Integrand size = 21, antiderivative size = 516 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {17}{180} c^2 x \sqrt {c+a^2 c x^2}+\frac {1}{60} c x \left (c+a^2 c x^2\right )^{3/2}-\frac {5 c^2 \sqrt {c+a^2 c x^2} \arctan (a x)}{8 a}-\frac {5 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}{36 a}-\frac {\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)}{15 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \arctan (a x)^2+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2}{8 a \sqrt {c+a^2 c x^2}}+\frac {259 c^{5/2} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {c+a^2 c x^2}}\right )}{360 a}+\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 i c^3 \sqrt {1+a^2 x^2} \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}-\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}}+\frac {5 c^3 \sqrt {1+a^2 x^2} \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )}{8 a \sqrt {c+a^2 c x^2}} \] Output:
17/180*c^2*x*(a^2*c*x^2+c)^(1/2)+1/60*c*x*(a^2*c*x^2+c)^(3/2)-5/8*c^2*(a^2 *c*x^2+c)^(1/2)*arctan(a*x)/a-5/36*c*(a^2*c*x^2+c)^(3/2)*arctan(a*x)/a-1/1 5*(a^2*c*x^2+c)^(5/2)*arctan(a*x)/a+5/16*c^2*x*(a^2*c*x^2+c)^(1/2)*arctan( a*x)^2+5/24*c*x*(a^2*c*x^2+c)^(3/2)*arctan(a*x)^2+1/6*x*(a^2*c*x^2+c)^(5/2 )*arctan(a*x)^2-5/8*I*c^3*(a^2*x^2+1)^(1/2)*arctan((1+I*a*x)/(a^2*x^2+1)^( 1/2))*arctan(a*x)^2/a/(a^2*c*x^2+c)^(1/2)+259/360*c^(5/2)*arctanh(a*c^(1/2 )*x/(a^2*c*x^2+c)^(1/2))/a+5/8*I*c^3*(a^2*x^2+1)^(1/2)*arctan(a*x)*polylog (2,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a/(a^2*c*x^2+c)^(1/2)-5/8*I*c^3*(a^2*x^ 2+1)^(1/2)*arctan(a*x)*polylog(2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))/a/(a^2*c*x ^2+c)^(1/2)-5/8*c^3*(a^2*x^2+1)^(1/2)*polylog(3,-I*(1+I*a*x)/(a^2*x^2+1)^( 1/2))/a/(a^2*c*x^2+c)^(1/2)+5/8*c^3*(a^2*x^2+1)^(1/2)*polylog(3,I*(1+I*a*x )/(a^2*x^2+1)^(1/2))/a/(a^2*c*x^2+c)^(1/2)
Time = 1.24 (sec) , antiderivative size = 771, normalized size of antiderivative = 1.49 \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (424 a x \sqrt {1+a^2 x^2}+368 a^3 x^3 \sqrt {1+a^2 x^2}-56 a^5 x^5 \sqrt {1+a^2 x^2}+8288 \coth ^{-1}\left (\frac {a x}{\sqrt {1+a^2 x^2}}\right )-11028 \sqrt {1+a^2 x^2} \arctan (a x)+504 a^2 x^2 \sqrt {1+a^2 x^2} \arctan (a x)+12 a^4 x^4 \sqrt {1+a^2 x^2} \arctan (a x)+11970 a x \sqrt {1+a^2 x^2} \arctan (a x)^2+7380 a^3 x^3 \sqrt {1+a^2 x^2} \arctan (a x)^2+1170 a^5 x^5 \sqrt {1+a^2 x^2} \arctan (a x)^2-7200 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2+1550 \arctan (a x) \cos (3 \arctan (a x))+3210 a^2 x^2 \arctan (a x) \cos (3 \arctan (a x))+1770 a^4 x^4 \arctan (a x) \cos (3 \arctan (a x))+110 a^6 x^6 \arctan (a x) \cos (3 \arctan (a x))-90 \arctan (a x) \cos (5 \arctan (a x))-270 a^2 x^2 \arctan (a x) \cos (5 \arctan (a x))-270 a^4 x^4 \arctan (a x) \cos (5 \arctan (a x))-90 a^6 x^6 \arctan (a x) \cos (5 \arctan (a x))+7200 i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-7200 