Integrand size = 50, antiderivative size = 641 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (b^2 c d-2 a c (c d-a h)-a b (c f+a j)+\left (b c (c d+a h)-a b^2 j-2 a c (c f-a j)\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {b c (c e+a i)-a b^2 k-2 a c (c g-a k)+\left (2 c^3 e-c^2 (b g+2 a i)-b^3 k+b c (b i+3 a k)\right ) x^2}{2 c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (b (c d+a h)+\frac {a b^2 j}{c}-2 a (c f+3 a j)+\frac {b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b (c d+a h)+\frac {a b^2 j}{c}-2 a (c f+3 a j)-\frac {b^2 c (c d-a h)-4 a c^2 (3 c d+a h)-a b^3 j+4 a b c (c f+2 a j)}{c \sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \sqrt {c} \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\left (4 c^3 e-c^2 (2 b g-4 a i)+b^3 k-6 a b c k\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {k \log \left (a+b x^2+c x^4\right )}{4 c^2} \] Output:
1/2*x*(b^2*c*d-2*a*c*(-a*h+c*d)-a*b*(a*j+c*f)+(b*c*(a*h+c*d)-a*b^2*j-2*a*c *(-a*j+c*f))*x^2)/a/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)-1/2*(b*c*(a*i+c*e)-a*b^ 2*k-2*a*c*(-a*k+c*g)+(2*c^3*e-c^2*(2*a*i+b*g)-b^3*k+b*c*(3*a*k+b*i))*x^2)/ c^2/(-4*a*c+b^2)/(c*x^4+b*x^2+a)+1/4*(b*(a*h+c*d)+a*b^2*j/c-2*a*(3*a*j+c*f )+(b^2*c*(-a*h+c*d)-4*a*c^2*(a*h+3*c*d)-a*b^3*j+4*a*b*c*(2*a*j+c*f))/c/(-4 *a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^ (1/2)/a/c^(1/2)/(-4*a*c+b^2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*(b*(a*h+c*d) +a*b^2*j/c-2*a*(3*a*j+c*f)-(b^2*c*(-a*h+c*d)-4*a*c^2*(a*h+3*c*d)-a*b^3*j+4 *a*b*c*(2*a*j+c*f))/c/(-4*a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4* a*c+b^2)^(1/2))^(1/2))*2^(1/2)/a/c^(1/2)/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2 ))^(1/2)+1/2*(4*c^3*e-c^2*(-4*a*i+2*b*g)+b^3*k-6*a*b*c*k)*arctanh((2*c*x^2 +b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(3/2)+1/4*k*ln(c*x^4+b*x^2+a)/c^2
Time = 4.09 (sec) , antiderivative size = 775, normalized size of antiderivative = 1.21 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {\frac {2 \left (2 a^3 c k-b c^2 d x \left (b+c x^2\right )+a \left (-b^3 k x^2+b^2 c x^2 (i+j x)+2 c^3 x (d+x (e+f x))+b c^2 (e+x (f-x (g+h x)))\right )+a^2 \left (-b^2 k+b c (i+x (j+3 k x))-2 c^2 (g+x (h+x (i+j x)))\right )\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\sqrt {2} \sqrt {c} \left (a b^3 j-b c \left (c \sqrt {b^2-4 a c} d+4 a c f+a \sqrt {b^2-4 a c} h+8 a^2 j\right )-b^2 \left (c^2 d-a c h+a \sqrt {b^2-4 a c} j\right )+2 a c \left (6 c^2 d+c \sqrt {b^2-4 a c} f+2 a c h+3 a \sqrt {b^2-4 a c} j\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (a b^3 j+b c \left (c \sqrt {b^2-4 a c} d-4 a c f+a \sqrt {b^2-4 a c} h-8 a^2 j\right )+2 a c \left (6 c^2 d-c \sqrt {b^2-4 a c} f+2 a c h-3 a \sqrt {b^2-4 a c} j\right )+b^2 \left (-c^2 d+a c h+a \sqrt {b^2-4 a c} j\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\left (-4 c^3 e+2 c^2 (b g-2 a i)+b^2 \left (-b+\sqrt {b^2-4 a c}\right ) k+a c \left (6 b k-4 \sqrt {b^2-4 a c} k\right )\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (4 c^3 e+c^2 (-2 b g+4 a i)+b^2 \left (b+\sqrt {b^2-4 a c}\right ) k-2 a c \left (3 b+2 \sqrt {b^2-4 a c}\right ) k\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 c^2} \] Input:
Integrate[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^6 + k*x^7)/(a + b *x^2 + c*x^4)^2,x]
Output:
