Optimal. Leaf size=41 \[ \frac {i \sqrt {c-i c \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 41, 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 = {3604, 37}
\begin {gather*} \frac {i \sqrt {c-i c \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 3604
Rubi steps
\begin {align*} \int \frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {a+i a \tan (e+f x)}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i \sqrt {c-i c \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.69, size = 41, normalized size = 1.00 \begin {gather*} \frac {i \sqrt {c-i c \tan (e+f x)}}{f \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 65, normalized size = 1.59
method | result | size |
risch | \(\frac {i \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}{\sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(50\) |
derivativedivides | \(-\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (1+i \tan \left (f x +e \right )\right )}{f a \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(65\) |
default | \(-\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (1+i \tan \left (f x +e \right )\right )}{f a \left (-\tan \left (f x +e \right )+i\right )^{2}}\) | \(65\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 37, normalized size = 0.90 \begin {gather*} \frac {i \, \sqrt {c} \sqrt {-i \, \tan \left (f x + e\right ) + 1}}{\sqrt {a} f \sqrt {i \, \tan \left (f x + e\right ) + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 67 vs. \(2 (33) = 66\).
time = 1.67, size = 67, normalized size = 1.63 \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 (i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-i \, f x - i \, e\right )}}{a f} \end {gather*}
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 \begin {gather*} \int \frac {\sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )}}\, dx \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 = 0.71, size = 34, normalized size = 0.83 \begin {gather*} \frac {\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{f\,\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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