Optimal. Leaf size=508 \[ -\frac {\sqrt {a-i b} (i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\sqrt {a+i b} (B-i (A-C)) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{5/2} d^{3/2} f}+\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f} \]
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Rubi [A]
time = 5.34, antiderivative size = 508, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 8, integrand size = 49, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {3728, 3736,
6857, 65, 223, 212, 95, 214} \begin {gather*} \frac {\left (a^3 C d^3-a^2 b d^2 (2 B d+3 c C)+a b^2 d \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )-\left (b^3 \left (-24 c d^2 (A-C)-6 B c^2 d+16 B d^3+c^3 C\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{5/2} d^{3/2} f}+\frac {\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (8 b d^2 (a B+A b-b C)-(b c-a d) (-a C d-6 b B d+b c C)\right )}{8 b^2 d f}-\frac {\sqrt {a-i b} (c-i d)^{3/2} (i A+B-i C) \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\sqrt {a+i b} (c+i d)^{3/2} (B-i (A-C)) \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {(-a C d-6 b B d+b c C) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 95
Rule 212
Rule 214
Rule 223
Rule 3728
Rule 3736
Rule 6857
Rubi steps
\begin {align*} \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\int \frac {(c+d \tan (e+f x))^{3/2} \left (\frac {1}{2} (-b c C+a (6 A-5 C) d)+3 (A b+a B-b C) d \tan (e+f x)-\frac {1}{2} (b c C-6 b B d-a C d) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{3 d}\\ &=-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left (-\frac {3}{4} \left (a^2 C d^2-2 a b d (4 A c-3 c C-3 B d)+b^2 c (c C+2 B d)\right )+6 b d (A b c+a B c-b c C+a A d-b B d-a C d) \tan (e+f x)+\frac {3}{4} \left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx}{6 b d}\\ &=\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\int \frac {\frac {3}{8} \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)-b^3 c \left (c^2 C+10 B c d+8 (A-C) d^2\right )-a b^2 d \left (13 c^2 C+20 B c d-8 C d^2-8 A \left (2 c^2-d^2\right )\right )\right )+6 b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) \tan (e+f x)+\frac {3}{8} \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx}{6 b^2 d}\\ &=\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\text {Subst}\left (\int \frac {\frac {3}{8} \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)-b^3 c \left (c^2 C+10 B c d+8 (A-C) d^2\right )-a b^2 d \left (13 c^2 C+20 B c d-8 C d^2-8 A \left (2 c^2-d^2\right )\right )\right )+6 b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) x+\frac {3}{8} \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{6 b^2 d f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\text {Subst}\left (\int \left (\frac {3 \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right )}{8 \sqrt {a+b x} \sqrt {c+d x}}+\frac {6 \left (-b^2 d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{6 b^2 d f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\text {Subst}\left (\int \frac {-b^2 d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )+b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{b^2 d f}+\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{16 b^2 d f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\text {Subst}\left (\int \left (\frac {-b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right )-i b^2 d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {b^2 d \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right )-i b^2 d \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{b^2 d f}+\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{8 b^3 d f}\\ &=\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\left ((i a+b) (A-i B-C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left ((i a-b) (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{8 b^3 d f}\\ &=\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{5/2} d^{3/2} f}+\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\left ((i a+b) (A-i B-C) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left ((i a-b) (A+i B-C) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}\\ &=-\frac {\sqrt {a-i b} (i A+B-i C) (c-i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}-\frac {\sqrt {a+i b} (B-i (A-C)) (c+i d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{8 b^{5/2} d^{3/2} f}+\frac {\left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 b^2 d f}-\frac {(b c C-6 b B d-a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 b d f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}\\ \end {align*}
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Mathematica [A]
time = 7.98, size = 867, normalized size = 1.71 \begin {gather*} \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}}{3 d f}+\frac {\frac {(-b c C+6 b B d+a C d) \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{4 b f}+\frac {\frac {3 \left (8 b (A b+a B-b C) d^2-(b c-a d) (b c C-6 b B d-a C d)\right ) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {\frac {6 b^2 d \left (b \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right )-\sqrt {-b^2} \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {6 b^2 d \left (b \left (2 a A c d-2 a c C d+A b \left (c^2-d^2\right )+a B \left (c^2-d^2\right )-b \left (c^2 C+2 B c d-C d^2\right )\right )+\sqrt {-b^2} \left (a \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+\frac {3 \sqrt {b} \sqrt {c-\frac {a d}{b}} \left (a^3 C d^3-a^2 b d^2 (3 c C+2 B d)+a b^2 d \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )-b^3 \left (c^3 C-6 B c^2 d-24 c (A-C) d^2+16 B d^3\right )\right ) \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{4 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 b}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )\, dx\]
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 (e + f x \right )}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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