((2*(2*a^3*c*k - b*c^2*d*x*(b + c*x^2) + a*(-(b^3*k*x^2) + b^2*c*x^2*(i + j*x) + 2*c^3*x*(d + x*(e + f*x)) + b*c^2*(e + x*(f - x*(g + h*x)))) + a^2* (-(b^2*k) + b*c*(i + x*(j + 3*k*x)) - 2*c^2*(g + x*(h + x*(i + j*x))))))/( a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*Sqrt[c]*(a*b^3*j - b*c*(c *Sqrt[b^2 - 4*a*c]*d + 4*a*c*f + a*Sqrt[b^2 - 4*a*c]*h + 8*a^2*j) - b^2*(c ^2*d - a*c*h + a*Sqrt[b^2 - 4*a*c]*j) + 2*a*c*(6*c^2*d + c*Sqrt[b^2 - 4*a* c]*f + 2*a*c*h + 3*a*Sqrt[b^2 - 4*a*c]*j))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt [b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c ]]) + (Sqrt[2]*Sqrt[c]*(a*b^3*j + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 4*a*c*f + a *Sqrt[b^2 - 4*a*c]*h - 8*a^2*j) + 2*a*c*(6*c^2*d - c*Sqrt[b^2 - 4*a*c]*f + 2*a*c*h - 3*a*Sqrt[b^2 - 4*a*c]*j) + b^2*(-(c^2*d) + a*c*h + a*Sqrt[b^2 - 4*a*c]*j))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b ^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((-4*c^3*e + 2*c^2*(b*g - 2*a*i) + b^2*(-b + Sqrt[b^2 - 4*a*c])*k + a*c*(6*b*k - 4*Sqrt[b^2 - 4*a*c ]*k))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((4*c^3 *e + c^2*(-2*b*g + 4*a*i) + b^2*(b + Sqrt[b^2 - 4*a*c])*k - 2*a*c*(3*b + 2 *Sqrt[b^2 - 4*a*c])*k)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c) ^(3/2))/(4*c^2)
Time = 1.54 (sec) , antiderivative size = 653, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {2202, 2194, 2191, 1142, 1083, 219, 1103, 2206, 25, 1480, 218}
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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\int \frac {x \left (k x^6+i x^4+g x^2+e\right )}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} \int \frac {k x^6+i x^4+g x^2+e}{\left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 2191 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\int \frac {\left (4 a-\frac {b^2}{c}\right ) k x^2+2 c e-b g+2 a i-\frac {a b k}{c}}{c x^4+b x^2+a}dx^2}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (-\frac {\frac {\left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{2 c^2}-\frac {k \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {k \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c^2}-\frac {\left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (-\frac {-\frac {k \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c^2}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \int \frac {j x^6+h x^4+f x^2+d}{\left (c x^4+b x^2+a\right )^2}dx+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {k \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )\) |
| \(\Big \downarrow \) 2206 |
| \(\displaystyle -\frac {\int -\frac {d b^2+\frac {a (c f+a j) b}{c}+\left (\frac {a j b^2}{c}+(c d+a h) b-2 a (c f+3 a j)\right ) x^2-2 a (3 c d+a h)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {k \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 \left (-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right )+c \left (-\frac {a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {d b^2+\frac {a (c f+a j) b}{c}+\left (\frac {a j b^2}{c}+(c d+a h) b-2 a (c f+3 a j)\right ) x^2-2 a (3 c d+a h)}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {k \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 \left (-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right )+c \left (-\frac {a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {a b^2 j}{c}+\frac {-a b^3 j+b^2 c (c d-a h)+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt {b^2-4 a c}}+b (a h+c d)-2 a (3 a j+c f)\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {a b^2 j}{c}-\frac {-a b^3 j+b^2 c (c d-a