Optimal. Leaf size=1384 \[ -\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )+3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )+3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c d^4 e^4+9 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.38, antiderivative size = 1384, normalized size of antiderivative = 1.00, number
of steps used = 31, number of rules used = 14, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.824, Rules
used = {6874, 1868, 1890, 281, 211, 1182, 1176, 631, 210, 1179, 642, 1262, 649, 266}
\begin {gather*} \frac {12 c d^2 \left (3 c d^4-a e^4\right ) \log (d+e x) e^7}{\left (c d^4+a e^4\right )^4}-\frac {3 c d^2 \left (3 c d^4-a e^4\right ) \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^4}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {\sqrt {c} \left (21 c^2 d^8-26 a c e^4 d^4+a^2 e^8\right ) \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 \left (7 c d^4-5 a e^4\right ) e^2+3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c e^4 d^4+a^2 e^8\right )\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) \text {ArcTan}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac {c \left (2 a d^2 \left (5 c d^4-3 a e^4\right ) e^3+x \left (2 c e^2 \left (3 c d^4-5 a e^4\right ) x^2 d^3+\left (c^2 d^8-12 a c e^4 d^4+3 a^2 e^8\right ) d-e \left (3 c^2 d^8-12 a c e^4 d^4+a^2 e^8\right ) x\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (c x^4+a\right )}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4+2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (3 c^2 d^8-36 a c e^4 d^4-2 \sqrt {a} \sqrt {c} e^2 \left (3 c d^4-5 a e^4\right ) d^2+9 a^2 e^8\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 210
Rule 211
Rule 266
Rule 281
Rule 631
Rule 642
Rule 649
Rule 1176
Rule 1179
Rule 1182
Rule 1262
Rule 1868
Rule 1890
Rule 6874
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )^2} \, dx &=\int \left (\frac {e^8}{\left (c d^4+a e^4\right )^2 (d+e x)^3}+\frac {8 c d^3 e^8}{\left (c d^4+a e^4\right )^3 (d+e x)^2}+\frac {12 c d^2 e^8 \left (3 c d^4-a e^4\right )}{\left (c d^4+a e^4\right )^4 (d+e x)}+\frac {c \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )^2}+\frac {c e^4 \left (3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3\right )}{\left (c d^4+a e^4\right )^4 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) x+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac {c \int \frac {d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2-2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \left (\frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4}+\frac {x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )+2 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x-2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {3 d \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )+4 c d^3 e^2 \left (7 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \int \frac {x \left (-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^4}-\frac {c \int \left (\frac {2 e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x}{a+c x^4}+\frac {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {\left (c e^4\right ) \text {Subst}\left (\int \frac {-e \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )-12 c d^2 e^3 \left (3 c d^4-a e^4\right ) x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac {c \int \frac {-3 d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^3}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \int \frac {x}{a+c x^4} \, dx}{2 a \left (c d^4+a e^4\right )^3}-\frac {\left (\sqrt {c} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{2 \left (c d^4+a e^4\right )^4}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}-\frac {\left (6 c^2 d^2 e^7 \left (3 c d^4-a e^4\right )\right ) \text {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^4}-\frac {\left (c e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^4}-\frac {\left (c e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right )\right ) \text {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^4+a e^4\right )^3}+\frac {\left (c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {\left (c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^4}+\frac {\left (c d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{4 \left (c d^4+a e^4\right )^4}-\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}-c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\sqrt {a} \sqrt {c}+c x^2}{a+c x^4} \, dx}{8 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac {\left (c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac {\left (c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}+2 x}{-\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {\frac {\sqrt {2} \sqrt [4]{a}}{\sqrt [4]{c}}-2 x}{-\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}-x^2} \, dx}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^3}+\frac {\left (c d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \int \frac {1}{\frac {\sqrt {a}}{\sqrt {c}}+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{c}}+x^2} \, dx}{16 a \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}+\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {\left (c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}\\ &=-\frac {e^7}{2 \left (c d^4+a e^4\right )^2 (d+e x)^2}-\frac {8 c d^3 e^7}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (2 a d^2 e^3 \left (5 c d^4-3 a e^4\right )+x \left (d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )-e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) x+2 c d^3 e^2 \left (3 c d^4-5 a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}-\frac {\sqrt {c} e^5 \left (21 c^2 d^8-26 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^4}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )^3}-\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac {c^{5/4} d e^4 \left (4 d^2 e^2 \left (7 c d^4-5 a e^4\right )+\frac {3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^4}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6+\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}+\frac {12 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^4}+\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}+\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d e^4 \left (4 \sqrt {a} \sqrt {c} d^2 e^2 \left (7 c d^4-5 a e^4\right )-3 \left (5 c^2 d^8-10 a c d^4 e^4+a^2 e^8\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^4}-\frac {c^{5/4} d \left (6 c d^6 e^2-10 a d^2 e^6-\frac {3 \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8\right )}{\sqrt {a} \sqrt {c}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{5/4} \left (c d^4+a e^4\right )^3}-\frac {3 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^4}\\ \end {align*}
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Mathematica [A]
time = 0.90, size = 996, normalized size = 0.72 \begin {gather*} \frac {-\frac {16 e^7 \left (c d^4+a e^4\right )^2}{(d+e x)^2}-\frac {256 c d^3 e^7 \left (c d^4+a e^4\right )}{d+e x}+\frac {8 c \left (c d^4+a e^4\right ) \left (-a^2 e^7 \left (6 d^2-3 d e x+e^2 x^2\right )+c^2 d^7 x \left (d^2-3 d e x+6 e^2 x^2\right )+2 a c d^3 e^3 \left (5 d^3-6 d^2 e x+6 d e^2 x^2-5 e^3 x^3\right )\right )}{a \left (a+c x^4\right )}-\frac {6 \sqrt {c} \left (\sqrt {2} c^{13/4} d^{13}-4 \sqrt [4]{a} c^3 d^{12} e+2 \sqrt {2} \sqrt {a} c^{11/4} d^{11} e^2+9 \sqrt {2} a c^{9/4} d^9 e^4-44 a^{5/4} c^2 d^8 e^5+36 \sqrt {2} a^{3/2} c^{7/4} d^7 e^6-49 \sqrt {2} a^2 c^{5/4} d^5 e^8+84 a^{9/4} c d^4 e^9-30 \sqrt {2} a^{5/2} c^{3/4} d^3 e^{10}+7 \sqrt {2} a^3 \sqrt [4]{c} d e^{12}-4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {6 \sqrt {c} \left (\sqrt {2} c^{13/4} d^{13}+4 \sqrt [4]{a} c^3 d^{12} e+2 \sqrt {2} \sqrt {a} c^{11/4} d^{11} e^2+9 \sqrt {2} a c^{9/4} d^9 e^4+44 a^{5/4} c^2 d^8 e^5+36 \sqrt {2} a^{3/2} c^{7/4} d^7 e^6-49 \sqrt {2} a^2 c^{5/4} d^5 e^8-84 a^{9/4} c d^4 e^9-30 \sqrt {2} a^{5/2} c^{3/4} d^3 e^{10}+7 \sqrt {2} a^3 \sqrt [4]{c} d e^{12}+4 a^{13/4} e^{13}\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+384 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log (d+e x)-\frac {3 \sqrt {2} c^{3/4} \left (c^3 d^{13}-2 \sqrt {a} c^{5/2} d^{11} e^2+9 a c^2 d^9 e^4-36 a^{3/2} c^{3/2} d^7 e^6-49 a^2 c d^5 e^8+30 a^{5/2} \sqrt {c} d^3 e^{10}+7 a^3 d e^{12}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {3 \sqrt {2} c^{3/4} \left (c^3 d^{13}-2 \sqrt {a} c^{5/2} d^{11} e^2+9 a c^2 d^9 e^4-36 a^{3/2} c^{3/2} d^7 e^6-49 a^2 c d^5 e^8+30 a^{5/2} \sqrt {c} d^3 e^{10}+7 a^3 d e^{12}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}-96 c d^2 e^7 \left (3 c d^4-a e^4\right ) \log \left (a+c x^4\right )}{32 \left (c d^4+a e^4\right )^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [A]
time = 0.33, size = 680, normalized size = 0.49
method | result | size |
default | \(-\frac {e^{7}}{2 \left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )^{2}}-\frac {8 c \,d^{3} e^{7}}{\left (e^{4} a +d^{4} c \right )^{3} \left (e x +d \right )}-\frac {12 e^{7} d^{2} c \left (e^{4} a -3 d^{4} c \right ) \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{4}}+\frac {c \left (\frac {-\frac {d^{3} e^{2} c \left (5 a^{2} e^{8}+2 a c \,d^{4} e^{4}-3 c^{2} d^{8}\right ) x^{3}}{2 a}-\frac {e \left (a^{3} e^{12}-11 a^{2} c \,d^{4} e^{8}-9 a \,c^{2} d^{8} e^{4}+3 c^{3} d^{12}\right ) x^{2}}{4 a}+\frac {d \left (3 a^{3} e^{12}-9 a^{2} c \,d^{4} e^{8}-11 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}\right ) x}{4 a}-\frac {d^{2} e^{3} \left (3 a^{2} e^{8}-2 a c \,d^{4} e^{4}-5 c^{2} d^{8}\right )}{2}}{c \,x^{4}+a}+\frac {\frac {3 \left (7 a^{3} d \,e^{12}-49 a^{2} c \,d^{5} e^{8}+9 a \,c^{2} d^{9} e^{4}+c^{3} d^{13}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a}+\frac {3 \left (-2 a^{3} e^{13}+42 a^{2} c \,d^{4} e^{9}-22 a \,c^{2} d^{8} e^{5}-2 c^{3} e \,d^{12}\right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{8 \sqrt {a c}}+\frac {3 \left (-30 a^{2} c \,d^{3} e^{10}+36 a \,c^{2} d^{7} e^{6}+2 e^{2} c^{3} d^{11}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{32 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+\frac {3 \left (16 a^{2} c \,d^{2} e^{11}-48 a \,c^{2} d^{6} e^{7}\right ) \ln \left (c \,x^{4}+a \right )}{16 c}}{a}\right )}{\left (e^{4} a +d^{4} c \right )^{4}}\) | \(680\) |
risch | \(\text {Expression too large to display}\) | \(1541\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 1321, normalized size = 0.95 \begin {gather*} \frac {3 \, c {\left (\frac {\sqrt {2} {\left (c^{4} d^{13} - 2 \, \sqrt {a} c^{\frac {7}{2}} d^{11} e^{2} + 9 \, a c^{3} d^{9} e^{4} - 36 \, a^{\frac {3}{2}} c^{\frac {5}{2}} d^{7} e^{6} - 48 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {9}{4}} d^{6} e^{7} - 49 \, a^{2} c^{2} d^{5} e^{8} + 30 \, a^{\frac {5}{2}} c^{\frac {3}{2}} d^{3} e^{10} + 16 \, \sqrt {2} a^{\frac {11}{4}} c^{\frac {5}{4}} d^{2} e^{11} + 7 \, a^{3} c d e^{12}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {\sqrt {2} {\left (c^{4} d^{13} - 2 \, \sqrt {a} c^{\frac {7}{2}} d^{11} e^{2} + 9 \, a c^{3} d^{9} e^{4} - 36 \, a^{\frac {3}{2}} c^{\frac {5}{2}} d^{7} e^{6} + 48 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {9}{4}} d^{6} e^{7} - 49 \, a^{2} c^{2} d^{5} e^{8} + 30 \, a^{\frac {5}{2}} c^{\frac {3}{2}} d^{3} e^{10} - 16 \, \sqrt {2} a^{\frac {11}{4}} c^{\frac {5}{4}} d^{2} e^{11} + 7 \, a^{3} c d e^{12}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {17}{4}} d^{13} + 4 \, \sqrt {a} c^{4} d^{12} e + 2 