Optimal. Leaf size=140 \[ \frac {2 \tan (e+f x)}{5 a^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.14, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3523, 45, 39} \[ \frac {2 \tan (e+f x)}{5 a^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 45
Rule 3523
Rubi steps
\begin {align*} \int \frac {1}{(a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{7/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(3 c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{5 f}\\ &=\frac {i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {(2 c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=\frac {i}{5 f (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}+\frac {i}{5 a f (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}+\frac {2 \tan (e+f x)}{5 a^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 = 2.89, size = 99, normalized size = 0.71 \[ \frac {\sqrt {c-i c \tan (e+f x)} (-4 \cos (2 (e+f x))+5 i \tan (e+f x)-3 i \sin (3 (e+f x)) \sec (e+f x)+12)}{20 a^2 c f (\tan (e+f x)-i) \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 122, normalized size = 0.87 \[ \frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-5 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 16 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 10 i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 16 i \, e^{\left (5 i \, f x + 5 i \, e\right )} + 20 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 6 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-5 i \, f x - 5 i \, e\right )}}{40 \, a^{3} c f} \]
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 \[ \int \frac {1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {5}{2}} \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.32, size = 118, normalized size = 0.84 \[ -\frac {\sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (4 i \left (\tan ^{4}\left (f x +e \right )\right )-2 \left (\tan ^{5}\left (f x +e \right )\right )+6 i \left (\tan ^{2}\left (f x +e \right )\right )-\left (\tan ^{3}\left (f x +e \right )\right )+2 i+\tan \left (f x +e \right )\right )}{5 f c \,a^{3} \left (\tan \left (f x +e \right )+i\right )^{2} \left (-\tan \left (f x +e \right )+i\right )^{4}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.96, size = 158, normalized size = 1.13 \[ \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,15{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,5{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,1{}\mathrm {i}+15\,\sin \left (2\,e+2\,f\,x\right )+5\,\sin \left (4\,e+4\,f\,x\right )+\sin \left (6\,e+6\,f\,x\right )-5{}\mathrm {i}\right )}{40\,a^3\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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