Optimal. Leaf size=75 \[ -\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {2 a^2 \, _2F_1(1,1+n;2+n;i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)} \]
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Rubi [A]
time = 0.08, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3624, 3618, 12,
66} \begin {gather*} -\frac {a^2 (d \tan (e+f x))^{n+1}}{d f (n+1)}+\frac {2 a^2 (d \tan (e+f x))^{n+1} \, _2F_1(1,n+1;n+2;i \tan (e+f x))}{d f (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 66
Rule 3618
Rule 3624
Rubi steps
\begin {align*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\int (d \tan (e+f x))^n \left (2 a^2+2 i a^2 \tan (e+f x)\right ) \, dx\\ &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {\left (4 i a^4\right ) \text {Subst}\left (\int \frac {2^{-n} \left (-\frac {i d x}{a^2}\right )^n}{-4 a^4+2 a^2 x} \, dx,x,2 i a^2 \tan (e+f x)\right )}{f}\\ &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {\left (i 2^{2-n} a^4\right ) \text {Subst}\left (\int \frac {\left (-\frac {i d x}{a^2}\right )^n}{-4 a^4+2 a^2 x} \, dx,x,2 i a^2 \tan (e+f x)\right )}{f}\\ &=-\frac {a^2 (d \tan (e+f x))^{1+n}}{d f (1+n)}+\frac {2 a^2 \, _2F_1(1,1+n;2+n;i \tan (e+f x)) (d \tan (e+f x))^{1+n}}{d f (1+n)}\\ \end {align*}
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Mathematica [F]
time = 2.60, size = 0, normalized size = 0.00 \begin {gather*} \int (d \tan (e+f x))^n (a+i a \tan (e+f x))^2 \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [F]
time = 0.50, size = 0, normalized size = 0.00 \[\int \left (d \tan \left (f x +e \right )\right )^{n} \left (a +i a \tan \left (f x +e \right )\right )^{2}\, dx\]
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*} - a^{2} \left (\int \left (- \left (d \tan {\left (e + f x \right )}\right )^{n}\right )\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (- 2 i \left (d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx\right ) \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 {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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