Optimal. Leaf size=192 \[ -\frac {2 d \left (-\left (c^2 (m+3)\right )+i c d m+d^2\right ) (a+i a \tan (e+f x))^m}{f m (m+2)}-\frac {d^2 (d m+i c (m+4)) (a+i a \tan (e+f x))^{m+1}}{a f (m+1) (m+2)}+\frac {(d+i c)^3 (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (m+2)} \]
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Rubi [A] time = 0.46, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3560, 3592, 3527, 3481, 68} \[ -\frac {2 d \left (c^2 (-(m+3))+i c d m+d^2\right ) (a+i a \tan (e+f x))^m}{f m (m+2)}-\frac {d^2 (d m+i c (m+4)) (a+i a \tan (e+f x))^{m+1}}{a f (m+1) (m+2)}+\frac {(d+i c)^3 (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (m+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3481
Rule 3527
Rule 3560
Rule 3592
Rubi steps
\begin {align*} \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx &=\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}-\frac {\int (a+i a \tan (e+f x))^m (c+d \tan (e+f x)) \left (-a \left (c^2 (2+m)-d (2 d+i c m)\right )+a d (i d m-c (4+m)) \tan (e+f x)\right ) \, dx}{a (2+m)}\\ &=-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}-\frac {\int (a+i a \tan (e+f x))^m \left (a \left (i c^2 d m-i d^3 m-c^3 (2+m)+c d^2 (6+m)\right )+2 a d \left (d^2+i c d m-c^2 (3+m)\right ) \tan (e+f x)\right ) \, dx}{a (2+m)}\\ &=-\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}+(c-i d)^3 \int (a+i a \tan (e+f x))^m \, dx\\ &=-\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}+\frac {\left (a (i c+d)^3\right ) \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{f}\\ &=-\frac {2 d \left (d^2+i c d m-c^2 (3+m)\right ) (a+i a \tan (e+f x))^m}{f m (2+m)}+\frac {(i c+d)^3 \, _2F_1\left (1,m;1+m;\frac {1}{2} (1+i \tan (e+f x))\right ) (a+i a \tan (e+f x))^m}{2 f m}-\frac {d^2 (d m+i c (4+m)) (a+i a \tan (e+f x))^{1+m}}{a f (1+m) (2+m)}+\frac {d (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^2}{f (2+m)}\\ \end {align*}
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Mathematica [F] time = 52.47, size = 0, normalized size = 0.00 \[ \int (a+i a \tan (e+f x))^m (c+d \tan (e+f x))^3 \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{3} + 3 i \, c^{2} d - 3 \, c d^{2} - i \, d^{3} + {\left (c^{3} - 3 i \, c^{2} d - 3 \, c d^{2} + i \, d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (3 \, c^{3} - 3 i \, c^{2} d + 3 \, c d^{2} - 3 i \, d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (3 \, c^{3} + 3 i \, c^{2} d + 3 \, c d^{2} + 3 i \, d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}\right )} \left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m}}{e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, e^{\left (2 i \, f x + 2 i \, e\right )} + 1}, x\right ) \]
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 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{3} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.48, size = 0, normalized size = 0.00 \[ \int \left (a +i a \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d \tan \left (f x + e\right ) + c\right )}^{3} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
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 \[ \int {\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^3 \,d x \]
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 \[ \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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