h)+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt {b^2-4 a c}}+b (a h+c d)-2 a (3 a j+c f)\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {k \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 \left (-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right )+c \left (-\frac {a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {a b^2 j}{c}+\frac {-a b^3 j+b^2 c (c d-a h)+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt {b^2-4 a c}}+b (a h+c d)-2 a (3 a j+c f)\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {a b^2 j}{c}-\frac {-a b^3 j+b^2 c (c d-a h)+4 a b c (2 a j+c f)-4 a c^2 (a h+3 c d)}{c \sqrt {b^2-4 a c}}+b (a h+c d)-2 a (3 a j+c f)\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {1}{2} \left (-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-c^2 (2 b g-4 a i)-6 a b c k+b^3 k+4 c^3 e\right )}{c^2 \sqrt {b^2-4 a c}}-\frac {k \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c^2}}{b^2-4 a c}-\frac {x^2 \left (-c^2 (2 a i+b g)+b c (3 a k+b i)+b^3 (-k)+2 c^3 e\right )-a b^2 k+b c (a i+c e)-2 a c (c g-a k)}{c^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\right )+\frac {x \left (x^2 \left (-a b^2 j+b c (a h+c d)-2 a c (c f-a j)\right )+c \left (-\frac {a b (a j+c f)}{c}-2 a (c d-a h)+b^2 d\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
Input:
Int[(d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^6 + k*x^7)/(a + b*x^2 + c*x^4)^2,x]
Output:
(x*(c*(b^2*d - 2*a*(c*d - a*h) - (a*b*(c*f + a*j))/c) + (b*c*(c*d + a*h) - a*b^2*j - 2*a*c*(c*f - a*j))*x^2))/(2*a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^ 4)) + (((b*(c*d + a*h) + (a*b^2*j)/c - 2*a*(c*f + 3*a*j) + (b^2*c*(c*d - a *h) - 4*a*c^2*(3*c*d + a*h) - a*b^3*j + 4*a*b*c*(c*f + 2*a*j))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt [2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b*(c*d + a*h) + (a*b^2*j)/c - 2*a*(c*f + 3*a*j) - (b^2*c*(c*d - a*h) - 4*a*c^2*(3*c*d + a*h) - a*b^3*j + 4*a*b*c*(c*f + 2*a*j))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x) /Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c] ]))/(2*a*(b^2 - 4*a*c)) + (-((b*c*(c*e + a*i) - a*b^2*k - 2*a*c*(c*g - a*k ) + (2*c^3*e - c^2*(b*g + 2*a*i) - b^3*k + b*c*(b*i + 3*a*k))*x^2)/(c^2*(b ^2 - 4*a*c)*(a + b*x^2 + c*x^4))) - (-(((4*c^3*e - c^2*(2*b*g - 4*a*i) + b ^3*k - 6*a*b*c*k)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c])) - ((b^2 - 4*a*c)*k*Log[a + b*x^2 + c*x^4])/(2*c^2))/(b^2 - 4*a* c))/2
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
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[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[P q, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(b*f - 2*a*g + (2*c*f - b*g)*x)*((a + b*x + c*x^2)^ (p + 1)/((p + 1)*(b^2 - 4*a*c))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int [(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (2*p + 3)* (2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^ 2 - 4*a*c, 0] && LtQ[p, -1]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b ^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c *x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x ^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.45 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.66
| method | result | size |
