Optimal. Leaf size=378 \[ -\frac {\sqrt [4]{-1} a^{3/2} \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}} \]
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Rubi [A]
time = 1.14, antiderivative size = 378, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {3637, 3676,
3678, 3682, 3625, 214, 3680, 65, 223, 212} \begin {gather*} -\frac {\sqrt [4]{-1} a^{3/2} \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\frac {a (7 d+5 i c) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a (c-3 i d) (3 d+5 i c) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 212
Rule 214
Rule 223
Rule 3625
Rule 3637
Rule 3676
Rule 3678
Rule 3680
Rule 3682
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{5/2} \, dx &=-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\frac {a \int \frac {\left (-\frac {1}{2} a (i c-13 d)-\frac {1}{2} a (c-11 i d) \tan (e+f x)\right ) (c+d \tan (e+f x))^{5/2}}{\sqrt {a+i a \tan (e+f x)}} \, dx}{3 d}\\ &=\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {\int \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \left (-\frac {1}{2} a^2 (7 c-5 i d) d-\frac {1}{2} a^2 d (5 i c+7 d) \tan (e+f x)\right ) \, dx}{3 a d}\\ &=\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {\int \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (-\frac {3}{4} a^3 d \left (11 c^2-14 i c d-7 d^2\right )-\frac {3}{4} a^3 d \left (18 c d+i \left (5 c^2-9 d^2\right )\right ) \tan (e+f x)\right ) \, dx}{6 a^2 d}\\ &=\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {\int \frac {\sqrt {a+i a \tan (e+f x)} \left (-\frac {3}{8} a^4 (3 c-i d) d \left (9 c^2-14 i c d-9 d^2\right )-\frac {3}{8} a^4 d \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx}{6 a^3 d}\\ &=\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\left (2 a (c-i d)^3\right ) \int \frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx-\frac {1}{16} \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right ) \int \frac {(a-i a \tan (e+f x)) \sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}} \, dx\\ &=\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\frac {\left (4 a^3 (i c+d)^3\right ) \text {Subst}\left (\int \frac {1}{a c-i a d-2 a^2 x^2} \, dx,x,\frac {\sqrt {c+d \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}}\right )}{f}-\frac {\left (a^2 \left (5 c^3-45 i c^2 d-55 c d^2+23 i d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\frac {\left (a \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c+i d-\frac {i d x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{8 f}\\ &=-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}+\frac {\left (a \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right )\right ) \text {Subst}\left (\int \frac {1}{1+\frac {i d x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{8 f}\\ &=-\frac {\sqrt [4]{-1} a^{3/2} \left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \tanh ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c+d \tan (e+f x)}}\right )}{8 \sqrt {d} f}-\frac {2 i \sqrt {2} a^{3/2} (c-i d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {c+d \tan (e+f x)}}{\sqrt {c-i d} \sqrt {a+i a \tan (e+f x)}}\right )}{f}+\frac {a (c-3 i d) (5 i c+3 d) \sqrt {a+i a \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{8 f}+\frac {a (5 i c+7 d) \sqrt {a+i a \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{12 f}+\frac {a^2 (c+i d) (c+d \tan (e+f x))^{5/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}-\frac {a^2 (c+d \tan (e+f x))^{7/2}}{3 d f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 8.95, size = 645, normalized size = 1.71 \begin {gather*} \frac {i \sec (e+f x) (\cos (e)-i \sin (e)) (\cos (f x)-i \sin (f x)) (a+i a \tan (e+f x))^{3/2} \left (-\frac {(3-3 i) \cos ^3(e+f x) \left (\left (5 i c^3+45 c^2 d-55 i c d^2-23 d^3\right ) \left (\log \left (\frac {(2+2 i) e^{\frac {i e}{2}} \left (c+i d-i c e^{i (e+f x)}-d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{\sqrt {d} \left (-5 i c^3-45 c^2 d+55 i c d^2+23 d^3\right ) \left (i+e^{i (e+f x)}\right )}\right )-\log \left (-\frac {(2+2 i) e^{\frac {i e}{2}} \left (c+i d+i c e^{i (e+f x)}+d e^{i (e+f x)}+(1+i) \sqrt {d} \sqrt {1+e^{2 i (e+f x)}} \sqrt {c-\frac {i d \left (-1+e^{2 i (e+f x)}\right )}{1+e^{2 i (e+f x)}}}\right )}{\sqrt {d} \left (-5 i c^3-45 c^2 d+55 i c d^2+23 d^3\right ) \left (-i+e^{i (e+f x)}\right )}\right )\right )+(32+32 i) (c-i d)^{5/2} \sqrt {d} \log \left (2 \left (\sqrt {c-i d} \cos (e+f x)+i \sqrt {c-i d} \sin (e+f x)+\sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))} \sqrt {c+d \tan (e+f x)}\right )\right )\right )}{\sqrt {d} \sqrt {1+\cos (2 (e+f x))+i \sin (2 (e+f x))}}+\left (33 c^2-68 i c d-19 d^2+\left (33 c^2-68 i c d-35 d^2\right ) \cos (2 (e+f x))+2 (13 c-7 i d) d \sin (2 (e+f x))\right ) \sqrt {c+d \tan (e+f x)}\right )}{48 f} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1502 vs. \(2 (305 ) = 610\).
