Optimal. Leaf size=155 \[ -\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}} \]
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Rubi [A]
time = 0.17, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps
used = 4, 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 {(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}}-\frac {(-7 B+2 i A) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \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))^{9/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {(a+i a x)^{3/2} (A+B x)}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}+\frac {(a (2 A+7 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 )}{9 f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}+\frac {(a (2 A+7 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 )}{63 c f}\\ &=-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(2 i A-7 B) (a+i a \tan (e+f x))^{5/2}}{315 c^2 f (c-i c \tan (e+f x))^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 3.63, size = 135, normalized size = 0.87 \begin {gather*} \frac {a^2 \cos (e+f x) (-45 i A+7 (-7 i A+2 B) \cos (2 (e+f x))-7 (2 A+7 i B) \sin (2 (e+f x))) (\cos (7 e+9 f x)+i \sin (7 e+9 f x)) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{630 c^5 f (\cos (f x)+i \sin (f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.42, size = 138, normalized size = 0.89
method | result | size |
risch | \(-\frac {a^{2} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (35 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+35 B \,{\mathrm e}^{8 i \left (f x +e \right )}+90 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+63 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-63 B \,{\mathrm e}^{4 i \left (f x +e \right )}\right )}{1260 c^{4} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(119\) |
derivativedivides | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (-47 A -33 i A \tan \left (f x +e \right )-12 A \left (\tan ^{2}\left (f x +e \right )\right )-7 i B -42 i B \left (\tan ^{2}\left (f x +e \right )\right )-42 B \tan \left (f x +e \right )+2 i A \left (\tan ^{3}\left (f x +e \right )\right )-7 B \left (\tan ^{3}\left (f x +e \right )\right )\right )}{315 f \,c^{5} \left (i+\tan \left (f x +e \right )\right )^{6}}\) | \(138\) |
default | \(-\frac {i \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (-47 A -33 i A \tan \left (f x +e \right )-12 A \left (\tan ^{2}\left (f x +e \right )\right )-7 i B -42 i B \left (\tan ^{2}\left (f x +e \right )\right )-42 B \tan \left (f x +e \right )+2 i A \left (\tan ^{3}\left (f x +e \right )\right )-7 B \left (\tan ^{3}\left (f x +e \right )\right )\right )}{315 f \,c^{5} \left (i+\tan \left (f x +e \right )\right )^{6}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.64, size = 210, normalized size = 1.35 \begin {gather*} \frac {{\left (35 \, {\left (-i \, A - B\right )} 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 ) - 90 i \, A 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 ) + 63 \, {\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 ) + 35 \, {\left (A - i \, B\right )} 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 ) + 90 \, A 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 ) + 63 \, {\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}}{1260 \, c^{\frac {9}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.96, size = 131, normalized size = 0.85 \begin {gather*} -\frac {{\left (35 \, {\left (i \, A + B\right )} a^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + 5 \, {\left (25 i \, A + 7 \, B\right )} a^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 9 \, {\left (17 i \, A - 7 \, B\right )} a^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 63 \, {\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}}}{1260 \, c^{5} f} \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]
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 = 11.00, size = 217, normalized size = 1.40 \begin {gather*} -\frac {a^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (4\,e+4\,f\,x\right )\,63{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,90{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,35{}\mathrm {i}-63\,B\,\cos \left (4\,e+4\,f\,x\right )+35\,B\,\cos \left (8\,e+8\,f\,x\right )-63\,A\,\sin \left (4\,e+4\,f\,x\right )-90\,A\,\sin \left (6\,e+6\,f\,x\right )-35\,A\,\sin \left (8\,e+8\,f\,x\right )-B\,\sin \left (4\,e+4\,f\,x\right )\,63{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,35{}\mathrm {i}\right )}{1260\,c^4\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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