Optimal. Leaf size=44 \[ \frac {\tan (e+f x)}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A] time = 0.10, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {3523, 39} \[ \frac {\tan (e+f x)}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 39
Rule 3523
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a 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 )}{f}\\ &=\frac {\tan (e+f x)}{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 = 1.42, size = 64, normalized size = 1.45 \[ \frac {\sin (e+f x) \sqrt {c-i c \tan (e+f x)} (\cos (e+f x)+i \sin (e+f x))}{c f \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 67, normalized size = 1.52 \[ \frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (4 i \, f x + 4 i \, e\right )} + i\right )} e^{\left (-i \, f x - i \, e\right )}}{2 \, a c f} \]
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 \[ \int \frac {1}{\sqrt {i \, a \tan \left (f x + e\right ) + a} \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.36, size = 82, normalized size = 1.86 \[ \frac {\sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (1+\tan ^{2}\left (f x +e \right )\right ) \tan \left (f x +e \right )}{f c a \left (\tan \left (f x +e \right )+i\right )^{2} \left (-\tan \left (f x +e \right )+i\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.66, size = 16, normalized size = 0.36 \[ \frac {\sin \left (f x + e\right )}{\sqrt {a} \sqrt {c} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.48, size = 112, normalized size = 2.55 \[ \frac {\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+\sin \left (2\,e+2\,f\,x\right )-\mathrm {i}\right )\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{2\,a\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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 \[ \int \frac {1}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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