3.5.51 \(\int \frac {x^4 \arctan (a x)^3}{(c+a^2 c x^2)^{5/2}} \, dx\) [451]

3.5.51.1 Optimal result

Integrand size = 24, antiderivative size = 622 \[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {2}{27 a^5 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {68}{9 a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {2 x^3 \arctan (a x)}{9 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {22 x \arctan (a x)}{3 a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^2 \arctan (a x)^2}{3 a^3 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {11 \arctan (a x)^2}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \arctan (a x)^3}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \arctan (a x)^3}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 i \sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {6 i \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}} \]

output

-2/27/a^5/c/(a^2*c*x^2+c)^(3/2)+2/9*x^3*arctan(a*x)/a^2/c/(a^2*c*x^2+c)^(3 
/2)-1/3*x^2*arctan(a*x)^2/a^3/c/(a^2*c*x^2+c)^(3/2)-1/3*x^3*arctan(a*x)^3/ 
a^2/c/(a^2*c*x^2+c)^(3/2)+68/9/a^5/c^2/(a^2*c*x^2+c)^(1/2)+22/3*x*arctan(a 
*x)/a^4/c^2/(a^2*c*x^2+c)^(1/2)-11/3*arctan(a*x)^2/a^5/c^2/(a^2*c*x^2+c)^( 
1/2)-x*arctan(a*x)^3/a^4/c^2/(a^2*c*x^2+c)^(1/2)-2*I*arctan((1+I*a*x)/(a^2 
*x^2+1)^(1/2))*arctan(a*x)^3*(a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2) 
+3*I*arctan(a*x)^2*polylog(2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^( 
1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2)-3*I*arctan(a*x)^2*polylog(2,I*(1+I*a*x)/( 
a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2)-6*arctan(a 
*x)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^5/c^2/(a 
^2*c*x^2+c)^(1/2)+6*arctan(a*x)*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*( 
a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2)-6*I*polylog(4,-I*(1+I*a*x)/(a 
^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^(1/2)+6*I*polylog 
(4,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))*(a^2*x^2+1)^(1/2)/a^5/c^2/(a^2*c*x^2+c)^ 
(1/2)
 

3.5.51.2 Mathematica [A] (verified)

Time = 2.25 (sec) , antiderivative size = 691, normalized size of antiderivative = 1.11 \[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=-\frac {\sqrt {c \left (1+a^2 x^2\right )} \left (189 i \pi ^4-\frac {12960}{\sqrt {1+a^2 x^2}}+216 i \pi ^3 \arctan (a x)-\frac {12960 a x \arctan (a x)}{\sqrt {1+a^2 x^2}}-648 i \pi ^2 \arctan (a x)^2+\frac {6480 \arctan (a x)^2}{\sqrt {1+a^2 x^2}}+864 i \pi \arctan (a x)^3+\frac {2160 a x \arctan (a x)^3}{\sqrt {1+a^2 x^2}}-432 i \arctan (a x)^4+32 \cos (3 \arctan (a x))-144 \arctan (a x)^2 \cos (3 \arctan (a x))-1296 \pi ^2 \arctan (a x) \log \left (1-i e^{-i \arctan (a x)}\right )+2592 \pi \arctan (a x)^2 \log \left (1-i e^{-i \arctan (a x)}\right )+216 \pi ^3 \log \left (1+i e^{-i \arctan (a x)}\right )-1728 \arctan (a x)^3 \log \left (1+i e^{-i \arctan (a x)}\right )-216 \pi ^3 \log \left (1+i e^{i \arctan (a x)}\right )+1296 \pi ^2 \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )-2592 \pi \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )+1728 \arctan (a x)^3 \log \left (1+i e^{i \arctan (a x)}\right )-216 \pi ^3 \log \left (\tan \left (\frac {1}{4} (\pi +2 \arctan (a x))\right )\right )-5184 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{-i \arctan (a x)}\right )-1296 i \pi (\pi -4 \arctan (a x)) \operatorname {PolyLog}\left (2,i e^{-i \arctan (a x)}\right )-1296 i \pi ^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )+5184 i \pi \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-5184 i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-10368 \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{-i \arctan (a x)}\right )+5184 \pi \operatorname {PolyLog}\left (3,i e^{-i \arctan (a x)}\right )-5184 \pi \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+10368 \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+10368 i \operatorname {PolyLog}\left (4,-i e^{-i \arctan (a x)}\right )+10368 i \operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )+96 \arctan (a x) \sin (3 \arctan (a x))-144 \arctan (a x)^3 \sin (3 \arctan (a x))\right )}{1728 a^5 c^3 \sqrt {1+a^2 x^2}} \]

