Optimal. Leaf size=182 \[ \frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}} \]
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Rubi [A]
time = 0.11, antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3604, 47, 37}
\begin {gather*} \frac {2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 3604
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac {c \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{21 a f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac {(2 c) \text {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{105 a^2 f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 2.81, size = 115, normalized size = 0.63 \begin {gather*} \frac {c \sec ^2(e+f x) (92+140 \cos (2 (e+f x))+35 i \sec (e+f x) \sin (3 (e+f x))+27 i \tan (e+f x)) (i+\tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{1260 a^4 f (-i+\tan (e+f x))^4 \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.40, size = 97, normalized size = 0.53
method | result | size |
derivativedivides | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (2 i \left (\tan ^{3}\left (f x +e \right )\right )-33 i \tan \left (f x +e \right )+12 \left (\tan ^{2}\left (f x +e \right )\right )-58\right )}{315 f \,a^{5} \left (-\tan \left (f x +e \right )+i\right )^{6}}\) | \(97\) |
default | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (2 i \left (\tan ^{3}\left (f x +e \right )\right )-33 i \tan \left (f x +e \right )+12 \left (\tan ^{2}\left (f x +e \right )\right )-58\right )}{315 f \,a^{5} \left (-\tan \left (f x +e \right )+i\right )^{6}}\) | \(97\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.61, size = 200, normalized size = 1.10 \begin {gather*} \frac {{\left (35 i \, c \cos \left (9 \, f x + 9 \, e\right ) + 135 i \, c \cos \left (\frac {7}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 189 i \, c \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 105 i \, c \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 35 \, c \sin \left (9 \, f x + 9 \, e\right ) + 135 \, c \sin \left (\frac {7}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 189 \, c \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 105 \, c \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right )\right )} \sqrt {c}}{2520 \, a^{\frac {9}{2}} f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.13, size = 110, normalized size = 0.60 \begin {gather*} \frac {{\left (105 i \, c e^{\left (8 i \, f x + 8 i \, e\right )} + 294 i \, c e^{\left (6 i \, f x + 6 i \, e\right )} + 324 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 170 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 35 i \, c\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}} e^{\left (-9 i \, f x - 9 i \, e\right )}}{2520 \, a^{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 = 7.30, size = 205, normalized size = 1.13 \begin {gather*} \frac {c\,\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}}\,\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}}\,\left (105\,\sin \left (2\,e+2\,f\,x\right )+294\,\sin \left (4\,e+4\,f\,x\right )+324\,\sin \left (6\,e+6\,f\,x\right )+170\,\sin \left (8\,e+8\,f\,x\right )+35\,\sin \left (10\,e+10\,f\,x\right )+\cos \left (2\,e+2\,f\,x\right )\,105{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,294{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,324{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,170{}\mathrm {i}+\cos \left (10\,e+10\,f\,x\right )\,35{}\mathrm {i}\right )}{5040\,a^5\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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