Optimal. Leaf size=43 \[ \frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.11, antiderivative size = 43, 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, 37} \[ \frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 3523
Rubi steps
\begin {align*} \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {\sqrt {c-i c x}}{(a+i a x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i (c-i c \tan (e+f x))^{3/2}}{3 f (a+i a \tan (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 2.25, size = 69, normalized size = 1.60 \[ \frac {c (1-i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{3 a f (\tan (e+f x)-i) \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.41, size = 67, normalized size = 1.56 \[ \frac {{\left (i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + i \, c\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-3 i \, f x - 3 i \, e\right )}}{3 \, a^{2} f} \]
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 \[ \int \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.22, size = 64, normalized size = 1.49 \[ \frac {\sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan ^{2}\left (f x +e \right )\right )}{3 f \,a^{2} \left (-\tan \left (f x +e \right )+i\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.05, size = 35, normalized size = 0.81 \[ \frac {{\left (i \, c \cos \left (3 \, f x + 3 \, e\right ) + c \sin \left (3 \, f x + 3 \, e\right )\right )} \sqrt {c}}{3 \, a^{\frac {3}{2}} f} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.52, size = 62, normalized size = 1.44 \[ -\frac {c\,\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,\sqrt {-c\,\left (-1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}}{3\,a\,f\,\left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\sqrt {a\,\left (1+\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}} \]
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 \[ \int \frac {\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}{\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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