input

Integrate[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]
 

output

-1/1728*(Sqrt[c*(1 + a^2*x^2)]*((189*I)*Pi^4 - 12960/Sqrt[1 + a^2*x^2] + ( 
216*I)*Pi^3*ArcTan[a*x] - (12960*a*x*ArcTan[a*x])/Sqrt[1 + a^2*x^2] - (648 
*I)*Pi^2*ArcTan[a*x]^2 + (6480*ArcTan[a*x]^2)/Sqrt[1 + a^2*x^2] + (864*I)* 
Pi*ArcTan[a*x]^3 + (2160*a*x*ArcTan[a*x]^3)/Sqrt[1 + a^2*x^2] - (432*I)*Ar 
cTan[a*x]^4 + 32*Cos[3*ArcTan[a*x]] - 144*ArcTan[a*x]^2*Cos[3*ArcTan[a*x]] 
 - 1296*Pi^2*ArcTan[a*x]*Log[1 - I/E^(I*ArcTan[a*x])] + 2592*Pi*ArcTan[a*x 
]^2*Log[1 - I/E^(I*ArcTan[a*x])] + 216*Pi^3*Log[1 + I/E^(I*ArcTan[a*x])] - 
 1728*ArcTan[a*x]^3*Log[1 + I/E^(I*ArcTan[a*x])] - 216*Pi^3*Log[1 + I*E^(I 
*ArcTan[a*x])] + 1296*Pi^2*ArcTan[a*x]*Log[1 + I*E^(I*ArcTan[a*x])] - 2592 
*Pi*ArcTan[a*x]^2*Log[1 + I*E^(I*ArcTan[a*x])] + 1728*ArcTan[a*x]^3*Log[1 
+ I*E^(I*ArcTan[a*x])] - 216*Pi^3*Log[Tan[(Pi + 2*ArcTan[a*x])/4]] - (5184 
*I)*ArcTan[a*x]^2*PolyLog[2, (-I)/E^(I*ArcTan[a*x])] - (1296*I)*Pi*(Pi - 4 
*ArcTan[a*x])*PolyLog[2, I/E^(I*ArcTan[a*x])] - (1296*I)*Pi^2*PolyLog[2, ( 
-I)*E^(I*ArcTan[a*x])] + (5184*I)*Pi*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcT 
an[a*x])] - (5184*I)*ArcTan[a*x]^2*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - 10 
368*ArcTan[a*x]*PolyLog[3, (-I)/E^(I*ArcTan[a*x])] + 5184*Pi*PolyLog[3, I/ 
E^(I*ArcTan[a*x])] - 5184*Pi*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + 10368*Ar 
cTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[a*x])] + (10368*I)*PolyLog[4, (-I)/E 
^(I*ArcTan[a*x])] + (10368*I)*PolyLog[4, (-I)*E^(I*ArcTan[a*x])] + 96*ArcT 
an[a*x]*Sin[3*ArcTan[a*x]] - 144*ArcTan[a*x]^3*Sin[3*ArcTan[a*x]]))/(a^...
 

3.5.51.3 Rubi [A] (verified)

Time = 3.66 (sec) , antiderivative size = 563, normalized size of antiderivative = 0.91, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.792, Rules used = {5499, 5479, 5475, 243, 53, 2009, 5465, 5429, 5499, 5425, 5423, 3042, 4669, 3011, 5433, 5429, 7163, 2720, 7143}

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.

