Optimal. Leaf size=97 \[ \frac {i 2^{-\frac {3}{2}-n} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (5+2 n);-\frac {1}{2};\frac {1}{2} (1+i \tan (c+d x))\right ) (e \sec (c+d x))^{-3-2 n} (1-i \tan (c+d x))^{\frac {3}{2}+n} (a+i a \tan (c+d x))^n}{3 d} \]
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Rubi [A]
time = 0.15, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3586, 3604, 7,
72, 71} \begin {gather*} \frac {i 2^{-n-\frac {3}{2}} (1-i \tan (c+d x))^{n+\frac {3}{2}} (a+i a \tan (c+d x))^n (e \sec (c+d x))^{-2 n-3} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (2 n+5);-\frac {1}{2};\frac {1}{2} (i \tan (c+d x)+1)\right )}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 7
Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps
\begin {align*} \int (e \sec (c+d x))^{-3-2 n} (a+i a \tan (c+d x))^n \, dx &=\left ((e \sec (c+d x))^{-3-2 n} (a-i a \tan (c+d x))^{\frac {1}{2} (3+2 n)} (a+i a \tan (c+d x))^{\frac {1}{2} (3+2 n)}\right ) \int (a-i a \tan (c+d x))^{\frac {1}{2} (-3-2 n)} (a+i a \tan (c+d x))^{\frac {1}{2} (-3-2 n)+n} \, dx\\ &=\frac {\left (a^2 (e \sec (c+d x))^{-3-2 n} (a-i a \tan (c+d x))^{\frac {1}{2} (3+2 n)} (a+i a \tan (c+d x))^{\frac {1}{2} (3+2 n)}\right ) \text {Subst}\left (\int (a-i a x)^{-1+\frac {1}{2} (-3-2 n)} (a+i a x)^{-1+\frac {1}{2} (-3-2 n)+n} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (a^2 (e \sec (c+d x))^{-3-2 n} (a-i a \tan (c+d x))^{\frac {1}{2} (3+2 n)} (a+i a \tan (c+d x))^{\frac {1}{2} (3+2 n)}\right ) \text {Subst}\left (\int \frac {(a-i a x)^{-1+\frac {1}{2} (-3-2 n)}}{(a+i a x)^{5/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{-\frac {5}{2}-n} (e \sec (c+d x))^{-3-2 n} (a-i a \tan (c+d x))^{-\frac {1}{2}-n+\frac {1}{2} (3+2 n)} \left (\frac {a-i a \tan (c+d x)}{a}\right )^{\frac {1}{2}+n} (a+i a \tan (c+d x))^{\frac {1}{2} (3+2 n)}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}-\frac {i x}{2}\right )^{-1+\frac {1}{2} (-3-2 n)}}{(a+i a x)^{5/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {i 2^{-\frac {3}{2}-n} \, _2F_1\left (-\frac {3}{2},\frac {1}{2} (5+2 n);-\frac {1}{2};\frac {1}{2} (1+i \tan (c+d x))\right ) (e \sec (c+d x))^{-3-2 n} (1-i \tan (c+d x))^{\frac {3}{2}+n} (a+i a \tan (c+d x))^n}{3 d}\\ \end {align*}
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Mathematica [A]
time = 13.59, size = 166, normalized size = 1.71 \begin {gather*} \frac {i 2^{-3-n} e^{-3 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} \left (1+e^{2 i (c+d x)}\right )^{-n} \, _2F_1\left (-\frac {3}{2},-3-n;-\frac {1}{2};-e^{2 i (c+d x)}\right ) \sec ^{3+n}(c+d x) (e \sec (c+d x))^{-3-2 n} (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.78, size = 0, normalized size = 0.00 \[\int \left (e \sec \left (d x +c \right )\right )^{-3-2 n} \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (e \sec {\left (c + d x \right )}\right )^{- 2 n - 3} \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n}{{\left (\frac {e}{\cos \left (c+d\,x\right )}\right )}^{2\,n+3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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