Optimal. Leaf size=65 \[ \frac {i \sqrt {c-i c \tan (e+f x)} (a+i a \tan (e+f x))^m \, _2F_1\left (1,m+\frac {1}{2};\frac {3}{2};\frac {1}{2} (1-i \tan (e+f x))\right )}{f} \]
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Rubi [A] time = 0.10, antiderivative size = 86, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3523, 70, 69} \[ \frac {i 2^m \sqrt {c-i c \tan (e+f x)} (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \, _2F_1\left (\frac {1}{2},1-m;\frac {3}{2};\frac {1}{2} (1-i \tan (e+f x))\right )}{f} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3523
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^m \sqrt {c-i c \tan (e+f x)} \, dx &=\frac {(a c) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-1+m}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {\left (2^{-1+m} c (a+i a \tan (e+f x))^m \left (\frac {a+i a \tan (e+f x)}{a}\right )^{-m}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-1+m}}{\sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac {i 2^m \, _2F_1\left (\frac {1}{2},1-m;\frac {3}{2};\frac {1}{2} (1-i \tan (e+f x))\right ) (1+i \tan (e+f x))^{-m} (a+i a \tan (e+f x))^m \sqrt {c-i c \tan (e+f x)}}{f}\\ \end {align*}
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Mathematica [B] time = 2.16, size = 141, normalized size = 2.17 \[ -\frac {i c 2^{m-\frac {1}{2}} \left (e^{i f x}\right )^m \left (\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}\right )^m \, _2F_1\left (\frac {1}{2},1;m+1;-e^{2 i (e+f x)}\right ) \sec ^{-m}(e+f x) (\cos (f x)+i \sin (f x))^{-m} (a+i a \tan (e+f x))^m}{f m \sqrt {\frac {c}{1+e^{2 i (e+f x)}}}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {2} \left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}, x\right ) \]
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 \[ \int \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.08, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{m} \sqrt {c -i c \tan \left (f x +e \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-i \, c \tan \left (f x + e\right ) + c} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,d x \]
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 \[ \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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