Optimal. Leaf size=90 \[ \frac {i \sqrt {c-i c \tan (e+f x)}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {i \sqrt {c-i c \tan (e+f x)}}{3 a f \sqrt {a+i a \tan (e+f x)}} \]
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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 {c-i c \tan (e+f x)}}{3 a f \sqrt {a+i a \tan (e+f x)}}+\frac {i \sqrt {c-i c \tan (e+f x)}}{3 f (a+i a \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 {c-i c \tan (e+f x)}}{(a+i a \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \text {Subst}\left (\int \frac {1}{(a+i a x)^{5/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i \sqrt {c-i c \tan (e+f x)}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {c \text {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{3 f}\\ &=\frac {i \sqrt {c-i c \tan (e+f x)}}{3 f (a+i a \tan (e+f x))^{3/2}}+\frac {i \sqrt {c-i c \tan (e+f x)}}{3 a f \sqrt {a+i a \tan (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 68, normalized size = 0.76 \begin {gather*} \frac {(2+i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)}}{3 a f (-i+\tan (e+f x)) \sqrt {a+i a \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 74, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (3 i \tan \left (f x +e \right )-\left (\tan ^{2}\left (f x +e \right )\right )+2\right )}{3 f \,a^{2} \left (-\tan \left (f x +e \right )+i\right )^{3}}\) | \(74\) |
default | \(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \left (3 i \tan \left (f x +e \right )-\left (\tan ^{2}\left (f x +e \right )\right )+2\right )}{3 f \,a^{2} \left (-\tan \left (f x +e \right )+i\right )^{3}}\) | \(74\) |
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 = 1.34, size = 80, normalized size = 0.89 \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 (3 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 4 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + i\right )} e^{\left (-3 i \, f x - 3 i \, e\right )}}{6 \, a^{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 c \left (\tan {\left (e + f x \right )} + i\right )}}{\left (i a \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 = 6.31, size = 135, normalized size = 1.50 \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}}\,\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}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,4{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,1{}\mathrm {i}+4\,\sin \left (2\,e+2\,f\,x\right )+\sin \left (4\,e+4\,f\,x\right )+3{}\mathrm {i}\right )}{12\,a^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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