Optimal. Leaf size=889 \[ -\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \text {ArcTan}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{128 c^{7/2} e}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \tanh ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e} \]
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Rubi [A]
time = 23.48, antiderivative size = 889, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3781, 6857,
626, 635, 212, 756, 654, 1004, 1050, 1044, 214, 211} \begin {gather*} \frac {\tan (d+e x) \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{4 c e}-\frac {5 b \left (c \tan ^2(d+e x)+b \tan (d+e x)+a\right )^{3/2}}{24 c^2 e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{64 c^3 e}-\frac {(b+2 c \tan (d+e x)) \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}{4 c e}-\frac {\sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \text {ArcTan}\left (\frac {b \sqrt {a^2-2 c a+b^2+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c-\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c-\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{8 c^{3/2} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{128 c^{7/2} e}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{e}-\frac {\sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \tanh ^{-1}\left (\frac {\sqrt {a^2-2 c a+b^2+c^2} b+\left (b^2+(a-c) \left (a-c-\sqrt {a^2-2 c a+b^2+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} \sqrt {a^2-\left (2 c+\sqrt {a^2-2 c a+b^2+c^2}\right ) a+b^2+c \left (c+\sqrt {a^2-2 c a+b^2+c^2}\right )} \sqrt {c \tan ^2(d+e x)+b \tan (d+e x)+a}}\right )}{\sqrt {2} \sqrt [4]{a^2-2 c a+b^2+c^2} e} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 212
Rule 214
Rule 626
Rule 635
Rule 654
Rule 756
Rule 1004
Rule 1044
Rule 1050
Rule 3781
Rule 6857
Rubi steps
\begin {align*} \int \tan ^4(d+e x) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4 \sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=\frac {\text {Subst}\left (\int \left (-\sqrt {a+b x+c x^2}+x^2 \sqrt {a+b x+c x^2}+\frac {\sqrt {a+b x+c x^2}}{1+x^2}\right ) \, dx,x,\tan (d+e x)\right )}{e}\\ &=-\frac {\text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int x^2 \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x+c x^2}}{1+x^2} \, dx,x,\tan (d+e x)\right )}{e}\\ &=-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}-\frac {\text {Subst}\left (\int \frac {-a+c-b x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\text {Subst}\left (\int \left (-a-\frac {5 b x}{2}\right ) \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{4 c e}+\frac {c \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{e}+\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{8 c e}\\ &=-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}+\frac {(2 c) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{4 c e}+\frac {\left (5 b^2-4 a c\right ) \text {Subst}\left (\int \sqrt {a+b x+c x^2} \, dx,x,\tan (d+e x)\right )}{16 c^2 e}-\frac {\text {Subst}\left (\int \frac {b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )-b \sqrt {a^2+b^2-2 a c+c^2} x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}+\frac {\text {Subst}\left (\int \frac {b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )+b \sqrt {a^2+b^2-2 a c+c^2} x}{\left (1+x^2\right ) \sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{2 \sqrt {a^2+b^2-2 a c+c^2} e}\\ &=\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x+c x^2}} \, dx,x,\tan (d+e x)\right )}{128 c^3 e}-\frac {\left (b \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {-b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}-\frac {\left (b \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )\right ) \text {Subst}\left (\int \frac {1}{2 b \sqrt {a^2+b^2-2 a c+c^2} \left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right )+b x^2} \, dx,x,\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}\\ &=-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \tan ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \tanh ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}-\frac {\left (\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right )\right ) \text {Subst}\left (\int \frac {1}{4 c-x^2} \, dx,x,\frac {b+2 c \tan (d+e x)}{\sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{64 c^3 e}\\ &=-\frac {\sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \tan ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}-\left (b^2+(a-c) \left (a-c+\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c-\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c-\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{e}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{8 c^{3/2} e}-\frac {\left (b^2-4 a c\right ) \left (5 b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{128 c^{7/2} e}-\frac {\sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \tanh ^{-1}\left (\frac {b \sqrt {a^2+b^2-2 a c+c^2}+\left (b^2+(a-c) \left (a-c-\sqrt {a^2+b^2-2 a c+c^2}\right )\right ) \tan (d+e x)}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} \sqrt {a^2+b^2+c \left (c+\sqrt {a^2+b^2-2 a c+c^2}\right )-a \left (2 c+\sqrt {a^2+b^2-2 a c+c^2}\right )} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {2} \sqrt [4]{a^2+b^2-2 a c+c^2} e}-\frac {(b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{4 c e}+\frac {\left (5 b^2-4 a c\right ) (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{64 c^3 e}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{24 c^2 e}+\frac {\tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{4 c e}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 5.24, size = 582, normalized size = 0.65 \begin {gather*} \frac {-4 i \sqrt {a-i b-c} \tanh ^{-1}\left (\frac {2 a-i b+(b-2 i c) \tan (d+e x)}{2 \sqrt {a-i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+4 i \sqrt {a+i b-c} \tanh ^{-1}\left (\frac {2 a+i b+(b+2 i c) \tan (d+e x)}{2 \sqrt {a+i b-c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )+\frac {2 (-i b+2 c) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}+\frac {2 (i b+2 c) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{\sqrt {c}}+\frac {\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )}{c^{3/2}}-\frac {2 (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}{c}-\frac {5 b \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{3 c^2}+\frac {2 \tan (d+e x) \left (a+b \tan (d+e x)+c \tan ^2(d+e x)\right )^{3/2}}{c}-\frac {\left (\frac {5 b^2}{2}-2 a c\right ) \left (\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac {b+2 c \tan (d+e x)}{2 \sqrt {c} \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}}\right )-2 \sqrt {c} (b+2 c \tan (d+e x)) \sqrt {a+b \tan (d+e x)+c \tan ^2(d+e x)}\right )}{8 c^{7/2}}}{8 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] result has leaf size over 500,000. Avoiding possible recursion issues.
time = 0.56, size = 17248526, normalized size = 19402.17 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {a + b \tan {\left (d + e x \right )} + c \tan ^{2}{\left (d + e x \right )}} \tan ^{4}{\left (d + e x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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