Optimal. Leaf size=264 \[ -\frac {d \left (c^2 \left (m^2-5 m+6\right )+2 i c d (3-m) m-d^2 \left (m^2-m+2\right )\right ) (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;-\frac {d (i \tan (e+f x)+1)}{i c-d}\right )}{2 f m \left (c^2+d^2\right )^3}-\frac {d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}-\frac {d (a+i a \tan (e+f x))^m}{2 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}-\frac {(a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m (d+i c)^3} \]
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Rubi [A] time = 0.88, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3561, 3598, 3600, 3481, 68, 3599} \[ -\frac {d \left (c^2 \left (m^2-5 m+6\right )+2 i c d (3-m) m-d^2 \left (m^2-m+2\right )\right ) (a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;-\frac {d (i \tan (e+f x)+1)}{i c-d}\right )}{2 f m \left (c^2+d^2\right )^3}-\frac {d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 f \left (c^2+d^2\right )^2 (c+d \tan (e+f x))}-\frac {d (a+i a \tan (e+f x))^m}{2 f \left (c^2+d^2\right ) (c+d \tan (e+f x))^2}-\frac {(a+i a \tan (e+f x))^m \, _2F_1\left (1,m;m+1;\frac {1}{2} (i \tan (e+f x)+1)\right )}{2 f m (d+i c)^3} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3481
Rule 3561
Rule 3598
Rule 3599
Rule 3600
Rubi steps
\begin {align*} \int \frac {(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}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}+\frac {\int \frac {(a+i a \tan (e+f x))^m (a (2 c+i d m)-a d (2-m) \tan (e+f x))}{(c+d \tan (e+f x))^2} \, dx}{2 a \left (c^2+d^2\right )}\\ &=-\frac {d (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int \frac {(a+i a \tan (e+f x))^m \left (a^2 \left (2 c^2+i c d (5-m) m-d^2 \left (2-m+m^2\right )\right )-a^2 d (1-m) (c (4-m)+i d m) \tan (e+f x)\right )}{c+d \tan (e+f x)} \, dx}{2 a^2 \left (c^2+d^2\right )^2}\\ &=-\frac {d (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}+\frac {\int (a+i a \tan (e+f x))^m \, dx}{(c-i d)^3}-\frac {\left (d \left (2 i c d (3-m) m+c^2 \left (6-5 m+m^2\right )-d^2 \left (2-m+m^2\right )\right )\right ) \int \frac {(a-i a \tan (e+f x)) (a+i a \tan (e+f x))^m}{c+d \tan (e+f x)} \, dx}{2 a (c+i d)^2 (i c+d)^3}\\ &=-\frac {d (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}-\frac {a \operatorname {Subst}\left (\int \frac {(a+x)^{-1+m}}{a-x} \, dx,x,i a \tan (e+f x)\right )}{(i c+d)^3 f}-\frac {\left (a d \left (2 i c d (3-m) m+c^2 \left (6-5 m+m^2\right )-d^2 \left (2-m+m^2\right )\right )\right ) \operatorname {Subst}\left (\int \frac {(a+i a x)^{-1+m}}{c+d x} \, dx,x,\tan (e+f x)\right )}{2 (c+i d)^2 (i c+d)^3 f}\\ &=-\frac {\, _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 (i c+d)^3 f m}-\frac {d \left (2 i c d (3-m) m+c^2 \left (6-5 m+m^2\right )-d^2 \left (2-m+m^2\right )\right ) \, _2F_1\left (1,m;1+m;-\frac {d (1+i \tan (e+f x))}{i c-d}\right ) (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right )^3 f m}-\frac {d (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right ) f (c+d \tan (e+f x))^2}-\frac {d (c (4-m)+i d m) (a+i a \tan (e+f x))^m}{2 \left (c^2+d^2\right )^2 f (c+d \tan (e+f x))}\\ \end {align*}
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Mathematica [F] time = 47.24, size = 0, normalized size = 0.00 \[ \int \frac {(a+i a \tan (e+f x))^m}{(c+d \tan (e+f x))^3} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (\frac {2 \, a e^{\left (2 i \, f x + 2 i \, e\right )}}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}\right )^{m} {\left (-i \, e^{\left (6 i \, f x + 6 i \, e\right )} - 3 i \, e^{\left (4 i \, f x + 4 i \, e\right )} - 3 i \, e^{\left (2 i \, f x + 2 i \, e\right )} - i\right )}}{-i \, c^{3} + 3 \, c^{2} d + 3 i \, c d^{2} - d^{3} + {\left (-i \, c^{3} - 3 \, c^{2} d + 3 i \, c d^{2} + d^{3}\right )} e^{\left (6 i \, f x + 6 i \, e\right )} + {\left (-3 i \, c^{3} - 3 \, c^{2} d - 3 i \, c d^{2} - 3 \, d^{3}\right )} e^{\left (4 i \, f x + 4 i \, e\right )} + {\left (-3 i \, c^{3} + 3 \, c^{2} d - 3 i \, c d^{2} + 3 \, d^{3}\right )} e^{\left (2 i \, f x + 2 i \, e\right )}}, x\right ) \]
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 \[ \int \frac {{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 4.98, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +i a \tan \left (f x +e \right )\right )^{m}}{\left (c +d \tan \left (f x +e \right )\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^m}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^3} \,d x \]
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 \[ \int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{m}}{\left (c + d \tan {\left (e + f x \right )}\right )^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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