Optimal. Leaf size=95 \[ -\frac {i c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a f}+\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{a f (c+i c \tan (e+f x))} \]
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Rubi [A]
time = 0.12, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {3603, 3568, 43,
65, 212} \begin {gather*} \frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{a f (c+i c \tan (e+f x))}-\frac {i c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 43
Rule 65
Rule 212
Rule 3568
Rule 3603
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^{3/2}}{a+i a \tan (e+f x)} \, dx &=\frac {\int \cos ^2(e+f x) (c-i c \tan (e+f x))^{5/2} \, dx}{a c}\\ &=\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {\sqrt {c+x}}{(c-x)^2} \, dx,x,-i c \tan (e+f x)\right )}{a f}\\ &=\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{a f (c+i c \tan (e+f x))}-\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {1}{(c-x) \sqrt {c+x}} \, dx,x,-i c \tan (e+f x)\right )}{2 a f}\\ &=\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{a f (c+i c \tan (e+f x))}-\frac {\left (i c^2\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,\sqrt {c-i c \tan (e+f x)}\right )}{a f}\\ &=-\frac {i c^{3/2} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right )}{\sqrt {2} a f}+\frac {i c^2 \sqrt {c-i c \tan (e+f x)}}{a f (c+i c \tan (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.91, size = 114, normalized size = 1.20 \begin {gather*} \frac {(-i c \cos (e+f x)-c \sin (e+f x)) \left (\sqrt {2} \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c-i c \tan (e+f x)}}{\sqrt {2} \sqrt {c}}\right ) (\cos (e+f x)+i \sin (e+f x))-2 \cos (e+f x) \sqrt {c-i c \tan (e+f x)}\right )}{2 a f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 77, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {2 i c^{2} \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}}{2 c +2 i c \tan \left (f x +e \right )}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{f a}\) | \(77\) |
default | \(\frac {2 i c^{2} \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}}{2 c +2 i c \tan \left (f x +e \right )}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )}{4 \sqrt {c}}\right )}{f a}\) | \(77\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 110, normalized size = 1.16 \begin {gather*} \frac {i \, {\left (\frac {\sqrt {2} c^{\frac {5}{2}} \log \left (-\frac {\sqrt {2} \sqrt {c} - \sqrt {-i \, c \tan \left (f x + e\right ) + c}}{\sqrt {2} \sqrt {c} + \sqrt {-i \, c \tan \left (f x + e\right ) + c}}\right )}{a} - \frac {4 \, \sqrt {-i \, c \tan \left (f x + e\right ) + c} c^{3}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )} a - 2 \, a c}\right )}}{4 \, c 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. 267 vs. \(2 (77) = 154\).
time = 0.71, size = 267, normalized size = 2.81 \begin {gather*} -\frac {{\left (\sqrt {2} a f \sqrt {-\frac {c^{3}}{a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, c^{2} + {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {c^{3}}{a^{2} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) - \sqrt {2} a f \sqrt {-\frac {c^{3}}{a^{2} f^{2}}} e^{\left (2 i \, f x + 2 i \, e\right )} \log \left (-\frac {2 \, {\left (i \, c^{2} - {\left (a f e^{\left (2 i \, f x + 2 i \, e\right )} + a f\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {-\frac {c^{3}}{a^{2} f^{2}}}\right )} e^{\left (-i \, f x - i \, e\right )}}{a f}\right ) + 2 \, \sqrt {2} {\left (-i \, c e^{\left (2 i \, f x + 2 i \, e\right )} - i \, c\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}\right )} e^{\left (-2 i \, f x - 2 i \, e\right )}}{4 \, 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*} - \frac {i \left (\int \frac {c \sqrt {- i c \tan {\left (e + f x \right )} + c}}{\tan {\left (e + f x \right )} - i}\, dx + \int \left (- \frac {i c \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}}{\tan {\left (e + f x \right )} - i}\right )\, dx\right )}{a} \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 = 0.29, size = 83, normalized size = 0.87 \begin {gather*} \frac {\sqrt {2}\,{\left (-c\right )}^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}}{2\,\sqrt {-c}}\right )\,1{}\mathrm {i}}{2\,a\,f}+\frac {c^2\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{a\,f\,\left (c+c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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