Optimal. Leaf size=1141 \[ \frac {8 c d^3 \log (d+e x) e^7}{\left (c d^4+a e^4\right )^3}-\frac {2 c d^3 \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^3}-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} d \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{\sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} \left (\sqrt {c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\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 )^3}+\frac {\sqrt [4]{c} \left (\sqrt {c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\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 )^3}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}-\frac {\sqrt {c} d \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{2 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac {c \left (4 a d^3 e^3+x \left (\left (c d^4-3 a e^4\right ) d^2-2 e \left (c d^4-a e^4\right ) x d+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (3 c d^4-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 )^2}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\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 )^2}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\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 )^2}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\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 )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.66, antiderivative size = 1141, 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 = {6742, 1854, 1876, 275, 205, 1168, 1162, 617, 204, 1165, 628, 1248, 635, 260} \[ \frac {8 c d^3 \log (d+e x) e^7}{\left (c d^4+a e^4\right )^3}-\frac {2 c d^3 \log \left (c x^4+a\right ) e^7}{\left (c d^4+a e^4\right )^3}-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} d \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{\sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} \left (\sqrt {c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\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 )^3}+\frac {\sqrt [4]{c} \left (\sqrt {c} \left (5 c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (7 c d^4-a e^4\right )\right ) \tan ^{-1}\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 )^3}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (7 c d^4-a e^4\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 )^3}-\frac {\sqrt {c} d \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{2 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac {c \left (4 a d^3 e^3+x \left (\left (c d^4-3 a e^4\right ) d^2-2 e \left (c d^4-a e^4\right ) x d+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (c x^4+a\right )}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (3 c d^4-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 )^2}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} \left (c d^4-3 a e^4\right ) d^2+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \tan ^{-1}\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 )^2}-\frac {\sqrt [4]{c} \left (3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\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 )^2}+\frac {\sqrt [4]{c} \left (3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\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 )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 205
Rule 260
Rule 275
Rule 617
Rule 628
Rule 635
Rule 1162
Rule 1165
Rule 1168
Rule 1248
Rule 1854
Rule 1876
Rule 6742
Rubi steps
\begin {align*} \int \frac {1}{(d+e x)^2 \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)^2}+\frac {8 c d^3 e^8}{\left (c d^4+a e^4\right )^3 (d+e x)}+\frac {c \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^2 \left (a+c x^4\right )^2}+\frac {c e^4 \left (d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2-8 c d^3 e^3 x^3\right )}{\left (c d^4+a e^4\right )^3 \left (a+c x^4\right )}\right ) \, dx\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \frac {d^2 \left (5 c d^4-3 a e^4\right )-2 d e \left (3 c d^4-a e^4\right ) x+e^2 \left (7 c d^4-a e^4\right ) x^2-8 c d^3 e^3 x^3}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac {c \int \frac {d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2-4 c d^3 e^3 x^3}{\left (a+c x^4\right )^2} \, dx}{\left (c d^4+a e^4\right )^2}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \left (\frac {x \left (-2 d e \left (3 c d^4-a e^4\right )-8 c d^3 e^3 x^2\right )}{a+c x^4}+\frac {d^2 \left (5 c d^4-3 a e^4\right )+e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{\left (c d^4+a e^4\right )^3}-\frac {c \int \frac {-3 d^2 \left (c d^4-3 a e^4\right )+4 d e \left (c d^4-a e^4\right ) x-e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^2}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \frac {x \left (-2 d e \left (3 c d^4-a e^4\right )-8 c d^3 e^3 x^2\right )}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \int \frac {d^2 \left (5 c d^4-3 a e^4\right )+e^2 \left (7 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{\left (c d^4+a e^4\right )^3}-\frac {c \int \left (\frac {4 d e \left (c d^4-a e^4\right ) x}{a+c x^4}+\frac {-3 d^2 \left (c d^4-3 a e^4\right )-e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4}\right ) \, dx}{4 a \left (c d^4+a e^4\right )^2}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\left (c e^4\right ) \operatorname {Subst}\left (\int \frac {-2 d e \left (3 c d^4-a e^4\right )-8 c d^3 