\(\displaystyle \int \frac {x^4 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 5499

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{5/2}}dx}{a^2}\)

\(\Big \downarrow \) 5479

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \int \frac {x^3 \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{5/2}}dx}{a^2}\)

\(\Big \downarrow \) 5475

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {2}{9} \int \frac {x^3}{\left (a^2 c x^2+c\right )^{5/2}}dx-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{9} \int \frac {x^2}{\left (a^2 c x^2+c\right )^{5/2}}dx^2-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {1}{9} \int \left (\frac {1}{a^2 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {1}{a^2 \left (a^2 c x^2+c\right )^{5/2}}\right )dx^2-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}\right )}{a^2}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (\frac {2 \int \frac {x \arctan (a x)^2}{\left (a^2 c x^2+c\right )^{3/2}}dx}{3 a^2 c}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 5465

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (\frac {2 \left (\frac {2 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 5429

\(\displaystyle \frac {\int \frac {x^2 \arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 5499

\(\displaystyle \frac {\frac {\int \frac {\arctan (a x)^3}{\sqrt {a^2 c x^2+c}}dx}{a^2 c}-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 5425

\(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^3}{\sqrt {a^2 x^2+1}}dx}{a^2 c \sqrt {a^2 c x^2+c}}-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 5423

\(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^3d\arctan (a x)}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {\sqrt {a^2 x^2+1} \int \arctan (a x)^3 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{a^3 c \sqrt {a^2 c x^2+c}}-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}}{a^2 c}-\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}\)

\(\Big \downarrow \) 4669

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (-3 \int \arctan (a x)^2 \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+3 \int \arctan (a x)^2 \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

\(\Big \downarrow \) 3011

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {\int \frac {\arctan (a x)^3}{\left (a^2 c x^2+c\right )^{3/2}}dx}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

\(\Big \downarrow \) 5433

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {-6 \int \frac {\arctan (a x)}{\left (a^2 c x^2+c\right )^{3/2}}dx+\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

\(\Big \downarrow \) 5429

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \int \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

\(\Big \downarrow \) 7163

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (i \int \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )d\arctan (a x)-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

\(\Big \downarrow \) 2720

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

\(\Big \downarrow \) 7143

\(\displaystyle -\frac {\frac {x^3 \arctan (a x)^3}{3 c \left (a^2 c x^2+c\right )^{3/2}}-a \left (-\frac {x^2 \arctan (a x)^2}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {2 \left (\frac {2 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a}-\frac {\arctan (a x)^2}{a^2 c \sqrt {a^2 c x^2+c}}\right )}{3 a^2 c}+\frac {2 x^3 \arctan (a x)}{9 a c \left (a^2 c x^2+c\right )^{3/2}}+\frac {1}{9} \left (\frac {2}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2}{3 a^4 c \left (a^2 c x^2+c\right )^{3/2}}\right )\right )}{a^2}+\frac {-\frac {\frac {x \arctan (a x)^3}{c \sqrt {a^2 c x^2+c}}+\frac {3 \arctan (a x)^2}{a c \sqrt {a^2 c x^2+c}}-6 \left (\frac {x \arctan (a x)}{c \sqrt {a^2 c x^2+c}}+\frac {1}{a c \sqrt {a^2 c x^2+c}}\right )}{a^2}+\frac {\sqrt {a^2 x^2+1} \left (3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,-i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )\right )-3 \left (i \arctan (a x)^2 \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-2 i \left (\operatorname {PolyLog}\left (4,i e^{i \arctan (a x)}\right )-i \arctan (a x) \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^3\right )}{a^3 c \sqrt {a^2 c x^2+c}}}{a^2 c}\)

input

Int[(x^4*ArcTan[a*x]^3)/(c + a^2*c*x^2)^(5/2),x]
 

