Optimal. Leaf size=85 \[ -\frac {i 2^{-\frac {1}{2}+n} \cos (c+d x) \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-n;\frac {1}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^n}{d} \]
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Rubi [A]
time = 0.13, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3586, 3604, 72,
71} \begin {gather*} -\frac {i 2^{n-\frac {1}{2}} \cos (c+d x) (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^n \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-n;\frac {1}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 71
Rule 72
Rule 3586
Rule 3604
Rubi steps
\begin {align*} \int \cos (c+d x) (a+i a \tan (c+d x))^n \, dx &=\left (\cos (c+d x) \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right ) \int \frac {(a+i a \tan (c+d x))^{-\frac {1}{2}+n}}{\sqrt {a-i a \tan (c+d x)}} \, dx\\ &=\frac {\left (a^2 \cos (c+d x) \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}\right ) \text {Subst}\left (\int \frac {(a+i a x)^{-\frac {3}{2}+n}}{(a-i a x)^{3/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{-\frac {3}{2}+n} a \cos (c+d x) \sqrt {a-i a \tan (c+d x)} (a+i a \tan (c+d x))^n \left (\frac {a+i a \tan (c+d x)}{a}\right )^{\frac {1}{2}-n}\right ) \text {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-\frac {3}{2}+n}}{(a-i a x)^{3/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {i 2^{-\frac {1}{2}+n} \cos (c+d x) \, _2F_1\left (-\frac {1}{2},\frac {3}{2}-n;\frac {1}{2};\frac {1}{2} (1-i \tan (c+d x))\right ) (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^n}{d}\\ \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. \(195\) vs. \(2(85)=170\).
time = 13.56, size = 195, normalized size = 2.29 \begin {gather*} -\frac {i 2^{-1+n} \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^{-1+n} \left (1+e^{2 i (c+d x)}\right )^{-1+n} \left ((1+2 n) \, _2F_1\left (-\frac {1}{2}+n,n;\frac {1}{2}+n;-e^{2 i (c+d x)}\right )+e^{2 i (c+d x)} (-1+2 n) \, _2F_1\left (n,\frac {1}{2}+n;\frac {3}{2}+n;-e^{2 i (c+d x)}\right )\right ) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d \left (-1+4 n^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.33, size = 0, normalized size = 0.00 \[\int \cos \left (d x +c \right ) \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{n} \cos {\left (c + d x \right )}\, 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 \cos \left (c+d\,x\right )\,{\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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