Optimal. Leaf size=381 \[ \frac {\left (3 i c^2-3 c d (3-n)-i d^2 \left (2 n^2-9 n+10\right )\right ) (c+d \tan (e+f x))^{n+1}}{24 f (c+i d)^3 \left (a^3+i a^3 \tan (e+f x)\right )}+\frac {\left (3 i c^3-c^2 d (9-6 n)-3 i c d^2 \left (2 n^2-6 n+3\right )+d^3 \left (-4 n^3+18 n^2-20 n+3\right )\right ) (c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c+i d}\right )}{48 a^3 f (n+1) (c+i d)^4}+\frac {(c+d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {c+d \tan (e+f x)}{c-i d}\right )}{16 a^3 f (n+1) (d+i c)}+\frac {(-d (7-2 n)+3 i c) (c+d \tan (e+f x))^{n+1}}{24 a f (c+i d)^2 (a+i a \tan (e+f x))^2}-\frac {(c+d \tan (e+f x))^{n+1}}{6 f (-d+i c) (a+i a \tan (e+f x))^3} \]
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Rubi [A] time = 1.06, antiderivative size = 381, normalized size of antiderivative = 1.00, 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 68
Rule 3537
Rule 3539
Rule 3559
Rule 3596
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*}
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Mathematica [F] time = 35.47, size = 0, normalized size = 0.00 \[ \int \frac {(c+d \tan (e+f x))^n}{(a+i a \tan (e+f x))^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {{\left (c - i \, d\right )} e^{\left (2 i \, f x + 2 i \, e\right )} + c + i \, d}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{n} {\left (e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1\right )} e^{\left (-6 i \, f x - 6 i \, e\right )}}{8 \, a^{3}}, x\right ) \]
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 \[ \int \frac {{\left (d \tan \left (f x + e\right ) + c\right )}^{n}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.27, size = 0, normalized size = 0.00 \[ \int \frac {\left (c +d \tan \left (f x +e \right )\right )^{n}}{\left (a +i a \tan \left (f x +e \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^3} \,d x \]
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 \[ \frac {i \int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{n}}{\tan ^{3}{\left (e + f x \right )} - 3 i \tan ^{2}{\left (e + f x \right )} - 3 \tan {\left (e + f x \right )} + i}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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