Optimal. Leaf size=137 \[ -\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \]
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Rubi [A] time = 0.14, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {3523, 45, 37} \[ -\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 37
Rule 45
Rule 3523
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {1}{(a+i a x)^{3/2} (c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}+\frac {(2 c) \operatorname {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}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}+\frac {2 \operatorname {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}{f \sqrt {a+i a \tan (e+f x)} (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a f (c-i c \tan (e+f x))^{3/2}}-\frac {2 i \sqrt {a+i a \tan (e+f x)}}{3 a c f \sqrt {c-i c \tan (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.82, size = 89, normalized size = 0.65 \[ \frac {i \sqrt {c-i c \tan (e+f x)} (\cos (2 (e+f x))+i \sin (2 (e+f x))) (-2 i \sin (2 (e+f x))+\cos (2 (e+f x))-3)}{6 c^2 f \sqrt {a+i a \tan (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 111, normalized size = 0.81 \[ \frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 7 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 4 i \, e^{\left (3 i \, f x + 3 i \, e\right )} - 3 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 4 i \, e^{\left (i \, f x + i \, e\right )} + 3 i\right )} e^{\left (-i \, f x - i \, e\right )}}{12 \, a c^{2} f} \]
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 \[ \int \frac {1}{\sqrt {i \, a \tan \left (f x + e\right ) + a} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 109, normalized size = 0.80 \[ \frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (-1+i \tan \left (f x +e \right )\right )}\, \left (2 i \left (\tan ^{3}\left (f x +e \right )\right )+2 \left (\tan ^{4}\left (f x +e \right )\right )+2 i \tan \left (f x +e \right )+3 \left (\tan ^{2}\left (f x +e \right )\right )+1\right )}{3 f \,c^{2} a \left (\tan \left (f x +e \right )+i\right )^{3} \left (-\tan \left (f x +e \right )+i\right )^{2}} \]
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 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.39, size = 117, normalized size = 0.85 \[ \frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}+2\,\sin \left (2\,e+2\,f\,x\right )-3{}\mathrm {i}\right )}{6\,a\,c\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
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 \[ \int \frac {1}{\sqrt {i a \left (\tan {\left (e + f x \right )} - i\right )} \left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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