Optimal. Leaf size=288 \[ -\frac {a^{5/2} c^{7/2} (-B+6 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^2 c^3 (6 A+i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a c^2 (6 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {c (-B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f} \]
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Rubi [A] time = 0.33, antiderivative size = 288, 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 = {3588, 80, 49, 38, 63, 217, 203} \[ -\frac {a^{5/2} c^{7/2} (-B+6 i A) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^2 c^3 (6 A+i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a c^2 (6 A+i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {c (-B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f} \]
Antiderivative was successfully verified.
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Rule 38
Rule 49
Rule 63
Rule 80
Rule 203
Rule 217
Rule 3588
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int (a+i a x)^{3/2} (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))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}+\frac {(a (6 A+i B) c) \operatorname {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=-\frac {(6 i A-B) c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}+\frac {\left (a (6 A+i B) c^2\right ) \operatorname {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{6 f}\\ &=\frac {a (6 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(6 i A-B) c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}+\frac {\left (a^2 (6 A+i B) c^3\right ) \operatorname {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{8 f}\\ &=\frac {a^2 (6 A+i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a (6 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(6 i A-B) c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}+\frac {\left (a^3 (6 A+i B) c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{16 f}\\ &=\frac {a^2 (6 A+i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a (6 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(6 i A-B) c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}-\frac {\left (a^2 (6 i A-B) c^4\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{8 f}\\ &=\frac {a^2 (6 A+i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a (6 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(6 i A-B) c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}-\frac {\left (a^2 (6 i A-B) c^4\right ) \operatorname {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 )}{8 f}\\ &=-\frac {a^{5/2} (6 i A-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 )}{8 f}+\frac {a^2 (6 A+i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a (6 A+i B) c^2 \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}-\frac {(6 i A-B) c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{7/2}}{6 f}\\ \end {align*}
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Mathematica [A] time = 17.34, size = 568, normalized size = 1.97 \[ \frac {c^4 (B-6 i A) \sqrt {e^{i f x}} e^{-i (3 e+f x)} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \tan ^{-1}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{8 f \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} \sec ^{\frac {7}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{5/2} (A \cos (e+f x)+B \sin (e+f x))}+\frac {\cos ^3(e+f x) (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left (\sec (e) \left (\frac {1}{30} c^3 \cos (2 e)-\frac {1}{30} i c^3 \sin (2 e)\right ) \sec ^4(e+f x) (-6 i A \cos (e)-5 i B \sin (e)+6 B \cos (e))+\sec (e) \left (\frac {1}{24} \cos (2 e)-\frac {1}{24} i \sin (2 e)\right ) \sec ^3(e+f x) \left (6 A c^3 \sin (f x)+i B c^3 \sin (f x)\right )+\sec (e) \left (\frac {1}{16} \cos (2 e)-\frac {1}{16} i \sin (2 e)\right ) \sec (e+f x) \left (6 A c^3 \sin (f x)+i B c^3 \sin (f x)\right )+(6 A+i B) \tan (e) \left (\frac {1}{24} c^3 \cos (2 e)-\frac {1}{24} i c^3 \sin (2 e)\right ) \sec ^2(e+f x)+(6 A+i B) \tan (e) \left (\frac {1}{16} c^3 \cos (2 e)-\frac {1}{16} i c^3 \sin (2 e)\right )-i B c^3 \sec (e) \left (\frac {1}{6} \cos (2 e)-\frac {1}{6} i \sin (2 e)\right ) \sin (f x) \sec ^5(e+f x)\right )}{f (\cos (f x)+i \sin (f x))^2 (A \cos (e+f x)+B \sin (e+f x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 753, normalized size = 2.61 \[ -\frac {15 \, \sqrt {\frac {{\left (36 \, A^{2} + 12 i \, A B - B^{2}\right )} a^{5} c^{7}}{f^{2}}} {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {2 \, {\left ({\left ({\left (-24 i \, A + 4 \, B\right )} a^{2} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-24 i \, A + 4 \, B\right )} a^{2} 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}} + 2 \, \sqrt {\frac {{\left (36 \, A^{2} + 12 i \, A B - B^{2}\right )} a^{5} c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-6 i \, A + B\right )} a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-6 i \, A + B\right )} a^{2} c^{3}}\right ) - 15 \, \sqrt {\frac {{\left (36 \, A^{2} + 12 i \, A B - B^{2}\right )} a^{5} c^{7}}{f^{2}}} {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (\frac {2 \, {\left ({\left ({\left (-24 i \, A + 4 \, B\right )} a^{2} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-24 i \, A + 4 \, B\right )} a^{2} 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}} - 2 \, \sqrt {\frac {{\left (36 \, A^{2} + 12 i \, A B - B^{2}\right )} a^{5} c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-6 i \, A + B\right )} a^{2} c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-6 i \, A + B\right )} a^{2} c^{3}}\right ) - 4 \, {\left ({\left (-90 i \, A + 15 \, B\right )} a^{2} c^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + {\left (-510 i \, A + 85 \, B\right )} a^{2} c^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + {\left (-1188 i \, A + 198 \, B\right )} a^{2} c^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + {\left (-348 i \, A + 1338 \, B\right )} a^{2} c^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + {\left (510 i \, A - 85 \, B\right )} a^{2} c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (90 i \, A - 15 \, B\right )} a^{2} 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}}}{480 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 478, normalized size = 1.66 \[ -\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, a^{2} c^{3} \left (40 i B \left (\tan ^{5}\left (f x +e \right )\right ) \sqrt {c a}\, \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}+48 i A \left (\tan ^{4}\left (f x +e \right )\right ) \sqrt {c a}\, \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}+70 i B \left (\tan ^{3}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-48 B \left (\tan ^{4}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+96 i A \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-60 A \left (\tan ^{3}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-15 i B \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) a c +15 i B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )-96 B \left (\tan ^{2}\left (f x +e \right )\right ) \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}+48 i A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}-90 A \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}}{\sqrt {c a}}\right ) a c -150 A \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\, \tan \left (f x +e \right )-48 B \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}\right )}{240 f \sqrt {c a \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {c a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 8.53, size = 2046, normalized size = 7.10 \[ \text {result too large to display} \]
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 \[ \int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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