i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-7200 \operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )+7200 \operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )+372 \sin (3 \arctan (a x))+636 a^2 x^2 \sin (3 \arctan (a x))+156 a^4 x^4 \sin (3 \arctan (a x))-108 a^6 x^6 \sin (3 \arctan (a x))-1425 \arctan (a x)^2 \sin (3 \arctan (a x))-3555 a^2 x^2 \arctan (a x)^2 \sin (3 \arctan (a x))-2835 a^4 x^4 \arctan (a x)^2 \sin (3 \arctan (a x))-705 a^6 x^6 \arctan (a x)^2 \sin (3 \arctan (a x))-52 \sin (5 \arctan (a x))-156 a^2 x^2 \sin (5 \arctan (a x))-156 a^4 x^4 \sin (5 \arctan (a x))-52 a^6 x^6 \sin (5 \arctan (a x))+45 \arctan (a x)^2 \sin (5 \arctan (a x))+135 a^2 x^2 \arctan (a x)^2 \sin (5 \arctan (a x))+135 a^4 x^4 \arctan (a x)^2 \sin (5 \arctan (a x))+45 a^6 x^6 \arctan (a x)^2 \sin (5 \arctan (a x))\right )}{11520 a \sqrt {1+a^2 x^2}} \] Input:
Integrate[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]
Output:
(c^2*Sqrt[c + a^2*c*x^2]*(424*a*x*Sqrt[1 + a^2*x^2] + 368*a^3*x^3*Sqrt[1 + a^2*x^2] - 56*a^5*x^5*Sqrt[1 + a^2*x^2] + 8288*ArcCoth[(a*x)/Sqrt[1 + a^2 *x^2]] - 11028*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 504*a^2*x^2*Sqrt[1 + a^2*x^ 2]*ArcTan[a*x] + 12*a^4*x^4*Sqrt[1 + a^2*x^2]*ArcTan[a*x] + 11970*a*x*Sqrt [1 + a^2*x^2]*ArcTan[a*x]^2 + 7380*a^3*x^3*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 + 1170*a^5*x^5*Sqrt[1 + a^2*x^2]*ArcTan[a*x]^2 - (7200*I)*ArcTan[E^(I*Arc Tan[a*x])]*ArcTan[a*x]^2 + 1550*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 3210*a^2* x^2*ArcTan[a*x]*Cos[3*ArcTan[a*x]] + 1770*a^4*x^4*ArcTan[a*x]*Cos[3*ArcTan [a*x]] + 110*a^6*x^6*ArcTan[a*x]*Cos[3*ArcTan[a*x]] - 90*ArcTan[a*x]*Cos[5 *ArcTan[a*x]] - 270*a^2*x^2*ArcTan[a*x]*Cos[5*ArcTan[a*x]] - 270*a^4*x^4*A rcTan[a*x]*Cos[5*ArcTan[a*x]] - 90*a^6*x^6*ArcTan[a*x]*Cos[5*ArcTan[a*x]] + (7200*I)*ArcTan[a*x]*PolyLog[2, (-I)*E^(I*ArcTan[a*x])] - (7200*I)*ArcTa n[a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - 7200*PolyLog[3, (-I)*E^(I*ArcTan[ a*x])] + 7200*PolyLog[3, I*E^(I*ArcTan[a*x])] + 372*Sin[3*ArcTan[a*x]] + 6 36*a^2*x^2*Sin[3*ArcTan[a*x]] + 156*a^4*x^4*Sin[3*ArcTan[a*x]] - 108*a^6*x ^6*Sin[3*ArcTan[a*x]] - 1425*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 3555*a^2*x ^2*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 2835*a^4*x^4*ArcTan[a*x]^2*Sin[3*Arc Tan[a*x]] - 705*a^6*x^6*ArcTan[a*x]^2*Sin[3*ArcTan[a*x]] - 52*Sin[5*ArcTan [a*x]] - 156*a^2*x^2*Sin[5*ArcTan[a*x]] - 156*a^4*x^4*Sin[5*ArcTan[a*x]] - 52*a^6*x^6*Sin[5*ArcTan[a*x]] + 45*ArcTan[a*x]^2*Sin[5*ArcTan[a*x]] + ...
Time = 1.87 (sec) , antiderivative size = 485, normalized size of antiderivative = 0.94, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.905, Rules used = {5415, 211, 211, 224, 219, 5415, 211, 224, 219, 5415, 224, 219, 5425, 5423, 3042, 4669, 3011, 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.
