11.03 (sec) , antiderivative size = 1185, normalized size of antiderivative = 0.97 \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\frac {a d x}{4 e^2 \left (d+e x^2\right )^2}-\frac {5 a x}{8 e^2 \left (d+e x^2\right )}+\frac {3 a \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{8 \sqrt {d} e^{5/2}}+b \left (-\frac {5 \left (\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}}\right )}{16 e^{5/2}}+\frac {5 \left (-\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}}\right )}{16 e^{5/2}}+\frac {i \sqrt {d} \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 e^2}-\frac {i \sqrt {d} \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 e^2}+\frac {3 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 \sqrt {d} e^{5/2}}-\frac {3 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 \sqrt {d} e^{5/2}}\right ) \]
(a*d*x)/(4*e^2*(d + e*x^2)^2) - (5*a*x)/(8*e^2*(d + e*x^2)) + (3*a*ArcTan[ (Sqrt[e]*x)/Sqrt[d]])/(8*Sqrt[d]*e^(5/2)) + b*((-5*(ArcCosh[c*x]/((-I)*Sqr t[d] + Sqrt[e]*x) + (c*Log[(2*e*(I*Sqrt[e] + c^2*Sqrt[d]*x - I*Sqrt[-(c^2* d) - e]*Sqrt[-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*e^(5/2)) + (5*(-(ArcCosh[c*x]/(I*Sq rt[d] + Sqrt[e]*x)) - (c*Log[(2*e*(-Sqrt[e] - I*c^2*Sqrt[d]*x + Sqrt[-(c^2 *d) - e]*Sqrt[-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*e^(5/2)) + ((I/16)*Sqrt[d]*((c*Sqr t[-1 + c*x]*Sqrt[1 + c*x])/((c^2*d + e)*((-I)*Sqrt[d] + Sqrt[e]*x)) - ArcC osh[c*x]/(Sqrt[e]*((-I)*Sqrt[d] + Sqrt[e]*x)^2) + (c^3*Sqrt[d]*(Log[4] + L og[(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))))/e^2 - ((I/16)*Sqrt[d]*((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))))/e^2 + (((3*I)/3 2)*(ArcCosh[c*x]*(-ArcCosh[c*x] + 2*(Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c *Sqrt[d] - Sqrt[-(c^2*d) - e])] + Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(I*c*Sq rt[d] + Sqrt[-(c^2*d) - e])])) + 2*PolyLog[2, (Sqrt[e]*E^ArcCosh[c*x])/...
Time = 4.14 (sec) , antiderivative size = 1224, 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.
| \(\displaystyle \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 6374 |
| \(\displaystyle \int \left (\frac {d^2 (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )^3}-\frac {2 d (a+b \text {arccosh}(c x))}{e^2 \left (d+e x^2\right )^2}+\frac {a+b \text {arccosh}(c x)}{e^2 \left (d+e x^2\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {b d \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^{5/2}}+\frac {b d \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^{5/2}}-\frac {5 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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}+\frac {5 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 \sqrt {c \sqrt {-d}-\sqrt {e}} \sqrt {\sqrt {-d} c+\sqrt {e}} e^{5/2}}-\frac {b \sqrt {-d} \sqrt {c x-1} \sqrt {c x+1} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {b \sqrt {-d} \sqrt {c x-1} \sqrt {c x+1} c}{16 e^2 \left (d c^2+e\right ) \left (\sqrt {e} x+\sqrt {-d}\right )}+\frac {5 (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {5 (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )}-\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {\sqrt {-d} (a+b \text {arccosh}(c x))}{16 e^{5/2} \left (\sqrt {e} x+\sqrt {-d}\right )^2}+\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 (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 \sqrt {-d} e^{5/2}}+\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 (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 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{c \sqrt {-d}-\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}-\frac {3 b \operatorname {PolyLog}\left (2,-\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}+\frac {3 b \operatorname {PolyLog}\left (2,\frac {\sqrt {e} e^{\text {arccosh}(c x)}}{\sqrt {-d} c+\sqrt {-d c^2-e}}\right )}{16 \sqrt {-d} e^{5/2}}\) |