output

-(((x^3*ArcTan[a*x]^3)/(3*c*(c + a^2*c*x^2)^(3/2)) - a*((-2/(3*a^4*c*(c + 
a^2*c*x^2)^(3/2)) + 2/(a^4*c^2*Sqrt[c + a^2*c*x^2]))/9 + (2*x^3*ArcTan[a*x 
])/(9*a*c*(c + a^2*c*x^2)^(3/2)) - (x^2*ArcTan[a*x]^2)/(3*a^2*c*(c + a^2*c 
*x^2)^(3/2)) + (2*(-(ArcTan[a*x]^2/(a^2*c*Sqrt[c + a^2*c*x^2])) + (2*(1/(a 
*c*Sqrt[c + a^2*c*x^2]) + (x*ArcTan[a*x])/(c*Sqrt[c + a^2*c*x^2])))/a))/(3 
*a^2*c)))/a^2) + (-(((3*ArcTan[a*x]^2)/(a*c*Sqrt[c + a^2*c*x^2]) + (x*ArcT 
an[a*x]^3)/(c*Sqrt[c + a^2*c*x^2]) - 6*(1/(a*c*Sqrt[c + a^2*c*x^2]) + (x*A 
rcTan[a*x])/(c*Sqrt[c + a^2*c*x^2])))/a^2) + (Sqrt[1 + a^2*x^2]*((-2*I)*Ar 
cTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^3 + 3*(I*ArcTan[a*x]^2*PolyLog[2, (-I) 
*E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, (-I)*E^(I*ArcTan[ 
a*x])] + PolyLog[4, (-I)*E^(I*ArcTan[a*x])])) - 3*(I*ArcTan[a*x]^2*PolyLog 
[2, I*E^(I*ArcTan[a*x])] - (2*I)*((-I)*ArcTan[a*x]*PolyLog[3, I*E^(I*ArcTa 
n[a*x])] + PolyLog[4, I*E^(I*ArcTan[a*x])]))))/(a^3*c*Sqrt[c + a^2*c*x^2]) 
)/(a^2*c)
 

3.5.51.3.1 Defintions of rubi rules used

rule 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) 
*(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
 

rule 3011

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) 
*(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + 
b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F]))   Int[(f + g*x)^( 
m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e 
, f, g, n}, x] && GtQ[m, 0]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4669

Int[csc[(e_.) + Pi*(k_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol 
] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*k*Pi)*E^(I*(e + f*x))]/f), x] + (-Si 
mp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 - E^(I*k*Pi)*E^(I*(e + f*x))], x], 
 x] + Simp[d*(m/f)   Int[(c + d*x)^(m - 1)*Log[1 + E^(I*k*Pi)*E^(I*(e + f*x 
))], x], x]) /; FreeQ[{c, d, e, f}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 5423

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[1/(c*Sqrt[d])   Subst[Int[(a + b*x)^p*Sec[x], x], x, ArcTan[ 
c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0] && Gt 
Q[d, 0]
 

rule 5425

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]   Int[(a + b*ArcTan[c*x])^ 
p/Sqrt[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] & 
& IGtQ[p, 0] &&  !GtQ[d, 0]
 

rule 5429

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)^2)^(3/2), x_Symbo 
l] :> Simp[b/(c*d*Sqrt[d + e*x^2]), x] + Simp[x*((a + b*ArcTan[c*x])/(d*Sqr 
t[d + e*x^2])), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d]
 

rule 5433

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)/((d_) + (e_.)*(x_)^2)^(3/2), x_ 
Symbol] :> Simp[b*p*((a + b*ArcTan[c*x])^(p - 1)/(c*d*Sqrt[d + e*x^2])), x] 
 + (Simp[x*((a + b*ArcTan[c*x])^p/(d*Sqrt[d + e*x^2])), x] - Simp[b^2*p*(p 
- 1)   Int[(a + b*ArcTan[c*x])^(p - 2)/(d + e*x^2)^(3/2), x], x]) /; FreeQ[ 
{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[p, 1]
 

rule 5465

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)*((d_) + (e_.)*(x_)^2)^(q_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(q + 1)*((a + b*ArcTan[c*x])^p/(2*e*(q + 
1))), x] - Simp[b*(p/(2*c*(q + 1)))   Int[(d + e*x^2)^q*(a + b*ArcTan[c*x]) 
^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, q}, x] && EqQ[e, c^2*d] && GtQ[p, 
 0] && NeQ[q, -1]
 

