Optimal. Leaf size=96 \[ \frac {i a 2^{n+\frac {7}{4}} (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^{n-1} \, _2F_1\left (\frac {3}{4},\frac {1}{4}-n;\frac {7}{4};\frac {1}{2} (1-i \tan (c+d x))\right )}{3 d} \]
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Rubi [A] time = 0.20, antiderivative size = 96, normalized size of antiderivative = 1.00, 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} \[ \frac {i a 2^{n+\frac {7}{4}} (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^{n-1} \text {Hypergeometric2F1}\left (\frac {3}{4},\frac {1}{4}-n,\frac {7}{4},\frac {1}{2} (1-i \tan (c+d x))\right )}{3 d} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^n \, dx &=\frac {(e \sec (c+d x))^{3/2} \int (a-i a \tan (c+d x))^{3/4} (a+i a \tan (c+d x))^{\frac {3}{4}+n} \, dx}{(a-i a \tan (c+d x))^{3/4} (a+i a \tan (c+d x))^{3/4}}\\ &=\frac {\left (a^2 (e \sec (c+d x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-\frac {1}{4}+n}}{\sqrt [4]{a-i a x}} \, dx,x,\tan (c+d x)\right )}{d (a-i a \tan (c+d x))^{3/4} (a+i a \tan (c+d x))^{3/4}}\\ &=\frac {\left (2^{-\frac {1}{4}+n} a^2 (e \sec (c+d x))^{3/2} (a+i a \tan (c+d x))^{-1+n} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{\frac {1}{4}-n}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-\frac {1}{4}+n}}{\sqrt [4]{a-i a x}} \, dx,x,\tan (c+d x)\right )}{d (a-i a \tan (c+d x))^{3/4}}\\ &=\frac {i 2^{\frac {7}{4}+n} a \, _2F_1\left (\frac {3}{4},\frac {1}{4}-n;\frac {7}{4};\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{3/2} (1+i \tan (c+d x))^{\frac {1}{4}-n} (a+i a \tan (c+d x))^{-1+n}}{3 d}\\ \end {align*}
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Mathematica [A] time = 8.93, size = 156, normalized size = 1.62 \[ -\frac {i 2^{n+\frac {5}{2}} e^{i (c+d x)} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{n+\frac {1}{2}} (e \sec (c+d x))^{3/2} \, _2F_1\left (\frac {1}{4},1;n+\frac {7}{4};-e^{2 i (c+d x)}\right ) \sec ^{-n-\frac {3}{2}}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (4 n+3)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, \sqrt {2} e \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} \sqrt {\frac {e}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (\frac {3}{2} i \, d x + \frac {3}{2} i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}, x\right ) \]
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 \[ \int \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.83, size = 0, normalized size = 0.00 \[ \int \left (e \sec \left (d x +c \right )\right )^{\frac {3}{2}} \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int \left (e \sec \left (d x + c\right )\right )^{\frac {3}{2}} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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 \[ \int \left (e \sec {\left (c + d x \right )}\right )^{\frac {3}{2}} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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