Optimal. Leaf size=58 \[ -\frac {2 a (i A+B)}{f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a B \sqrt {c-i c \tan (e+f x)}}{c f} \]
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Rubi [A]
time = 0.06, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3669, 45}
\begin {gather*} -\frac {2 a (B+i A)}{f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a B \sqrt {c-i c \tan (e+f x)}}{c f} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 3669
Rubi steps
\begin {align*} \int \frac {(a+i a \tan (e+f x)) (A+B \tan (e+f x))}{\sqrt {c-i c \tan (e+f x)}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {A+B x}{(c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {(a c) \text {Subst}\left (\int \left (\frac {A-i B}{(c-i c x)^{3/2}}+\frac {i B}{c \sqrt {c-i c x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {2 a (i A+B)}{f \sqrt {c-i c \tan (e+f x)}}-\frac {2 a B \sqrt {c-i c \tan (e+f x)}}{c f}\\ \end {align*}
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Mathematica [A]
time = 0.83, size = 82, normalized size = 1.41 \begin {gather*} \frac {2 a (\cos (f x)-i \sin (f x)) ((A-2 i B) \cos (e+f x)-B \sin (e+f x)) (-i \cos (e+2 f x)+\sin (e+2 f x)) \sqrt {c-i c \tan (e+f x)}}{c f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 53, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {2 i a \left (i B \sqrt {c -i c \tan \left (f x +e \right )}-\frac {c \left (-i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f c}\) | \(53\) |
default | \(\frac {2 i a \left (i B \sqrt {c -i c \tan \left (f x +e \right )}-\frac {c \left (-i B +A \right )}{\sqrt {c -i c \tan \left (f x +e \right )}}\right )}{f c}\) | \(53\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 50, normalized size = 0.86 \begin {gather*} \frac {2 i \, {\left (i \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} B a - \frac {{\left (A - i \, B\right )} a c}{\sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}}{c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.11, size = 57, normalized size = 0.98 \begin {gather*} \frac {\sqrt {2} {\left ({\left (-i \, A - B\right )} a e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-i \, A - 3 \, B\right )} a\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{c 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*} i a \left (\int \left (- \frac {i A}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx + \int \frac {A \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \frac {B \tan ^{2}{\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\, dx + \int \left (- \frac {i B \tan {\left (e + f x \right )}}{\sqrt {- i c \tan {\left (e + f x \right )} + c}}\right )\, dx\right ) \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 = 10.16, size = 164, normalized size = 2.83 \begin {gather*} -\frac {a\,\sqrt {\frac {2\,c}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,\left (A\,1{}\mathrm {i}+3\,B+A\,\left (\frac {{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}}{2}\right )\,1{}\mathrm {i}-A\,\left (\frac {{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )+B\,\left (\frac {{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}}{2}\right )+B\,\left (\frac {{\mathrm {e}}^{-e\,2{}\mathrm {i}-f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}-\frac {{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}}{2}\right )\,1{}\mathrm {i}\right )}{c\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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