| risch | \(\frac {-\frac {\left (2 a^{2} c j -a \,b^{2} j +a b c h -2 a \,c^{2} f +b \,c^{2} d \right ) x^{3}}{2 a \left (4 a c -b^{2}\right ) c}+\frac {\left (3 a b c k -2 a \,c^{2} i -b^{3} k +b^{2} c i -b \,c^{2} g +2 c^{3} e \right ) x^{2}}{2 \left (4 a c -b^{2}\right ) c^{2}}+\frac {\left (a^{2} b j -2 a^{2} c h +a b c f +2 a \,c^{2} d -b^{2} c d \right ) x}{2 a c \left (4 a c -b^{2}\right )}+\frac {2 a^{2} c k -a \,b^{2} k +a b c i -2 a \,c^{2} g +b \,c^{2} e}{2 \left (4 a c -b^{2}\right ) c^{2}}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (2 k \,\textit {\_R}^{3}+\frac {\left (6 a^{2} c j -a \,b^{2} j -a b c h +2 a \,c^{2} f -b \,c^{2} d \right ) \textit {\_R}^{2}}{a \left (4 a c -b^{2}\right )}-\frac {2 \left (a b k -2 a c i +b c g -2 e \,c^{2}\right ) \textit {\_R}}{4 a c -b^{2}}-\frac {a^{2} b j -2 a^{2} c h +a b c f -6 a \,c^{2} d +b^{2} c d}{a \left (4 a c -b^{2}\right )}\right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{3} c +\textit {\_R} b}}{4 c}\) | \(422\) |
| default | \(\text {Expression too large to display}\) | \(1072\) |
Input:
int((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x,method =_RETURNVERBOSE)
Output:
(-1/2/a*(2*a^2*c*j-a*b^2*j+a*b*c*h-2*a*c^2*f+b*c^2*d)/(4*a*c-b^2)/c*x^3+1/ 2*(3*a*b*c*k-2*a*c^2*i-b^3*k+b^2*c*i-b*c^2*g+2*c^3*e)/(4*a*c-b^2)/c^2*x^2+ 1/2*(a^2*b*j-2*a^2*c*h+a*b*c*f+2*a*c^2*d-b^2*c*d)/a/c/(4*a*c-b^2)*x+1/2*(2 *a^2*c*k-a*b^2*k+a*b*c*i-2*a*c^2*g+b*c^2*e)/(4*a*c-b^2)/c^2)/(c*x^4+b*x^2+ a)+1/4/c*sum((2*k*_R^3+1/a*(6*a^2*c*j-a*b^2*j-a*b*c*h+2*a*c^2*f-b*c^2*d)/( 4*a*c-b^2)*_R^2-2*(a*b*k-2*a*c*i+b*c*g-2*c^2*e)/(4*a*c-b^2)*_R-1/a*(a^2*b* j-2*a^2*c*h+a*b*c*f-6*a*c^2*d+b^2*c*d)/(4*a*c-b^2))/(2*_R^3*c+_R*b)*ln(x-_ R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate((k*x**7+j*x**6+i*x**5+h*x**4+g*x**3+f*x**2+e*x+d)/(c*x**4+b*x**2 +a)**2,x)
Output:
Timed out
\[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {k x^{7} + j x^{6} + i x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \] Input:
integrate((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
Output:
-1/2*(a*b*c^2*e - 2*a^2*c^2*g + a^2*b*c*i - (b*c^3*d - 2*a*c^3*f + a*b*c^2 *h - (a*b^2*c - 2*a^2*c^2)*j)*x^3 + (2*a*c^3*e - a*b*c^2*g + (a*b^2*c - 2* a^2*c^2)*i - (a*b^3 - 3*a^2*b*c)*k)*x^2 - (a^2*b^2 - 2*a^3*c)*k + (a*b*c^2 *f - 2*a^2*c^2*h + a^2*b*c*j - (b^2*c^2 - 2*a*c^3)*d)*x)/(a^2*b^2*c^2 - 4* a^3*c^3 + (a*b^2*c^3 - 4*a^2*c^4)*x^4 + (a*b^3*c^2 - 4*a^2*b*c^3)*x^2) - 1 /2*integrate(-(2*(a*b^2 - 4*a^2*c)*k*x^3 + a*b*c*f - 2*a^2*c*h + a^2*b*j + (b*c^2*d - 2*a*c^2*f + a*b*c*h + (a*b^2 - 6*a^2*c)*j)*x^2 + (b^2*c - 6*a* c^2)*d - 2*(2*a*c^2*e - a*b*c*g + 2*a^2*c*i - a^2*b*k)*x)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 16214 vs. \(2 (590) = 1180\).
Time = 2.27 (sec) , antiderivative size = 16214, normalized size of antiderivative = 25.29 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
Output:
1/4*k*log(abs(c*x^4 + b*x^2 + a))/c^2 + 1/16*((a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)^2*(2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*d - 2*(a^2*b^4*c^3 - 8*a^3*b^2* c^4 + 16*a^4*c^5)^2*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*s qrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sq rt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*a*c^4)*f + (a^2*b^4*c^3 - 8*a^3 *b^2*c^4 + 16*a^4*c^5)^2*(2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4 *a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)* sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*h + (a^2*b^ 4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5)^2*(2*a*b^4*c^2 - 20*a^2*b^2*c^3 + 48*a ^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c - 2...
