Optimal. Leaf size=63 \[ -\frac {2 i \sqrt {a} \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f} \]
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Rubi [A]
time = 0.08, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3604, 65, 223,
209} \begin {gather*} -\frac {2 i \sqrt {a} \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 223
Rule 3604
Rubi steps
\begin {align*} \int \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {(2 i c) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{f}\\ &=-\frac {(2 i c) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{f}\\ &=-\frac {2 i \sqrt {a} \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}\\ \end {align*}
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Mathematica [A]
time = 0.94, size = 74, normalized size = 1.17 \begin {gather*} -\frac {i \sqrt {2} c e^{-i (e+f x)} \text {ArcTan}\left (e^{i (e+f x)}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {\frac {c}{1+e^{2 i (e+f x)}}} f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 96, normalized size = 1.52
method | result | size |
derivativedivides | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a c \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right )}{f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(96\) |
default | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a c \ln \left (\frac {c a \tan \left (f x +e \right )+\sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}{\sqrt {a c}}\right )}{f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\) | \(96\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 114 vs. \(2 (47) = 94\).
time = 0.61, size = 114, normalized size = 1.81 \begin {gather*} \frac {\sqrt {a} \sqrt {c} {\left (-2 i \, \arctan \left (\cos \left (f x + e\right ), \sin \left (f x + e\right ) + 1\right ) - 2 i \, \arctan \left (\cos \left (f x + e\right ), -\sin \left (f x + e\right ) + 1\right ) + \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} + 2 \, \sin \left (f x + e\right ) + 1\right ) - \log \left (\cos \left (f x + e\right )^{2} + \sin \left (f x + e\right )^{2} - 2 \, \sin \left (f x + e\right ) + 1\right )\right )}}{2 \, f} \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. 229 vs. \(2 (47) = 94\).
time = 1.29, size = 229, normalized size = 3.63 \begin {gather*} \frac {1}{2} \, \sqrt {\frac {a c}{f^{2}}} \log \left (\frac {4 \, {\left (2 \, \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 (e^{\left (3 i \, f x + 3 i \, e\right )} + e^{\left (i \, f x + i \, e\right )}\right )} - {\left (i \, f e^{\left (2 i \, f x + 2 i \, e\right )} - i \, f\right )} \sqrt {\frac {a c}{f^{2}}}\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right ) - \frac {1}{2} \, \sqrt {\frac {a c}{f^{2}}} \log \left (\frac {4 \, {\left (2 \, \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 (e^{\left (3 i \, f x + 3 i \, e\right )} + e^{\left (i \, f x + i \, e\right )}\right )} - {\left (-i \, f e^{\left (2 i \, f x + 2 i \, e\right )} + i \, f\right )} \sqrt {\frac {a c}{f^{2}}}\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right ) \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 \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 \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 = 5.77, size = 62, normalized size = 0.98 \begin {gather*} -\frac {\sqrt {a}\,\sqrt {c}\,\mathrm {atan}\left (\frac {\sqrt {c}\,\left (\sqrt {a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-\sqrt {a}\right )}{\sqrt {a}\,\left (\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}-\sqrt {c}\right )}\right )\,4{}\mathrm {i}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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