Optimal. Leaf size=279 \[ -\frac {a^{3/2} (5 i A-2 B) c^{7/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f} \]
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Rubi [A]
time = 0.23, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3669, 81, 51,
38, 65, 223, 209} \begin {gather*} -\frac {a^{3/2} c^{7/2} (-2 B+5 i A) \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a c^3 (5 A+2 i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {c^2 (-2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {c (-2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 38
Rule 51
Rule 65
Rule 81
Rule 209
Rule 223
Rule 3669
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac {(a c) \text {Subst}\left (\int \sqrt {a+i a x} (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {(a (5 A+2 i B) c) \text {Subst}\left (\int \sqrt {a+i a x} (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {\left (a (5 A+2 i B) c^2\right ) \text {Subst}\left (\int \sqrt {a+i a x} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {\left (a (5 A+2 i B) c^3\right ) \text {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {\left (a^2 (5 A+2 i B) c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}-\frac {\left (a (5 i A-2 B) c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{4 f}\\ &=\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}-\frac {\left (a (5 i A-2 B) c^4\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{4 f}\\ &=-\frac {a^{3/2} (5 i A-2 B) c^{7/2} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}\\ \end {align*}
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Mathematica [A]
time = 5.48, size = 257, normalized size = 0.92 \begin {gather*} \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) \left (\frac {(-5 i A+2 B) c^4 e^{-2 i (e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \text {ArcTan}\left (e^{i (e+f x)}\right )}{\sqrt {\frac {c}{1+e^{2 i (e+f x)}}}}-\frac {1}{240} c^3 \sec ^{\frac {7}{2}}(e+f x) (320 (A+i B) \cos (2 (e+f x))+30 (i A+6 B) \sin (2 (e+f x))+(5 A+2 i B) (64+15 i \sin (4 (e+f x)))) (i+\tan (e+f x)) \sqrt {c-i c \tan (e+f x)}\right )}{4 f \sec ^{\frac {5}{2}}(e+f x) (A \cos (e+f x)+B \sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 412, normalized size = 1.48
method | result | size |
derivativedivides | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} a \left (60 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+24 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{4}\left (f x +e \right )\right )+80 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+30 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )-30 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +30 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-32 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+80 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}-75 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -45 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-56 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{120 f \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}\) | \(412\) |
default | \(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} a \left (60 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+24 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{4}\left (f x +e \right )\right )+80 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+30 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )-30 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +30 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-32 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+80 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}-75 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -45 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-56 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{120 f \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}\) | \(412\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 1745 vs. \(2 (224) = 448\).
time = 3.01, size = 1745, normalized size = 6.25 \begin {gather*} \text {Too large to display} \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. 713 vs. \(2 (224) = 448\).
time = 3.96, size = 713, normalized size = 2.56 \begin {gather*} \frac {15 \, \sqrt {\frac {{\left (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} + \sqrt {\frac {{\left (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-5 i \, A + 2 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-5 i \, A + 2 \, B\right )} a c^{3}}\right ) - 15 \, \sqrt {\frac {{\left (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} - \sqrt {\frac {{\left (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-5 i \, A + 2 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-5 i \, A + 2 \, B\right )} a c^{3}}\right ) - 4 \, {\left (15 \, {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 70 \, {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 128 \, {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 10 \, {\left (29 i \, A - 50 \, B\right )} a c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (-5 i \, A + 2 \, B\right )} a c^{3} e^{\left (i \, f x + i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{240 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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