Optimal. Leaf size=200 \[ \frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}+\frac {6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac {8 \tan (e+f x)}{35 a^2 c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {16 \tan (e+f x)}{35 a^3 c^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.12, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3604, 47, 40,
39} \begin {gather*} \frac {16 \tan (e+f x)}{35 a^3 c^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {8 \tan (e+f x)}{35 a^2 c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 39
Rule 40
Rule 47
Rule 3604
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{9/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}+\frac {(6 c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{7/2} (c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{7 f}\\ &=\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}+\frac {6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac {24 \text {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{35 a f}\\ &=\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}+\frac {6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac {8 \tan (e+f x)}{35 a^2 c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {16 \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{35 a^2 c f}\\ &=\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}+\frac {6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac {8 \tan (e+f x)}{35 a^2 c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {16 \tan (e+f x)}{35 a^3 c^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 5.53, size = 115, normalized size = 0.58 \begin {gather*} -\frac {i (-350+175 \cos (2 (e+f x))+14 \cos (4 (e+f x))+\cos (6 (e+f x))+350 i \sin (2 (e+f x))+56 i \sin (4 (e+f x))+6 i \sin (6 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{1120 a^3 c^3 f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 151, normalized size = 0.76
method | result | size |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (16 i \left (\tan ^{7}\left (f x +e \right )\right )-16 \left (\tan ^{8}\left (f x +e \right )\right )+56 i \left (\tan ^{5}\left (f x +e \right )\right )-56 \left (\tan ^{6}\left (f x +e \right )\right )+70 i \left (\tan ^{3}\left (f x +e \right )\right )-70 \left (\tan ^{4}\left (f x +e \right )\right )+30 i \tan \left (f x +e \right )-35 \left (\tan ^{2}\left (f x +e \right )\right )-5\right )}{35 f \,a^{4} c^{3} \left (-\tan \left (f x +e \right )+i\right )^{5} \left (\tan \left (f x +e \right )+i\right )^{4}}\) | \(151\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (16 i \left (\tan ^{7}\left (f x +e \right )\right )-16 \left (\tan ^{8}\left (f x +e \right )\right )+56 i \left (\tan ^{5}\left (f x +e \right )\right )-56 \left (\tan ^{6}\left (f x +e \right )\right )+70 i \left (\tan ^{3}\left (f x +e \right )\right )-70 \left (\tan ^{4}\left (f x +e \right )\right )+30 i \tan \left (f x +e \right )-35 \left (\tan ^{2}\left (f x +e \right )\right )-5\right )}{35 f \,a^{4} c^{3} \left (-\tan \left (f x +e \right )+i\right )^{5} \left (\tan \left (f x +e \right )+i\right )^{4}}\) | \(151\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.89, size = 167, normalized size = 0.84 \begin {gather*} \frac {\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}} {\left (-7 i \, e^{\left (14 i \, f x + 14 i \, e\right )} - 77 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 595 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 320 i \, e^{\left (9 i \, f x + 9 i \, e\right )} + 175 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 320 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 875 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 217 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 47 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 5 i\right )} e^{\left (-7 i \, f x - 7 i \, e\right )}}{2240 \, a^{4} c^{3} 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 = 6.70, size = 186, normalized size = 0.93 \begin {gather*} \frac {\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 (\cos \left (2\,e+2\,f\,x\right )\,630{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,168{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,42{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,5{}\mathrm {i}+770\,\sin \left (2\,e+2\,f\,x\right )+182\,\sin \left (4\,e+4\,f\,x\right )+42\,\sin \left (6\,e+6\,f\,x\right )+5\,\sin \left (8\,e+8\,f\,x\right )-525{}\mathrm {i}\right )}{2240\,a^4\,c^2\,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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