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {15}{4}} d^{11} e^{2} + 9 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {13}{4}} d^{9} e^{4} + 44 \, a^{\frac {3}{2}} c^{3} d^{8} e^{5} + 36 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {11}{4}} d^{7} e^{6} - 49 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {9}{4}} d^{5} e^{8} - 84 \, a^{\frac {5}{2}} c^{2} d^{4} e^{9} - 30 \, \sqrt {2} a^{\frac {11}{4}} c^{\frac {7}{4}} d^{3} e^{10} + 7 \, \sqrt {2} a^{\frac {13}{4}} c^{\frac {5}{4}} d e^{12} + 4 \, a^{\frac {7}{2}} c e^{13}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {17}{4}} d^{13} - 4 \, \sqrt {a} c^{4} d^{12} e + 2 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {15}{4}} d^{11} e^{2} + 9 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {13}{4}} d^{9} e^{4} - 44 \, a^{\frac {3}{2}} c^{3} d^{8} e^{5} + 36 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {11}{4}} d^{7} e^{6} - 49 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {9}{4}} d^{5} e^{8} + 84 \, a^{\frac {5}{2}} c^{2} d^{4} e^{9} - 30 \, \sqrt {2} a^{\frac {11}{4}} c^{\frac {7}{4}} d^{3} e^{10} + 7 \, \sqrt {2} a^{\frac {13}{4}} c^{\frac {5}{4}} d e^{12} - 4 \, a^{\frac {7}{2}} c e^{13}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{32 \, {\left (a c^{4} d^{16} + 4 \, a^{2} c^{3} d^{12} e^{4} + 6 \, a^{3} c^{2} d^{8} e^{8} + 4 \, a^{4} c d^{4} e^{12} + a^{5} e^{16}\right )}} + \frac {12 \, {\left (3 \, c^{2} d^{6} e^{7} - a c d^{2} e^{11}\right )} \log \left (x e + d\right )}{c^{4} d^{16} + 4 \, a c^{3} d^{12} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{8} + 4 \, a^{3} c d^{4} e^{12} + a^{4} e^{16}} + \frac {10 \, a c^{2} d^{8} e^{3} - 40 \, a^{2} c d^{4} e^{7} + 6 \, {\left (c^{3} d^{7} e^{4} - 7 \, a c^{2} d^{3} e^{8}\right )} x^{5} + 3 \, {\left (3 \, c^{3} d^{8} e^{3} - 14 \, a c^{2} d^{4} e^{7} - a^{2} c e^{11}\right )} x^{4} + {\left (c^{3} d^{9} e^{2} + 2 \, a c^{2} d^{5} e^{6} + a^{2} c d e^{10}\right )} x^{3} - 2 \, a^{3} e^{11} - {\left (c^{3} d^{10} e + 2 \, a c^{2} d^{6} e^{5} + a^{2} c d^{2} e^{9}\right )} x^{2} + {\left (c^{3} d^{11} + 8 \, a c^{2} d^{7} e^{4} - 41 \, a^{2} c d^{3} e^{8}\right )} x}{4 \, {\left (a^{2} c^{3} d^{14} + 3 \, a^{3} c^{2} d^{10} e^{4} + 3 \, a^{4} c d^{6} e^{8} + a^{5} d^{2} e^{12} + {\left (a c^{4} d^{12} e^{2} + 3 \, a^{2} c^{3} d^{8} e^{6} + 3 \, a^{3} c^{2} d^{4} e^{10} + a^{4} c e^{14}\right )} x^{6} + 2 \, {\left (a c^{4} d^{13} e + 3 \, a^{2} c^{3} d^{9} e^{5} + 3 \, a^{3} c^{2} d^{5} e^{9} + a^{4} c d e^{13}\right )} x^{5} + {\left (a c^{4} d^{14} + 3 \, a^{2} c^{3} d^{10} e^{4} + 3 \, a^{3} c^{2} d^{6} e^{8} + a^{4} c d^{2} e^{12}\right )} x^{4} + {\left (a^{2} c^{3} d^{12} e^{2} + 3 \, a^{3} c^{2} d^{8} e^{6} + 3 \, a^{4} c d^{4} e^{10} + a^{5} e^{14}\right )} x^{2} + 2 \, {\left (a^{2} c^{3} d^{13} e + 3 \, a^{3} c^{2} d^{9} e^{5} + 3 \, a^{4} c d^{5} e^{9} + a^{5} d e^{13}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 3.98, size = 1488, normalized size = 1.08 \begin {gather*} \frac {3 \, {\left (2 \, \sqrt {2} \sqrt {a c} c^{3} d^{4} e + \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{5} + 4 \, \sqrt {2} a c^{3} d^{2} e^{3} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e^{2} - 9 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{4} + 2 \, \sqrt {2} \sqrt {a c} a c^{2} e^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{8} - 8 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{3} d^{7} e + 16 \, \sqrt {2} \sqrt {a c} a^{2} c^{3} d^{6} e^{2} + 34 \, \sqrt {2} a^{3} c^{3} d^{4} e^{4} - 40 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} c d^{5} e^{3} - 40 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{3} c^{2} d^{3} e^{5} + 16 \, \sqrt {2} \sqrt {a c} a^{3} c^{2} d^{2} e^{6} + \sqrt {2} a^{4} c^{2} e^{8} - 8 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{3} d