rule 5475

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((f_.)*(x_))^(m_)*((d_) + (e_.) 
*(x_)^2)^(q_), x_Symbol] :> Simp[b*p*(f*x)^m*(d + e*x^2)^(q + 1)*((a + b*Ar 
cTan[c*x])^(p - 1)/(c*d*m^2)), x] + (-Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(q + 
 1)*((a + b*ArcTan[c*x])^p/(c^2*d*m)), x] + Simp[f^2*((m - 1)/(c^2*d*m)) 
Int[(f*x)^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*ArcTan[c*x])^p, x], x] - Simp[ 
b^2*p*((p - 1)/m^2)   Int[(f*x)^m*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 2) 
, x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && EqQ[m + 2* 
q + 2, 0] && LtQ[q, -1] && GtQ[p, 1]
 

rule 5479

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_ 
.)*(x_)^2)^(q_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*((a + 
 b*ArcTan[c*x])^p/(d*f*(m + 1))), x] - Simp[b*c*(p/(f*(m + 1)))   Int[(f*x) 
^(m + 1)*(d + e*x^2)^q*(a + b*ArcTan[c*x])^(p - 1), x], x] /; FreeQ[{a, b, 
c, d, e, f, m, q}, x] && EqQ[e, c^2*d] && EqQ[m + 2*q + 3, 0] && GtQ[p, 0] 
&& NeQ[m, -1]
 

rule 5499

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_.)*(x_)^(m_)*((d_) + (e_.)*(x_)^2 
)^(q_), x_Symbol] :> Simp[1/e   Int[x^(m - 2)*(d + e*x^2)^(q + 1)*(a + b*Ar 
cTan[c*x])^p, x], x] - Simp[d/e   Int[x^(m - 2)*(d + e*x^2)^q*(a + b*ArcTan 
[c*x])^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IntegersQ 
[p, 2*q] && LtQ[q, -1] && IGtQ[m, 1] && NeQ[p, -1]
 

rule 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d 
, e, n, p}, x] && EqQ[b*d, a*e]
 

rule 7163

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_. 
)*(x_))))^(p_.)], x_Symbol] :> Simp[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a 
+ b*x)))^p]/(b*c*p*Log[F])), x] - Simp[f*(m/(b*c*p*Log[F]))   Int[(e + f*x) 
^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c 
, d, e, f, n, p}, x] && GtQ[m, 0]
 

3.5.51.4 Maple [F]

\[\int \frac {x^{4} \arctan \left (a x \right )^{3}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}d x\]

input

int(x^4*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x)
 

output

int(x^4*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x)
 

3.5.51.5 Fricas [F]

\[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^4*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="fricas")
 

output

integral(sqrt(a^2*c*x^2 + c)*x^4*arctan(a*x)^3/(a^6*c^3*x^6 + 3*a^4*c^3*x^ 
4 + 3*a^2*c^3*x^2 + c^3), x)
 

3.5.51.6 Sympy [F]

\[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \operatorname {atan}^{3}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]

input

integrate(x**4*atan(a*x)**3/(a**2*c*x**2+c)**(5/2),x)
 

output

Integral(x**4*atan(a*x)**3/(c*(a**2*x**2 + 1))**(5/2), x)
 

3.5.51.7 Maxima [F]

\[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^4*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="maxima")
 

output

integrate(x^4*arctan(a*x)^3/(a^2*c*x^2 + c)^(5/2), x)
 

3.5.51.8 Giac [F]

\[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int { \frac {x^{4} \arctan \left (a x\right )^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}} \,d x } \]

input

integrate(x^4*arctan(a*x)^3/(a^2*c*x^2+c)^(5/2),x, algorithm="giac")
 

output

sage0*x
 

3.5.51.9 Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \arctan (a x)^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\mathrm {atan}\left (a\,x\right )}^3}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]

input

int((x^4*atan(a*x)^3)/(c + a^2*c*x^2)^(5/2),x)
 

output

int((x^4*atan(a*x)^3)/(c + a^2*c*x^2)^(5/2), x)