Optimal. Leaf size=381 \[ \frac{\left (-c^2 d (9-6 n)+3 i c^3-3 i c d^2 \left (2 n^2-6 n+3\right )+d^3 \left (-4 n^3+18 n^2-20 n+3\right )\right ) (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \tan (e+f x)}{c+i d}\right )}{48 a^3 f (n+1) (c+i d)^4}+\frac{\left (3 i c^2-3 c d (3-n)-i d^2 \left (2 n^2-9 n+10\right )\right ) (c+d \tan (e+f x))^{n+1}}{24 f (c+i d)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{(c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right )}{16 a^3 f (n+1) (d+i c)}+\frac{(-d (7-2 n)+3 i c) (c+d \tan (e+f x))^{n+1}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{(c+d \tan (e+f x))^{n+1}}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]
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Rubi [A] time = 1.06033, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {3559, 3596, 3539, 3537, 68} \[ \frac{\left (-c^2 d (9-6 n)+3 i c^3-3 i c d^2 \left (2 n^2-6 n+3\right )+d^3 \left (-4 n^3+18 n^2-20 n+3\right )\right ) (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \tan (e+f x)}{c+i d}\right )}{48 a^3 f (n+1) (c+i d)^4}+\frac{\left (3 i c^2-3 c d (3-n)-i d^2 \left (2 n^2-9 n+10\right )\right ) (c+d \tan (e+f x))^{n+1}}{24 f (c+i d)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{(c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{c+d \tan (e+f x)}{c-i d}\right )}{16 a^3 f (n+1) (d+i c)}+\frac{(-d (7-2 n)+3 i c) (c+d \tan (e+f x))^{n+1}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac{(c+d \tan (e+f x))^{n+1}}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]
Antiderivative was successfully verified.
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Rule 3559
Rule 3596
Rule 3539
Rule 3537
Rule 68
Rubi steps
\begin{align*} \int \frac{(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx &=-\frac{(c+d \tan (e+f x))^{1+n}}{6 (i c-d) f (a+i a \tan (e+f x))^3}-\frac{\int \frac{(c+d \tan (e+f x))^n (-a (3 i c-d (5-n))-i a d (2-n) \tan (e+f x))}{(a+i a \tan (e+f x))^2} \, dx}{6 a^2 (i c-d)}\\ &=-\frac{(c+d \tan (e+f x))^{1+n}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-d (7-2 n)) (c+d \tan (e+f x))^{1+n}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}-\frac{\int \frac{(c+d \tan (e+f x))^n \left (-a^2 \left (6 c^2+3 i c d (5-n)-d^2 \left (13-9 n+2 n^2\right )\right )-a^2 d (3 c+i d (7-2 n)) (1-n) \tan (e+f x)\right )}{a+i a \tan (e+f x)} \, dx}{24 a^4 (c+i d)^2}\\ &=-\frac{(c+d \tan (e+f x))^{1+n}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-d (7-2 n)) (c+d \tan (e+f x))^{1+n}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (3 i c^2-3 c d (3-n)-i d^2 \left (10-9 n+2 n^2\right )\right ) (c+d \tan (e+f x))^{1+n}}{24 (c+i d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}-\frac{\int (c+d \tan (e+f x))^n \left (2 a^3 \left (3 i c^3-3 c^2 d (3-n)-3 i c d^2 \left (3-3 n+n^2\right )+d^3 \left (3-10 n+9 n^2-2 n^3\right )\right )-2 a^3 d n \left (3 i c^2-3 c d (3-n)-i d^2 \left (10-9 n+2 n^2\right )\right ) \tan (e+f x)\right ) \, dx}{48 a^6 (i c-d)^3}\\ &=-\frac{(c+d \tan (e+f x))^{1+n}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-d (7-2 n)) (c+d \tan (e+f x))^{1+n}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (3 i c^2-3 c d (3-n)-i d^2 \left (10-9 n+2 n^2\right )\right ) (c+d \tan (e+f x))^{1+n}}{24 (c+i d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{\int (1+i \tan (e+f x)) (c+d \tan (e+f x))^n \, dx}{16 a^3}-\frac{\left (3 i c^3-3 c^2 d (3-2 n)-3 i c d^2 \left (3-6 n+2 n^2\right )+d^3 \left (3-20 n+18 n^2-4 n^3\right )\right ) \int (1-i \tan (e+f x)) (c+d \tan (e+f x))^n \, dx}{48 a^3 (i c-d)^3}\\ &=-\frac{(c+d \tan (e+f x))^{1+n}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-d (7-2 n)) (c+d \tan (e+f x))^{1+n}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (3 i c^2-3 c d (3-n)-i d^2 \left (10-9 n+2 n^2\right )\right ) (c+d \tan (e+f x))^{1+n}}{24 (c+i d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}+\frac{i \operatorname{Subst}\left (\int \frac{(c-i d x)^n}{-1+x} \, dx,x,i \tan (e+f x)\right )}{16 a^3 f}+\frac{\left (i \left (3 i c^3-3 c^2 d (3-2 n)-3 i c d^2 \left (3-6 n+2 n^2\right )+d^3 \left (3-20 n+18 n^2-4 n^3\right )\right )\right ) \operatorname{Subst}\left (\int \frac{(c+i d x)^n}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{48 a^3 (i c-d)^3 f}\\ &=\frac{\, _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}}{16 a^3 (i c+d) f (1+n)}+\frac{\left (3 i c^3-c^2 d (9-6 n)-3 i c d^2 \left (3-6 n+2 n^2\right )+d^3 \left (3-20 n+18 n^2-4 n^3\right )\right ) \, _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}}{48 a^3 (c+i d)^4 f (1+n)}-\frac{(c+d \tan (e+f x))^{1+n}}{6 (i c-d) f (a+i a \tan (e+f x))^3}+\frac{(3 i c-d (7-2 n)) (c+d \tan (e+f x))^{1+n}}{24 a (c+i d)^2 f (a+i a \tan (e+f x))^2}+\frac{\left (3 i c^2-3 c d (3-n)-i d^2 \left (10-9 n+2 n^2\right )\right ) (c+d \tan (e+f x))^{1+n}}{24 (c+i d)^3 f \left (a^3+i a^3 \tan (e+f x)\right )}\\ \end{align*}
Mathematica [F] time = 26.5132, size = 0, normalized size = 0. \[ \int \frac{(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.265, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( c+d\tan \left ( fx+e \right ) \right ) ^{n}}{ \left ( a+ia\tan \left ( fx+e \right ) \right ) ^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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}{\left (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\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{8 \, a^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d \tan \left (f x + e\right ) + c\right )}^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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