Optimal. Leaf size=121 \[ \frac {i a 2^{\frac {n+3}{2}} (1+i \tan (c+d x))^{\frac {1}{2} (-n-1)} (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{3-n} \, _2F_1\left (\frac {1}{2} (-n-1),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{d (3-n)} \]
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Rubi [A] time = 0.24, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3505, 3523, 70, 69} \[ \frac {i a 2^{\frac {n+3}{2}} (1+i \tan (c+d x))^{\frac {1}{2} (-n-1)} (a+i a \tan (c+d x))^{n-1} (e \sec (c+d x))^{3-n} \text {Hypergeometric2F1}\left (\frac {1}{2} (-n-1),\frac {3-n}{2},\frac {5-n}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{d (3-n)} \]
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-n} (a+i a \tan (c+d x))^n \, dx &=\left ((e \sec (c+d x))^{3-n} (a-i a \tan (c+d x))^{\frac {1}{2} (-3+n)} (a+i a \tan (c+d x))^{\frac {1}{2} (-3+n)}\right ) \int (a-i a \tan (c+d x))^{\frac {3-n}{2}} (a+i a \tan (c+d x))^{\frac {3-n}{2}+n} \, dx\\ &=\frac {\left (a^2 (e \sec (c+d x))^{3-n} (a-i a \tan (c+d x))^{\frac {1}{2} (-3+n)} (a+i a \tan (c+d x))^{\frac {1}{2} (-3+n)}\right ) \operatorname {Subst}\left (\int (a-i a x)^{-1+\frac {3-n}{2}} (a+i a x)^{-1+\frac {3-n}{2}+n} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{\frac {1}{2}+\frac {n}{2}} a^2 (e \sec (c+d x))^{3-n} (a-i a \tan (c+d x))^{\frac {1}{2} (-3+n)} (a+i a \tan (c+d x))^{\frac {1}{2}+\frac {1}{2} (-3+n)+\frac {n}{2}} \left (\frac {a+i a \tan (c+d x)}{a}\right )^{-\frac {1}{2}-\frac {n}{2}}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{2}+\frac {i x}{2}\right )^{-1+\frac {3-n}{2}+n} (a-i a x)^{-1+\frac {3-n}{2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i 2^{\frac {3+n}{2}} a \, _2F_1\left (\frac {1}{2} (-1-n),\frac {3-n}{2};\frac {5-n}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (e \sec (c+d x))^{3-n} (1+i \tan (c+d x))^{\frac {1}{2} (-1-n)} (a+i a \tan (c+d x))^{-1+n}}{d (3-n)}\\ \end {align*}
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Mathematica [A] time = 12.26, size = 116, normalized size = 0.96 \[ \frac {8 e^3 (\tan (d x)+i) \sec (d x) (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-n} \, _2F_1\left (3,\frac {3-n}{2};\frac {5-n}{2};i \sin (2 (c+d x))-\cos (2 (c+d x))\right )}{d (n-3) (\cos (c)+i \sin (c))^3 (\tan (d x)-i)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ \frac {{\left ({\left ({\left (-i \, n - i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, n e^{\left (2 i \, d x + 2 i \, c\right )} - i \, n + i\right )} \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{-n + 3} e^{\left (i \, d n x + i \, c n + n \log \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right ) + n \log \left (\frac {a}{e}\right )\right )} + 8 \, d e^{\left (2 i \, d x + 2 i \, c\right )} {\rm integral}\left (-\frac {1}{8} \, {\left (n^{2} + {\left (n^{2} - 1\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + 2 \, {\left (n^{2} - 1\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )} \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{-n + 3} e^{\left (i \, d n x + i \, c n - 2 i \, d x + n \log \left (\frac {2 \, e e^{\left (i \, d x + i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right ) + n \log \left (\frac {a}{e}\right ) - 2 i \, c\right )}, x\right )\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, d} \]
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 \[ \int \left (e \sec \left (d x + c\right )\right )^{-n + 3} {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.08, size = 0, normalized size = 0.00 \[ \int \left (e \sec \left (d x +c \right )\right )^{3-n} \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int {\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{3-n}\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
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 \[ \int \left (e \sec {\left (c + d x \right )}\right )^{3 - n} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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