e^{7}\right )}} + \frac {3 \, {\left (2 \, \sqrt {2} \sqrt {a c} c^{3} d^{4} e + \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{5} - 4 \, \sqrt {2} a c^{3} d^{2} e^{3} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e^{2} - 9 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d e^{4} + 2 \, \sqrt {2} \sqrt {a c} a c^{2} e^{5}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} c^{4} d^{8} + 8 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{3} d^{7} e + 16 \, \sqrt {2} \sqrt {a c} a^{2} c^{3} d^{6} e^{2} + 34 \, \sqrt {2} a^{3} c^{3} d^{4} e^{4} + 40 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} c d^{5} e^{3} + 40 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{3} c^{2} d^{3} e^{5} + 16 \, \sqrt {2} \sqrt {a c} a^{3} c^{2} d^{2} e^{6} + \sqrt {2} a^{4} c^{2} e^{8} + 8 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{3} d e^{7}\right )}} + \frac {3 \, {\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{4} d^{13} - 2 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c^{2} d^{11} e^{2} + 9 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{3} d^{9} e^{4} - 36 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a c d^{7} e^{6} - 49 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{5} e^{8} + 30 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d^{3} e^{10} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{3} c d e^{12}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{5} d^{16} + 4 \, a^{3} c^{4} d^{12} e^{4} + 6 \, a^{4} c^{3} d^{8} e^{8} + 4 \, a^{5} c^{2} d^{4} e^{12} + a^{6} c e^{16}\right )}} - \frac {3 \, {\left (\sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{4} d^{13} - 2 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c^{2} d^{11} e^{2} + 9 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{3} d^{9} e^{4} - 36 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a c d^{7} e^{6} - 49 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{5} e^{8} + 30 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d^{3} e^{10} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a^{3} c d e^{12}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{5} d^{16} + 4 \, a^{3} c^{4} d^{12} e^{4} + 6 \, a^{4} c^{3} d^{8} e^{8} + 4 \, a^{5} c^{2} d^{4} e^{12} + a^{6} c e^{16}\right )}} - \frac {3 \, {\left (3 \, c^{2} d^{6} e^{7} - a c d^{2} e^{11}\right )} \log \left ({\left | c x^{4} + a \right |}\right )}{c^{4} d^{16} + 4 \, a c^{3} d^{12} e^{4} + 6 \, a^{2} c^{2} d^{8} e^{8} + 4 \, a^{3} c d^{4} e^{12} + a^{4} e^{16}} + \frac {12 \, {\left (3 \, c^{2} d^{6} e^{8} - a c d^{2} e^{12}\right )} \log \left ({\left | x e + d \right |}\right )}{c^{4} d^{16} e + 4 \, a c^{3} d^{12} e^{5} + 6 \, a^{2} c^{2} d^{8} e^{9} + 4 \, a^{3} c d^{4} e^{13} + a^{4} e^{17}} + \frac {10 \, a c^{3} d^{12} e^{3} - 30 \, a^{2} c^{2} d^{8} e^{7} - 42 \, a^{3} c d^{4} e^{11} + 6 \, {\left (c^{4} d^{11} e^{4} - 6 \, a c^{3} d^{7} e^{8} - 7 \, a^{2} c^{2} d^{3} e^{12}\right )} x^{5} + 3 \, {\left (3 \, c^{4} d^{12} e^{3} - 11 \, a c^{3} d^{8} e^{7} - 15 \, a^{2} c^{2} d^{4} e^{11} - a^{3} c e^{15}\right )} x^{4} - 2 \, a^{4} e^{15} + {\left (c^{4} d^{13} e^{2} + 3 \, a c^{3} d^{9} e^{6} + 3 \, a^{2} c^{2} d^{5} e^{10} + a^{3} c d e^{14}\right )} x^{3} - {\left (c^{4} d^{14} e + 3 \, a c^{3} d^{10} e^{5} + 3 \, a^{2} c^{2} d^{6} e^{9} + a^{3} c d^{2} e^{13}\right )} x^{2} + {\left (c^{4} d^{15} + 9 \, a c^{3} d^{11} e^{4} - 33 \, a^{2} c^{2} d^{7} e^{8} - 41 \, a^{3} c d^{3} e^{12}\right )} x}{4 \, {\left (c d^{4} + a e^{4}\right )}^{4} {\left (c x^{4} + a\right )} {\left (x e + d\right )}^{2} a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.04, size = 2500, normalized size = 1.81 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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