Optimal. Leaf size=157 \[ \frac {a^3 (-d (2 n+5)+i c) (c+d \tan (e+f x))^{n+1}}{d^2 f (n+1) (n+2)}+\frac {4 a^3 (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{n+1}}{d f (n+2)} \]
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Rubi [A] time = 0.34, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3556, 3592, 3537, 68} \[ \frac {a^3 (-d (2 n+5)+i c) (c+d \tan (e+f x))^{n+1}}{d^2 f (n+1) (n+2)}+\frac {4 a^3 (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c-i d}\right )}{f (n+1) (d+i c)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{n+1}}{d f (n+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3537
Rule 3556
Rule 3592
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx &=-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}+\frac {a \int (a+i a \tan (e+f x)) (c+d \tan (e+f x))^n (a (i c+d (3+2 n))+a (c+i d (5+2 n)) \tan (e+f x)) \, dx}{d (2+n)}\\ &=\frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}+\frac {a \int (c+d \tan (e+f x))^n \left (4 a^2 d (2+n)+4 i a^2 d (2+n) \tan (e+f x)\right ) \, dx}{d (2+n)}\\ &=\frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}+\frac {\left (16 i a^5 d (2+n)\right ) \operatorname {Subst}\left (\int \frac {\left (c-\frac {i x}{4 a^2 (2+n)}\right )^n}{-16 a^4 d^2 (2+n)^2+4 a^2 d (2+n) x} \, dx,x,4 i a^2 d (2+n) \tan (e+f x)\right )}{f}\\ &=\frac {a^3 (i c-d (5+2 n)) (c+d \tan (e+f x))^{1+n}}{d^2 f (1+n) (2+n)}+\frac {4 a^3 \, _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)}-\frac {\left (a^3+i a^3 \tan (e+f x)\right ) (c+d \tan (e+f x))^{1+n}}{d f (2+n)}\\ \end {align*}
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Mathematica [F] time = 21.46, size = 0, normalized size = 0.00 \[ \int (a+i a \tan (e+f x))^3 (c+d \tan (e+f x))^n \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {8 \, a^{3} \left (\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} e^{\left (6 i \, f x + 6 i \, e\right )}}{e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.48, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{3} \left (c +d \tan \left (f x +e \right )\right )^{n}\, dx \]
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 \[ \int {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (d \tan \left (f x + e\right ) + c\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n \,d x \]
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 \[ - i a^{3} \left (\int i \left (c + d \tan {\left (e + f x \right )}\right )^{n}\, dx + \int \left (- 3 \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan {\left (e + f x \right )}\right )\, dx + \int \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan ^{3}{\left (e + f x \right )}\, dx + \int \left (- 3 i \left (c + d \tan {\left (e + f x \right )}\right )^{n} \tan ^{2}{\left (e + f x \right )}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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