Optimal. Leaf size=94 \[ \frac{i a 2^{n+\frac{5}{4}} \sqrt{e \sec (c+d x)} (1+i \tan (c+d x))^{\frac{3}{4}-n} (a+i a \tan (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{3}{4}-n,\frac{5}{4},\frac{1}{2} (1-i \tan (c+d x))\right )}{d} \]
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Rubi [A] time = 0.174919, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3505, 3523, 70, 69} \[ \frac{i a 2^{n+\frac{5}{4}} \sqrt{e \sec (c+d x)} (1+i \tan (c+d x))^{\frac{3}{4}-n} (a+i a \tan (c+d x))^{n-1} \text{Hypergeometric2F1}\left (\frac{1}{4},\frac{3}{4}-n,\frac{5}{4},\frac{1}{2} (1-i \tan (c+d x))\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3505
Rule 3523
Rule 70
Rule 69
Rubi steps
\begin{align*} \int \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^n \, dx &=\frac{\sqrt{e \sec (c+d x)} \int \sqrt [4]{a-i a \tan (c+d x)} (a+i a \tan (c+d x))^{\frac{1}{4}+n} \, dx}{\sqrt [4]{a-i a \tan (c+d x)} \sqrt [4]{a+i a \tan (c+d x)}}\\ &=\frac{\left (a^2 \sqrt{e \sec (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(a+i a x)^{-\frac{3}{4}+n}}{(a-i a x)^{3/4}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [4]{a-i a \tan (c+d x)} \sqrt [4]{a+i a \tan (c+d x)}}\\ &=\frac{\left (2^{-\frac{3}{4}+n} a^2 \sqrt{e \sec (c+d x)} (a+i a \tan (c+d x))^{-1+n} \left (\frac{a+i a \tan (c+d x)}{a}\right )^{\frac{3}{4}-n}\right ) \operatorname{Subst}\left (\int \frac{\left (\frac{1}{2}+\frac{i x}{2}\right )^{-\frac{3}{4}+n}}{(a-i a x)^{3/4}} \, dx,x,\tan (c+d x)\right )}{d \sqrt [4]{a-i a \tan (c+d x)}}\\ &=\frac{i 2^{\frac{5}{4}+n} a \, _2F_1\left (\frac{1}{4},\frac{3}{4}-n;\frac{5}{4};\frac{1}{2} (1-i \tan (c+d x))\right ) \sqrt{e \sec (c+d x)} (1+i \tan (c+d x))^{\frac{3}{4}-n} (a+i a \tan (c+d x))^{-1+n}}{d}\\ \end{align*}
Mathematica [A] time = 8.34891, size = 156, normalized size = 1.66 \[ -\frac{i 2^{n+\frac{3}{2}} e^{i (c+d x)} \left (e^{i d x}\right )^n \left (\frac{e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n-\frac{1}{2}} \sqrt{e \sec (c+d x)} \sec ^{-n-\frac{1}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{-n} \text{Hypergeometric2F1}\left (\frac{3}{4},1,n+\frac{5}{4},-e^{2 i (c+d x)}\right ) (a+i a \tan (c+d x))^n}{d (4 n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.194, size = 0, normalized size = 0. \begin{align*} \int \sqrt{e\sec \left ( dx+c \right ) } \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \sec \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{2} \left (\frac{2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \sqrt{\frac{e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac{1}{2} i \, d x + \frac{1}{2} i \, c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{e \sec \left (d x + c\right )}{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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