Optimal. Leaf size=61 \[ \frac {a (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)} \]
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Rubi [A] time = 0.07, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {3537, 68} \[ \frac {a (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3537
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n \, dx &=\frac {\left (i a^2\right ) \operatorname {Subst}\left (\int \frac {\left (c-\frac {i d x}{a}\right )^n}{-a^2+a x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=\frac {a \, _2F_1\left (1,1+n;2+n;\frac {c+d \tan (e+f x)}{c-i d}\right ) (c+d \tan (e+f x))^{1+n}}{(i c+d) f (1+n)}\\ \end {align*}
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Mathematica [F] time = 1.96, size = 0, normalized size = 0.00 \[ \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {2 \, a \left (\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.31, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right ) \left (c +d \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ i a \left (\int \left (- i \left (c + d \tan {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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