| \(\displaystyle \int \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {5}{6} c \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx+\frac {1}{15} c \int \left (a^2 c x^2+c\right )^{3/2}dx+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {5}{6} c \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx+\frac {1}{15} c \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {5}{6} c \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {5}{6} c \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{6} c \int \left (a^2 c x^2+c\right )^{3/2} \arctan (a x)^2dx+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \int \sqrt {a^2 c x^2+c}dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{6} c \left (\frac {1}{2} c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \int \sqrt {a^2 c x^2+c} \arctan (a x)^2dx+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 5415 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+c \int \frac {1}{1-\frac {a^2 c x^2}{a^2 c x^2+c}}d\frac {x}{\sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {1}{2} c \int \frac {\arctan (a x)^2}{\sqrt {a^2 c x^2+c}}dx+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 5425 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \frac {\arctan (a x)^2}{\sqrt {a^2 x^2+1}}dx}{2 \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 5423 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \sqrt {a^2 x^2+1} \arctan (a x)^2d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \int \arctan (a x)^2 \csc \left (\arctan (a x)+\frac {\pi }{2}\right )d\arctan (a x)}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 4669 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (-2 \int \arctan (a x) \log \left (1-i e^{i \arctan (a x)}\right )d\arctan (a x)+2 \int \arctan (a x) \log \left (1+i e^{i \arctan (a x)}\right )d\arctan (a x)-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-i \int \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )d\arctan (a x)\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\int e^{-i \arctan (a x)} \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )de^{i \arctan (a x)}\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {5}{6} c \left (\frac {3}{4} c \left (\frac {c \sqrt {a^2 x^2+1} \left (2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,-i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,-i e^{i \arctan (a x)}\right )\right )-2 \left (i \arctan (a x) \operatorname {PolyLog}\left (2,i e^{i \arctan (a x)}\right )-\operatorname {PolyLog}\left (3,i e^{i \arctan (a x)}\right )\right )-2 i \arctan \left (e^{i \arctan (a x)}\right ) \arctan (a x)^2\right )}{2 a \sqrt {a^2 c x^2+c}}+\frac {1}{2} x \arctan (a x)^2 \sqrt {a^2 c x^2+c}-\frac {\arctan (a x) \sqrt {a^2 c x^2+c}}{a}+\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{a}\right )+\frac {1}{4} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}{6 a}+\frac {1}{6} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )\right )+\frac {1}{6} x \arctan (a x)^2 \left (a^2 c x^2+c\right )^{5/2}-\frac {\arctan (a x) \left (a^2 c x^2+c\right )^{5/2}}{15 a}+\frac {1}{15} c \left (\frac {3}{4} c \left (\frac {\sqrt {c} \text {arctanh}\left (\frac {a \sqrt {c} x}{\sqrt {a^2 c x^2+c}}\right )}{2 a}+\frac {1}{2} x \sqrt {a^2 c x^2+c}\right )+\frac {1}{4} x \left (a^2 c x^2+c\right )^{3/2}\right )\) |
Input:
Int[(c + a^2*c*x^2)^(5/2)*ArcTan[a*x]^2,x]
Output:
-1/15*((c + a^2*c*x^2)^(5/2)*ArcTan[a*x])/a + (x*(c + a^2*c*x^2)^(5/2)*Arc Tan[a*x]^2)/6 + (c*((x*(c + a^2*c*x^2)^(3/2))/4 + (3*c*((x*Sqrt[c + a^2*c* x^2])/2 + (Sqrt[c]*ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a)))/4)) /15 + (5*c*(-1/6*((c + a^2*c*x^2)^(3/2)*ArcTan[a*x])/a + (x*(c + a^2*c*x^2 )^(3/2)*ArcTan[a*x]^2)/4 + (c*((x*Sqrt[c + a^2*c*x^2])/2 + (Sqrt[c]*ArcTan h[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/(2*a)))/6 + (3*c*(-((Sqrt[c + a^2*c* x^2]*ArcTan[a*x])/a) + (x*Sqrt[c + a^2*c*x^2]*ArcTan[a*x]^2)/2 + (Sqrt[c]* ArcTanh[(a*Sqrt[c]*x)/Sqrt[c + a^2*c*x^2]])/a + (c*Sqrt[1 + a^2*x^2]*((-2* I)*ArcTan[E^(I*ArcTan[a*x])]*ArcTan[a*x]^2 + 2*(I*ArcTan[a*x]*PolyLog[2, ( -I)*E^(I*ArcTan[a*x])] - PolyLog[3, (-I)*E^(I*ArcTan[a*x])]) - 2*(I*ArcTan [a*x]*PolyLog[2, I*E^(I*ArcTan[a*x])] - PolyLog[3, I*E^(I*ArcTan[a*x])]))) /(2*a*Sqrt[c + a^2*c*x^2])))/4))/6
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
