Optimal. Leaf size=152 \[ \frac {8 \sqrt [4]{-1} a^3 d^{3/2} \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {8 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f} \]
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Rubi [A]
time = 0.18, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3637, 3673,
3609, 3614, 211} \begin {gather*} \frac {8 \sqrt [4]{-1} a^3 d^{3/2} \text {ArcTan}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}+\frac {8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}+\frac {8 a^3 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 \left (a^3+i a^3 \tan (e+f x)\right ) (d \tan (e+f x))^{5/2}}{7 d f} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 3609
Rule 3614
Rule 3637
Rule 3673
Rubi steps
\begin {align*} \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x))^3 \, dx &=-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f}+\frac {(2 a) \int (d \tan (e+f x))^{3/2} (a+i a \tan (e+f x)) (6 a d+8 i a d \tan (e+f x)) \, dx}{7 d}\\ &=-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f}+\frac {(2 a) \int (d \tan (e+f x))^{3/2} \left (14 a^2 d+14 i a^2 d \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f}+\frac {(2 a) \int \sqrt {d \tan (e+f x)} \left (-14 i a^2 d^2+14 a^2 d^2 \tan (e+f x)\right ) \, dx}{7 d}\\ &=\frac {8 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f}+\frac {(2 a) \int \frac {-14 a^2 d^3-14 i a^2 d^3 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{7 d}\\ &=\frac {8 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f}+\frac {\left (112 a^5 d^5\right ) \text {Subst}\left (\int \frac {1}{-14 a^2 d^4+14 i a^2 d^3 x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{f}\\ &=\frac {8 \sqrt [4]{-1} a^3 d^{3/2} \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {8 a^3 d \sqrt {d \tan (e+f x)}}{f}+\frac {8 i a^3 (d \tan (e+f x))^{3/2}}{3 f}-\frac {32 a^3 (d \tan (e+f x))^{5/2}}{35 d f}-\frac {2 (d \tan (e+f x))^{5/2} \left (a^3+i a^3 \tan (e+f x)\right )}{7 d f}\\ \end {align*}
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Mathematica [A]
time = 2.47, size = 138, normalized size = 0.91 \begin {gather*} \frac {a^3 d \left (-1680 \tanh ^{-1}\left (\sqrt {\frac {-1+e^{2 i (e+f x)}}{1+e^{2 i (e+f x)}}}\right )+\sec ^3(e+f x) (1197 \cos (e+f x)+483 \cos (3 (e+f x))+95 i \sin (e+f x)+155 i \sin (3 (e+f x))) \sqrt {i \tan (e+f x)}\right ) \sqrt {d \tan (e+f x)}}{210 f \sqrt {i \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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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. 341 vs. \(2 (126 ) = 252\).
time = 0.16, size = 342, normalized size = 2.25
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (-\frac {i \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {4 i d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 d^{3} \sqrt {d \tan \left (f x +e \right )}-4 d^{4} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(342\) |
default | \(\frac {2 a^{3} \left (-\frac {i \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {4 i d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 d^{3} \sqrt {d \tan \left (f x +e \right )}-4 d^{4} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {i \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(342\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.56, size = 248, normalized size = 1.63 \begin {gather*} -\frac {105 \, a^{3} d^{3} {\left (\frac {\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\left (2 i + 2\right ) \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\left (i - 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\left (i - 1\right ) \, \sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}\right )} - \frac {2 \, {\left (-15 i \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} a^{3} - 63 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{3} d + 140 i \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} d^{2} + 420 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{3}\right )}}{d}}{105 \, d f} \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. 447 vs. \(2 (130) = 260\).
time = 0.38, size = 447, normalized size = 2.94 \begin {gather*} -\frac {105 \, \sqrt {-\frac {64 i \, a^{6} d^{3}}{f^{2}}} {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-8 i \, a^{3} d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + \sqrt {-\frac {64 i \, a^{6} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3} d}\right ) - 105 \, \sqrt {-\frac {64 i \, a^{6} d^{3}}{f^{2}}} {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {{\left (-8 i \, a^{3} d^{2} e^{\left (2 i \, f x + 2 i \, e\right )} - \sqrt {-\frac {64 i \, a^{6} d^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, a^{3} d}\right ) - 16 \, {\left (319 \, a^{3} d e^{\left (6 i \, f x + 6 i \, e\right )} + 646 \, a^{3} d e^{\left (4 i \, f x + 4 i \, e\right )} + 551 \, a^{3} d e^{\left (2 i \, f x + 2 i \, e\right )} + 164 \, a^{3} d\right )} \sqrt {\frac {-i \, d e^{\left (2 i \, f x + 2 i \, e\right )} + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{420 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - i a^{3} \left (\int i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (- 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )}\right )\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 3 i \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.77, size = 194, normalized size = 1.28 \begin {gather*} -\frac {2}{105} \, {\left (\frac {420 i \, \sqrt {2} a^{3} \sqrt {d} \arctan \left (\frac {8 \, \sqrt {d^{2}} \sqrt {d \tan \left (f x + e\right )}}{4 i \, \sqrt {2} d^{\frac {3}{2}} + 4 \, \sqrt {2} \sqrt {d^{2}} \sqrt {d}}\right )}{f {\left (\frac {i \, d}{\sqrt {d^{2}}} + 1\right )}} + \frac {15 i \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6} \tan \left (f x + e\right )^{3} + 63 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6} \tan \left (f x + e\right )^{2} - 140 i \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6} \tan \left (f x + e\right ) - 420 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{21} f^{6}}{d^{21} f^{7}}\right )} d \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.77, size = 119, normalized size = 0.78 \begin {gather*} \frac {a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,8{}\mathrm {i}}{3\,f}-\frac {6\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,d\,f}-\frac {a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}\,2{}\mathrm {i}}{7\,d^2\,f}+\frac {8\,a^3\,d\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}-\frac {\sqrt {16{}\mathrm {i}}\,a^3\,{\left (-d\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {16{}\mathrm {i}}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{4\,\sqrt {-d}}\right )\,2{}\mathrm {i}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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