Time = 28.53 (sec) , antiderivative size = 53538, normalized size of antiderivative = 83.52 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int((d + e*x + f*x^2 + g*x^3 + h*x^4 + i*x^5 + j*x^6 + k*x^7)/(a + b*x^2 + c*x^4)^2,x)
Output:
((b*c^2*e - 2*a*c^2*g - a*b^2*k + 2*a^2*c*k + a*b*c*i)/(2*c^2*(4*a*c - b^2 )) + (x^2*(2*c^3*e - b^3*k - b*c^2*g - 2*a*c^2*i + b^2*c*i + 3*a*b*c*k))/( 2*c^2*(4*a*c - b^2)) + (x*(2*a*c^2*d - b^2*c*d - 2*a^2*c*h + a^2*b*j + a*b *c*f))/(2*a*c*(4*a*c - b^2)) - (x^3*(b*c^2*d - 2*a*c^2*f - a*b^2*j + 2*a^2 *c*j + a*b*c*h))/(2*a*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) + symsum(log(r oot(1572864*a^8*b^2*c^9*z^4 - 983040*a^7*b^4*c^8*z^4 + 327680*a^6*b^6*c^7* z^4 - 61440*a^5*b^8*c^6*z^4 + 6144*a^4*b^10*c^5*z^4 - 256*a^3*b^12*c^4*z^4 - 1048576*a^9*c^10*z^4 - 1572864*a^8*b^2*c^7*k*z^3 + 983040*a^7*b^4*c^6*k *z^3 - 327680*a^6*b^6*c^5*k*z^3 + 61440*a^5*b^8*c^4*k*z^3 - 6144*a^4*b^10* c^3*k*z^3 + 256*a^3*b^12*c^2*k*z^3 + 1048576*a^9*c^8*k*z^3 + 98304*a^8*b*c ^6*i*k*z^2 + 98304*a^7*b*c^7*e*k*z^2 + 57344*a^7*b*c^7*f*j*z^2 + 32768*a^7 *b*c^7*g*i*z^2 + 57344*a^6*b*c^8*d*h*z^2 + 32768*a^6*b*c^8*e*g*z^2 - 32*a* b^10*c^4*d*f*z^2 - 90112*a^7*b^3*c^5*i*k*z^2 + 30720*a^6*b^5*c^4*i*k*z^2 - 4608*a^5*b^7*c^3*i*k*z^2 + 256*a^4*b^9*c^2*i*k*z^2 - 49152*a^7*b^2*c^6*g* k*z^2 + 45056*a^6*b^4*c^5*g*k*z^2 + 24576*a^7*b^2*c^6*h*j*z^2 - 15360*a^5* b^6*c^4*g*k*z^2 - 3072*a^5*b^6*c^4*h*j*z^2 + 2304*a^4*b^8*c^3*g*k*z^2 + 20 48*a^6*b^4*c^5*h*j*z^2 + 576*a^4*b^8*c^3*h*j*z^2 - 128*a^3*b^10*c^2*g*k*z^ 2 - 32*a^3*b^10*c^2*h*j*z^2 - 90112*a^6*b^3*c^6*e*k*z^2 - 49152*a^6*b^3*c^ 6*f*j*z^2 + 30720*a^5*b^5*c^5*e*k*z^2 - 24576*a^6*b^3*c^6*g*i*z^2 + 15360* a^5*b^5*c^5*f*j*z^2 + 6144*a^5*b^5*c^5*g*i*z^2 - 4608*a^4*b^7*c^4*e*k*z...
Time = 1.07 (sec) , antiderivative size = 11710, normalized size of antiderivative = 18.27 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5+j x^6+k x^7}{\left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int((k*x^7+j*x^6+i*x^5+h*x^4+g*x^3+f*x^2+e*x+d)/(c*x^4+b*x^2+a)^2,x)
Output:
(24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*s qrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b**2* c*k - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqr t(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4* b*c**2*i - 4*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan( (sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a **3*b**4*k + 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*at an((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b) )*a**3*b**3*c*k*x**2 + 8*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a ) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqr t(a) + b))*a**3*b**2*c**2*g - 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c )*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqr t(c)*sqrt(a) + b))*a**3*b**2*c**2*i*x**2 + 24*sqrt(2*sqrt(c)*sqrt(a) + b)* sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)* x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**2*k*x**4 - 16*sqrt(2*sqrt(c)* sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**3*e - 16*sqrt(2*sqr t(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b*c**3*i*x**4 - 4*s qrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqr...