Optimal. Leaf size=60 \[ \frac {i a^3 \, _2F_1\left (4,-3+n;-2+n;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-3+n}}{16 d (3-n)} \]
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Rubi [A]
time = 0.04, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 70}
\begin {gather*} \frac {i a^3 (a+i a \tan (c+d x))^{n-3} \, _2F_1\left (4,n-3;n-2;\frac {1}{2} (i \tan (c+d x)+1)\right )}{16 d (3-n)} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3568
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x))^n \, dx &=-\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {(a+x)^{-4+n}}{(a-x)^4} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=\frac {i a^3 \, _2F_1\left (4,-3+n;-2+n;\frac {1}{2} (1+i \tan (c+d x))\right ) (a+i a \tan (c+d x))^{-3+n}}{16 d (3-n)}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(143\) vs. \(2(60)=120\).
time = 10.84, size = 143, normalized size = 2.38 \begin {gather*} -\frac {i 2^{-7+n} e^{-6 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 )^7 \, _2F_1\left (1,4;-2+n;-e^{2 i (c+d x)}\right ) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (-3+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.06, size = 0, normalized size = 0.00 \[\int \left (\cos ^{6}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.02 \begin {gather*} \int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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