Optimal. Leaf size=90 \[ -\frac {i \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 c f \sqrt {c-i c \tan (e+f x)}} \]
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Rubi [A]
time = 0.08, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3604, 47, 37}
\begin {gather*} -\frac {i \sqrt {a+i a \tan (e+f x)}}{3 c f \sqrt {c-i c \tan (e+f x)}}-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 37
Rule 47
Rule 3604
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \tan (e+f x)}}{(c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {a \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} (c-i c x)^{3/2}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 f (c-i c \tan (e+f x))^{3/2}}-\frac {i \sqrt {a+i a \tan (e+f x)}}{3 c f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 1.19, size = 89, normalized size = 0.99 \begin {gather*} \frac {(2+2 \cos (2 (e+f x))-i \sin (2 (e+f x))) (-i \cos (2 (e+f x))+\sin (2 (e+f x))) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{6 c^2 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 70, normalized size = 0.78
method | result | size |
risch | \(-\frac {i \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{2 i \left (f x +e \right )}+3\right )}{6 c \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(64\) |
derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (3 i \tan \left (f x +e \right )+\tan ^{2}\left (f x +e \right )-2\right )}{3 f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{3}}\) | \(70\) |
default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (3 i \tan \left (f x +e \right )+\tan ^{2}\left (f x +e \right )-2\right )}{3 f \,c^{2} \left (\tan \left (f x +e \right )+i\right )^{3}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.93, size = 81, normalized size = 0.90 \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 (5 i \, f x + 5 i \, e\right )} - 4 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 3 i \, e^{\left (i \, f x + i \, e\right )}\right )}}{6 \, c^{2} 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 a \left (\tan {\left (e + f x \right )} - i\right )}}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, 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 = 5.05, size = 114, normalized size = 1.27 \begin {gather*} -\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}-\sin \left (2\,e+2\,f\,x\right )+3{}\mathrm {i}\right )}{6\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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