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]]]
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]
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]
Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))^(p_)*((d_) + (e_.)*(x_)^2)^(q_.), x_ Symbol] :> Simp[(-b)*p*(d + e*x^2)^q*((a + b*ArcTan[c*x])^(p - 1)/(2*c*q*(2 *q + 1))), x] + (Simp[x*(d + e*x^2)^q*((a + b*ArcTan[c*x])^p/(2*q + 1)), x] + Simp[2*d*(q/(2*q + 1)) Int[(d + e*x^2)^(q - 1)*(a + b*ArcTan[c*x])^p, x], x] + Simp[b^2*d*p*((p - 1)/(2*q*(2*q + 1))) Int[(d + e*x^2)^(q - 1)*( a + b*ArcTan[c*x])^(p - 2), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[q, 0] && GtQ[p, 1]
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]
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]
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]
Time = 3.53 (sec) , antiderivative size = 342, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(\frac {c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (120 a^{5} \arctan \left (a x \right )^{2} x^{5}-48 x^{4} \arctan \left (a x \right ) a^{4}+390 a^{3} \arctan \left (a x \right )^{2} x^{3}+12 a^{3} x^{3}-196 x^{2} a^{2} \arctan \left (a x \right )+495 a \arctan \left (a x \right )^{2} x +80 a x -598 \arctan \left (a x \right )\right )}{720 a}-\frac {i c^{2} \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (225 i \arctan \left (a x \right )^{2} \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-225 i \arctan \left (a x \right )^{2} \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+450 \arctan \left (a x \right ) \operatorname {polylog}\left (2, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-450 \arctan \left (a x \right ) \operatorname {polylog}\left (2, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+450 i \operatorname {polylog}\left (3, \frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-450 i \operatorname {polylog}\left (3, -\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+1036 \arctan \left (\frac {i a x +1}{\sqrt {a^{2} x^{2}+1}}\right )\right )}{720 a \sqrt {a^{2} x^{2}+1}}\) | \(342\) |
Input:
int((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x,method=_RETURNVERBOSE)
Output:
1/720*c^2/a*(c*(a*x-I)*(a*x+I))^(1/2)*(120*a^5*arctan(a*x)^2*x^5-48*x^4*ar ctan(a*x)*a^4+390*a^3*arctan(a*x)^2*x^3+12*a^3*x^3-196*x^2*a^2*arctan(a*x) +495*a*arctan(a*x)^2*x+80*a*x-598*arctan(a*x))-1/720*I*c^2*(c*(a*x-I)*(a*x +I))^(1/2)*(225*I*arctan(a*x)^2*ln(1-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-225*I* arctan(a*x)^2*ln(1+I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+450*arctan(a*x)*polylog( 2,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-450*arctan(a*x)*polylog(2,-I*(1+I*a*x)/(a ^2*x^2+1)^(1/2))+450*I*polylog(3,I*(1+I*a*x)/(a^2*x^2+1)^(1/2))-450*I*poly log(3,-I*(1+I*a*x)/(a^2*x^2+1)^(1/2))+1036*arctan((1+I*a*x)/(a^2*x^2+1)^(1 /2)))/a/(a^2*x^2+1)^(1/2)
\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="fricas")
Output:
integral((a^4*c^2*x^4 + 2*a^2*c^2*x^2 + c^2)*sqrt(a^2*c*x^2 + c)*arctan(a* x)^2, x)
\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}\, dx \] Input:
integrate((a**2*c*x**2+c)**(5/2)*atan(a*x)**2,x)
Output:
Integral((c*(a**2*x**2 + 1))**(5/2)*atan(a*x)**2, x)
\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{2} \,d x } \] Input:
integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="maxima")
Output:
integrate((a^2*c*x^2 + c)^(5/2)*arctan(a*x)^2, x)
Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((a^2*c*x^2+c)^(5/2)*arctan(a*x)^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\int {\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \] Input:
int(atan(a*x)^2*(c + a^2*c*x^2)^(5/2),x)
Output:
int(atan(a*x)^2*(c + a^2*c*x^2)^(5/2), x)
\[ \int \left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2 \, dx=\sqrt {c}\, c^{2} \left (\left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{4}d x \right ) a^{4}+2 \left (\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2} x^{2}d x \right ) a^{2}+\int \sqrt {a^{2} x^{2}+1}\, \mathit {atan} \left (a x \right )^{2}d x \right ) \] Input:
int((a^2*c*x^2+c)^(5/2)*atan(a*x)^2,x)
Output:
sqrt(c)*c**2*(int(sqrt(a**2*x**2 + 1)*atan(a*x)**2*x**4,x)*a**4 + 2*int(sq rt(a**2*x**2 + 1)*atan(a*x)**2*x**2,x)*a**2 + int(sqrt(a**2*x**2 + 1)*atan (a*x)**2,x))