-1/16*(b*c*Sqrt[-d]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(e^2*(c^2*d + e)*(Sqrt[- d] - Sqrt[e]*x)) - (b*c*Sqrt[-d]*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(16*e^2*(c^ 2*d + e)*(Sqrt[-d] + Sqrt[e]*x)) - (Sqrt[-d]*(a + b*ArcCosh[c*x]))/(16*e^( 5/2)*(Sqrt[-d] - Sqrt[e]*x)^2) + (5*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqr t[-d] - Sqrt[e]*x)) + (Sqrt[-d]*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d ] + Sqrt[e]*x)^2) - (5*(a + b*ArcCosh[c*x]))/(16*e^(5/2)*(Sqrt[-d] + Sqrt[ e]*x)) - (b*c^3*d*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] + Sqrt[e])^(3/2)*e^(5/2)) - (5*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] - S qrt[e]]*Sqrt[1 + c*x])/(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[-1 + c*x])])/(8*Sq rt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (b*c^3*d*Ar cTanh[(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] + Sqrt[e] )^(3/2)*e^(5/2)) + (5*b*c*ArcTanh[(Sqrt[c*Sqrt[-d] + Sqrt[e]]*Sqrt[1 + c*x ])/(Sqrt[c*Sqrt[-d] - Sqrt[e]]*Sqrt[-1 + c*x])])/(8*Sqrt[c*Sqrt[-d] - Sqrt [e]]*Sqrt[c*Sqrt[-d] + Sqrt[e]]*e^(5/2)) + (3*(a + b*ArcCosh[c*x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] - Sqrt[-(c^2*d) - e])])/(16*Sqrt[-d] *e^(5/2)) - (3*(a + b*ArcCosh[c*x])*Log[1 + (Sqrt[e]*E^ArcCosh[c*x])/(c*Sq rt[-d] - Sqrt[-(c^2*d) - e])])/(16*Sqrt[-d]*e^(5/2)) + (3*(a + b*ArcCosh[c *x])*Log[1 - (Sqrt[e]*E^ArcCosh[c*x])/(c*Sqrt[-d] + Sqrt[-(c^2*d) - e])...
3.6.11.3.1 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 = 49.46 (sec) , antiderivative size = 1752, normalized size of antiderivative = 1.43
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1752\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1754\) |
| default | \(\text {Expression too large to display}\) | \(1754\) |
a*((-5/8*x^3/e-3/8*d/e^2*x)/(e*x^2+d)^2+3/8/e^2/(d*e)^(1/2)*arctan(e*x/(d* e)^(1/2)))+b/c^5*(-1/8*c^6*((c*x-1)^(1/2)*(c*x+1)^(1/2)*c^4*d^2+(c*x+1)^(1 /2)*(c*x-1)^(1/2)*c^4*d*e*x^2+3*arccosh(c*x)*d^2*c^5*x+5*arccosh(c*x)*d*c^ 5*e*x^3+3*arccosh(c*x)*d*c^3*e*x+5*arccosh(c*x)*e^2*c^3*x^3)/e^2/(c^2*d+e) /(c^2*e*x^2+c^2*d)^2-5/8*(-(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^6*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/e^4+5/8*(-(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^6/e^4/(c^2*d+e)-5/8*((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^6*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/e^4+5/8*((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)*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^6/e^4/(c^2*d+e)-3/16/(c^2*d+e)/e*c^6*sum(1/_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((_R 1-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))+3/16/(c^2*d+e)/e*c^6*sum(_R1/(_R1^2*e+2*c^2*d+e)*(arccosh(c*x)*...
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x^{4}}{{\left (e x^{2} + d\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {x^4 (a+b \text {arccosh}(c x))}{\left (d+e x^2\right )^3} \, dx=\int \frac {x^4\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (e\,x^2+d\right )}^3} \,d x \]