Optimal. Leaf size=272 \[ -\frac {5 \sqrt {a} (4 i A-3 B) c^{7/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}-\frac {5 (4 i A-3 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {5 (4 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f} \]
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Rubi [A]
time = 0.21, antiderivative size = 272, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3669, 81, 52,
65, 223, 209} \begin {gather*} -\frac {5 \sqrt {a} c^{7/2} (-3 B+4 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 {5 c^3 (-3 B+4 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {5 c^2 (-3 B+4 i A) \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {c (-3 B+4 i A) \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 81
Rule 209
Rule 223
Rule 3669
Rubi steps
\begin {align*} \int \sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(A+B x) (c-i c x)^{5/2}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}+\frac {(a (4 A+3 i B) c) \text {Subst}\left (\int \frac {(c-i c x)^{5/2}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{4 f}\\ &=-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}+\frac {\left (5 a (4 A+3 i B) c^2\right ) \text {Subst}\left (\int \frac {(c-i c x)^{3/2}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{12 f}\\ &=-\frac {5 (4 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}+\frac {\left (5 a (4 A+3 i B) c^3\right ) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{\sqrt {a+i a x}} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=-\frac {5 (4 i A-3 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {5 (4 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}+\frac {\left (5 a (4 A+3 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 {5 (4 i A-3 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {5 (4 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}-\frac {\left (5 (4 i A-3 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 {5 (4 i A-3 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {5 (4 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}-\frac {\left (5 (4 i A-3 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 {5 \sqrt {a} (4 i A-3 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 {5 (4 i A-3 B) c^3 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {5 (4 i A-3 B) c^2 \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(4 i A-3 B) c \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{5/2}}{12 f}+\frac {B \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{7/2}}{4 f}\\ \end {align*}
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Mathematica [A]
time = 3.38, size = 257, normalized size = 0.94 \begin {gather*} \frac {\sqrt {a+i a \tan (e+f x)} (A+B \tan (e+f x)) \left (\frac {5 (-4 i A+3 B) c^4 e^{-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}{24} c^3 \sec ^{\frac {7}{2}}(e+f x) (64 (-4 i A+3 B) \cos (e+f x)+96 (-i A+B) \cos (3 (e+f x))-6 (12 A+13 i B+(12 A+17 i B) \cos (2 (e+f x))) \sin (e+f x)) \sqrt {c-i c \tan (e+f x)}\right )}{4 f \sec ^{\frac {3}{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 = 0.52, size = 349, normalized size = 1.28
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} \left (6 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+8 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+45 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -45 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-24 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )-88 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+60 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -36 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+72 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{24 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(349\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} \left (6 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+8 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+45 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -45 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-24 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )-88 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+60 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -36 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+72 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{24 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(349\) |
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. 1419 vs. \(2 (216) = 432\).
time = 1.53, size = 1419, normalized size = 5.22 \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. 631 vs. \(2 (216) = 432\).
time = 2.94, size = 631, normalized size = 2.32 \begin {gather*} \frac {15 \, \sqrt {\frac {{\left (16 \, A^{2} + 24 i \, A B - 9 \, B^{2}\right )} a c^{7}}{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 {4 \, {\left (2 \, {\left ({\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (4 i \, A - 3 \, B\right )} 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 (16 \, A^{2} + 24 i \, A B - 9 \, B^{2}\right )} a c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (4 i \, A - 3 \, B\right )} c^{3}}\right ) - 15 \, \sqrt {\frac {{\left (16 \, A^{2} + 24 i \, A B - 9 \, B^{2}\right )} a c^{7}}{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 {4 \, {\left (2 \, {\left ({\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (4 i \, A - 3 \, B\right )} 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 (16 \, A^{2} + 24 i \, A B - 9 \, B^{2}\right )} a c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (4 i \, A - 3 \, B\right )} c^{3}}\right ) - 4 \, {\left (15 \, {\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 55 \, {\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 73 \, {\left (4 i \, A - 3 \, B\right )} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 3 \, {\left (44 i \, A - 49 \, B\right )} 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}}}{48 \, {\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(-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 )\,\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,{\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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