time = 0.53, size = 1503, normalized size = 3.98
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1503\) |
default | \(\text {Expression too large to display}\) | \(1503\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1533 vs. \(2 (302) = 604\).
time = 1.69, size = 1533, normalized size = 4.06 \begin {gather*} \frac {48 \, \sqrt {2} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {-\frac {a^{3} c^{5} - 5 i \, a^{3} c^{4} d - 10 \, a^{3} c^{3} d^{2} + 10 i \, a^{3} c^{2} d^{3} + 5 \, a^{3} c d^{4} - i \, a^{3} d^{5}}{f^{2}}} \log \left (\frac {{\left (i \, \sqrt {2} f \sqrt {-\frac {a^{3} c^{5} - 5 i \, a^{3} c^{4} d - 10 \, a^{3} c^{3} d^{2} + 10 i \, a^{3} c^{2} d^{3} + 5 \, a^{3} c d^{4} - i \, a^{3} d^{5}}{f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (a c^{2} - 2 i \, a c d - a d^{2} + {\left (a c^{2} - 2 i \, a c d - a d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a c^{2} - 2 i \, a c d - a d^{2}}\right ) - 48 \, \sqrt {2} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {-\frac {a^{3} c^{5} - 5 i \, a^{3} c^{4} d - 10 \, a^{3} c^{3} d^{2} + 10 i \, a^{3} c^{2} d^{3} + 5 \, a^{3} c d^{4} - i \, a^{3} d^{5}}{f^{2}}} \log \left (\frac {{\left (-i \, \sqrt {2} f \sqrt {-\frac {a^{3} c^{5} - 5 i \, a^{3} c^{4} d - 10 \, a^{3} c^{3} d^{2} + 10 i \, a^{3} c^{2} d^{3} + 5 \, a^{3} c d^{4} - i \, a^{3} d^{5}}{f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (a c^{2} - 2 i \, a c d - a d^{2} + {\left (a c^{2} - 2 i \, a c d - a d^{2}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a c^{2} - 2 i \, a c d - a d^{2}}\right ) + 2 \, \sqrt {2} {\left ({\left (33 i \, a c^{2} + 94 \, a c d - 49 i \, a d^{2}\right )} e^{\left (5 i \, f x + 5 i \, e\right )} - 2 \, {\left (-33 i \, a c^{2} - 68 \, a c d + 19 i \, a d^{2}\right )} e^{\left (3 i \, f x + 3 i \, e\right )} - 3 \, {\left (-11 i \, a c^{2} - 14 \, a c d + 7 i \, a d^{2}\right )} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + 3 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-25 i \, a^{3} c^{6} - 450 \, a^{3} c^{5} d + 2575 i \, a^{3} c^{4} d^{2} + 5180 \, a^{3} c^{3} d^{3} - 5095 i \, a^{3} c^{2} d^{4} - 2530 \, a^{3} c d^{5} + 529 i \, a^{3} d^{6}}{d f^{2}}} \log \left (\frac {{\left (2 i \, d f \sqrt {\frac {-25 i \, a^{3} c^{6} - 450 \, a^{3} c^{5} d + 2575 i \, a^{3} c^{4} d^{2} + 5180 \, a^{3} c^{3} d^{3} - 5095 i \, a^{3} c^{2} d^{4} - 2530 \, a^{3} c d^{5} + 529 i \, a^{3} d^{6}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (-5 i \, a c^{3} - 45 \, a c^{2} d + 55 i \, a c d^{2} + 23 \, a d^{3} + {\left (-5 i \, a c^{3} - 45 \, a c^{2} d + 55 i \, a c d^{2} + 23 \, a d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{-5 i \, a c^{3} - 45 \, a c^{2} d + 55 i \, a c d^{2} + 23 \, a d^{3}}\right ) - 3 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-25 i \, a^{3} c^{6} - 450 \, a^{3} c^{5} d + 2575 i \, a^{3} c^{4} d^{2} + 5180 \, a^{3} c^{3} d^{3} - 5095 i \, a^{3} c^{2} d^{4} - 2530 \, a^{3} c d^{5} + 529 i \, a^{3} d^{6}}{d f^{2}}} \log \left (\frac {{\left (-2 i \, d f \sqrt {\frac {-25 i \, a^{3} c^{6} - 450 \, a^{3} c^{5} d + 2575 i \, a^{3} c^{4} d^{2} + 5180 \, a^{3} c^{3} d^{3} - 5095 i \, a^{3} c^{2} d^{4} - 2530 \, a^{3} c d^{5} + 529 i \, a^{3} d^{6}}{d f^{2}}} e^{\left (i \, f x + i \, e\right )} + \sqrt {2} {\left (-5 i \, a c^{3} - 45 \, a c^{2} d + 55 i \, a c d^{2} + 23 \, a d^{3} + {\left (-5 i \, a c^{3} - 45 \, a c^{2} d + 55 i \, a c d^{2} + 23 \, a d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \sqrt {\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-i \, f x - i \, e\right )}}{-5 i \, a c^{3} - 45 \, a c^{2} d + 55 i \, a c d^{2} + 23 \, a d^{3}}\right )}{48 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \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 [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 {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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