Optimal. Leaf size=94 \[ -\frac {i 2^{n-\frac {3}{2}} \cos ^3(c+d x) (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^{n+1} \, _2F_1\left (-\frac {3}{2},\frac {5}{2}-n;-\frac {1}{2};\frac {1}{2} (1-i \tan (c+d x))\right )}{3 a d} \]
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Rubi [A] time = 0.19, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3505, 3523, 70, 69} \[ -\frac {i 2^{n-\frac {3}{2}} \cos ^3(c+d x) (1+i \tan (c+d x))^{\frac {1}{2}-n} (a+i a \tan (c+d x))^{n+1} \text {Hypergeometric2F1}\left (-\frac {3}{2},\frac {5}{2}-n,-\frac {1}{2},\frac {1}{2} (1-i \tan (c+d x))\right )}{3 a d} \]
Antiderivative was successfully verified.
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Rule 69
Rule 70
Rule 3505
Rule 3523
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+i a \tan (c+d x))^n \, dx &=\left (\cos ^3(c+d x) (a-i a \tan (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}\right ) \int \frac {(a+i a \tan (c+d x))^{-\frac {3}{2}+n}}{(a-i a \tan (c+d x))^{3/2}} \, dx\\ &=\frac {\left (a^2 \cos ^3(c+d x) (a-i a \tan (c+d x))^{3/2} (a+i a \tan (c+d x))^{3/2}\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-\frac {5}{2}+n}}{(a-i a x)^{5/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\left (2^{-\frac {5}{2}+n} \cos ^3(c+d x) (a-i a \tan (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}{2}-n}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1}{2}+\frac {i x}{2}\right )^{-\frac {5}{2}+n}}{(a-i a x)^{5/2}} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {i 2^{-\frac {3}{2}+n} \cos ^3(c+d x) \, _2F_1\left (-\frac {3}{2},\frac {5}{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))^{1+n}}{3 a d}\\ \end {align*}
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Mathematica [A] time = 13.57, size = 149, normalized size = 1.59 \[ -\frac {i 2^{n-3} e^{-3 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^4 \left (e^{i d x}\right )^n \left (\frac {e^{i (c+d x)}}{1+e^{2 i (c+d x)}}\right )^n \, _2F_1\left (1,\frac {5}{2};n-\frac {1}{2};-e^{2 i (c+d x)}\right ) \sec ^{-n}(c+d x) (\cos (d x)+i \sin (d x))^{-n} (a+i a \tan (c+d x))^n}{d (2 n-3)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{8} \, \left (\frac {2 \, a e^{\left (2 i \, d x + 2 i \, c\right )}}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}\right )^{n} {\left (e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 7.96, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{3}\left (d x +c \right )\right ) \left (a +i a \tan \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{n} \cos \left (d x + c\right )^{3}\,{d x} \]
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 \[ \int {\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^n \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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