Optimal. Leaf size=261 \[ -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{429 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{3003 c^3 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{15015 c^4 f (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.22, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47,
37} \begin {gather*} -\frac {2 (-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{15015 c^4 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{3003 c^3 f (c-i c \tan (e+f x))^{7/2}}-\frac {(-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{429 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {(-9 B+4 i A) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 79
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{13/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{3/2} (A+B x)}{(c-i c x)^{15/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}+\frac {(a (4 A+9 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{13/2}} \, dx,x,\tan (e+f x)\right )}{13 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}+\frac {(3 a (4 A+9 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{143 c f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{429 c^2 f (c-i c \tan (e+f x))^{9/2}}+\frac {(2 a (4 A+9 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{429 c^2 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{429 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{3003 c^3 f (c-i c \tan (e+f x))^{7/2}}+\frac {(2 a (4 A+9 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{3/2}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{3003 c^3 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{13 f (c-i c \tan (e+f x))^{13/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{143 c f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{429 c^2 f (c-i c \tan (e+f x))^{9/2}}-\frac {2 (4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{3003 c^3 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-9 B) (a+i a \tan (e+f x))^{5/2}}{15015 c^4 f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 11.32, size = 183, normalized size = 0.70 \begin {gather*} -\frac {i a^2 \cos (e+f x) (5005 A+780 (9 A+i B) \cos (2 (e+f x))+231 (9 A+4 i B) \cos (4 (e+f x))-1560 i A \sin (2 (e+f x))+3510 B \sin (2 (e+f x))-924 i A \sin (4 (e+f x))+2079 B \sin (4 (e+f x))) (\cos (9 e+11 f x)+i \sin (9 e+11 f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{120120 c^7 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.47, size = 183, normalized size = 0.70
method | result | size |
risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (1155 i A \,{\mathrm e}^{12 i \left (f x +e \right )}+1155 B \,{\mathrm e}^{12 i \left (f x +e \right )}+5460 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+2730 B \,{\mathrm e}^{10 i \left (f x +e \right )}+10010 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+8580 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-4290 B \,{\mathrm e}^{6 i \left (f x +e \right )}+3003 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-3003 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{240240 c^{6} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(169\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (18 i B \left (\tan ^{5}\left (f x +e \right )\right )+64 i A \left (\tan ^{4}\left (f x +e \right )\right )+8 A \left (\tan ^{5}\left (f x +e \right )\right )-531 i B \left (\tan ^{3}\left (f x +e \right )\right )-144 B \left (\tan ^{4}\left (f x +e \right )\right )-544 i A \left (\tan ^{2}\left (f x +e \right )\right )-236 A \left (\tan ^{3}\left (f x +e \right )\right )-1704 i B \tan \left (f x +e \right )+1224 B \left (\tan ^{2}\left (f x +e \right )\right )-1763 i A +911 A \tan \left (f x +e \right )+213 B \right )}{15015 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(183\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (18 i B \left (\tan ^{5}\left (f x +e \right )\right )+64 i A \left (\tan ^{4}\left (f x +e \right )\right )+8 A \left (\tan ^{5}\left (f x +e \right )\right )-531 i B \left (\tan ^{3}\left (f x +e \right )\right )-144 B \left (\tan ^{4}\left (f x +e \right )\right )-544 i A \left (\tan ^{2}\left (f x +e \right )\right )-236 A \left (\tan ^{3}\left (f x +e \right )\right )-1704 i B \tan \left (f x +e \right )+1224 B \left (\tan ^{2}\left (f x +e \right )\right )-1763 i A +911 A \tan \left (f x +e \right )+213 B \right )}{15015 f \,c^{7} \left (i+\tan \left (f x +e \right )\right )^{8}}\) | \(183\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.64, size = 352, normalized size = 1.35 \begin {gather*} \frac {{\left (1155 \, {\left (-i \, A - B\right )} a^{2} \cos \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 2730 \, {\left (-2 i \, A - B\right )} a^{2} \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 10010 i \, A a^{2} \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 4290 \, {\left (-2 i \, A + B\right )} a^{2} \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3003 \, {\left (-i \, A + B\right )} a^{2} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1155 \, {\left (A - i \, B\right )} a^{2} \sin \left (\frac {13}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 2730 \, {\left (2 \, A - i \, B\right )} a^{2} \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 10010 \, A a^{2} \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 4290 \, {\left (2 \, A + i \, B\right )} a^{2} \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 3003 \, {\left (A + i \, B\right )} a^{2} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{240240 \, c^{\frac {13}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.10, size = 175, normalized size = 0.67 \begin {gather*} -\frac {{\left (1155 \, {\left (i \, A + B\right )} a^{2} e^{\left (15 i \, f x + 15 i \, e\right )} + 105 \, {\left (63 i \, A + 37 \, B\right )} a^{2} e^{\left (13 i \, f x + 13 i \, e\right )} + 910 \, {\left (17 i \, A + 3 \, B\right )} a^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + 1430 \, {\left (13 i \, A - 3 \, B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 429 \, {\left (27 i \, A - 17 \, B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 3003 \, {\left (i \, A - B\right )} a^{2} e^{\left (5 i \, f x + 5 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}}}{240240 \, c^{7} f} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 13.73, size = 191, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {a+\frac {a\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}\,\left (\frac {a^2\,{\mathrm {e}}^{e\,6{}\mathrm {i}+f\,x\,6{}\mathrm {i}}\,\left (2\,A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{56\,c^6\,f}+\frac {a^2\,{\mathrm {e}}^{e\,10{}\mathrm {i}+f\,x\,10{}\mathrm {i}}\,\left (2\,A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{88\,c^6\,f}+\frac {A\,a^2\,{\mathrm {e}}^{e\,8{}\mathrm {i}+f\,x\,8{}\mathrm {i}}\,1{}\mathrm {i}}{24\,c^6\,f}+\frac {a^2\,{\mathrm {e}}^{e\,4{}\mathrm {i}+f\,x\,4{}\mathrm {i}}\,\left (A+B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{80\,c^6\,f}+\frac {a^2\,{\mathrm {e}}^{e\,12{}\mathrm {i}+f\,x\,12{}\mathrm {i}}\,\left (A-B\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{208\,c^6\,f}\right )}{\sqrt {c-\frac {c\,\sin \left (e+f\,x\right )\,1{}\mathrm {i}}{\cos \left (e+f\,x\right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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