e^3 x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^4+a e^4\right )^3}-\frac {c \int \frac {-3 d^2 \left (c d^4-3 a e^4\right )-e^2 \left (3 c d^4-a e^4\right ) x^2}{a+c x^4} \, dx}{4 a \left (c d^4+a e^4\right )^2}-\frac {\left (c d e \left (c d^4-a e^4\right )\right ) \int \frac {x}{a+c x^4} \, dx}{a \left (c d^4+a e^4\right )^2}-\frac {\left (e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}+\frac {\left (e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {\left (4 c^2 d^3 e^7\right ) \operatorname {Subst}\left (\int \frac {x}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac {\left (c d e^5 \left (3 c d^4-a e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{\left (c d^4+a e^4\right )^3}-\frac {\left (c d e \left (c d^4-a e^4\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+c x^2} \, dx,x,x^2\right )}{2 a \left (c d^4+a e^4\right )^2}-\frac {\left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}+\frac {\left (e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}+\frac {\left (e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac {\sqrt {c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^2}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3}+\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right )\right ) \operatorname {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 )^3}-\frac {\left (\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right )\right ) \operatorname {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 )^3}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac {\sqrt {c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}+\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}-\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3}+\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right )\right ) \operatorname {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 )^2}-\frac {\left (\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\right )\right ) \operatorname {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 )^2}\\ &=-\frac {e^7}{\left (c d^4+a e^4\right )^2 (d+e x)}+\frac {c \left (4 a d^3 e^3+x \left (d^2 \left (c d^4-3 a e^4\right )-2 d e \left (c d^4-a e^4\right ) x+e^2 \left (3 c d^4-a e^4\right ) x^2\right )\right )}{4 a \left (c d^4+a e^4\right )^2 \left (a+c x^4\right )}-\frac {\sqrt {c} d e^5 \left (3 c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 a^{3/2} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}-\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}+\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6+\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6+\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^3}+\frac {8 c d^3 e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^3}+\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}+\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {\sqrt [4]{c} \left (3 c d^4 e^2-a e^6-\frac {3 \sqrt {c} d^2 \left (c d^4-3 a e^4\right )}{\sqrt {a}}\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 )^2}-\frac {\sqrt [4]{c} e^4 \left (7 c d^4 e^2-a e^6-\frac {\sqrt {c} d^2 \left (5 c d^4-3 a e^4\right )}{\sqrt {a}}\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} \sqrt [4]{a} \left (c d^4+a e^4\right )^3}-\frac {2 c d^3 e^7 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^3}\\ \end {align*}
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Mathematica [A] time = 0.86, size = 807, normalized size = 0.71 \[ \frac {256 c d^3 \log (d+e x) e^7-64 c d^3 \log \left (c x^4+a\right ) e^7-\frac {32 \left (c d^4+a e^4\right ) e^7}{d+e x}+\frac {8 c \left (c d^4+a e^4\right ) \left (c x \left (d^2-2 e x d+3 e^2 x^2\right ) d^4+a e^3 \left (4 d^3-3 e x d^2+2 e^2 x^2 d-e^3 x^3\right )\right )}{a \left (c x^4+a\right )}+\frac {2 \sqrt [4]{c} \left (-3 \sqrt {2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} e d^9-3 \sqrt {2} \sqrt {a} c^2 e^2 d^8-14 \sqrt {2} a c^{3/2} e^4 d^6+48 a^{5/4} c^{5/4} e^5 d^5-30 \sqrt {2} a^{3/2} c e^6 d^4+21 \sqrt {2} a^2 \sqrt {c} e^8 d^2-24 a^{9/4} \sqrt [4]{c} e^9 d+5 \sqrt {2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \sqrt [4]{c} \left (3 \sqrt {2} c^{5/2} d^{10}+8 \sqrt [4]{a} c^{9/4} e d^9+3 \sqrt {2} \sqrt {a} c^2 e^2 d^8+14 \sqrt {2} a c^{3/2} e^4 d^6+48 a^{5/4} c^{5/4} e^5 d^5+30 \sqrt {2} a^{3/2} c e^6 d^4-21 \sqrt {2} a^2 \sqrt {c} e^8 d^2-24 a^{9/4} \sqrt [4]{c} e^9 d-5 \sqrt {2} a^{5/2} e^{10}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}-\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt {a} c^2 e^2 d^8+14 a c^{3/2} e^4 d^6-30 a^{3/2} c e^6 d^4-21 a^2 \sqrt {c} e^8 d^2+5 a^{5/2} e^{10}\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{5/2} d^{10}-3 \sqrt {a} c^2 e^2 d^8+14 a c^{3/2} e^4 d^6-30 a^{3/2} c e^6 d^4-21 a^2 \sqrt {c} e^8 d^2+5 a^{5/2} e^{10}\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{a^{7/4}}}{32 \left (c d^4+a e^4\right )^3} \]
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 103.64, size = 1104, normalized size = 0.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 1636, normalized size = 1.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 2.54, size = 961, normalized size = 0.84 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.10, size = 2246, normalized size = 1.97 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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