Optimal. Leaf size=560 \[ \frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (b^2 d (m+3) (m+4) (d (A-C)+B c)-2 (b c-a d) (3 a C d-b (B d (m+4)+3 c C))\right )}{b^3 f (m+2) (m+3) (m+4)}+\frac {(a+b \tan (e+f x))^{m+1} \left (b c (m+2) \left (b^2 d (m+3) (m+4) (d (A-C)+B c)-2 (b c-a d) (3 a C d-b (B d (m+4)+3 c C))\right )+d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (b^2 d (m+3) (m+4) (d (A-C)+B c)-2 (b c-a d) (3 a C d-b (B d (m+4)+3 c C))\right )\right )\right )}{b^4 f (m+1) (m+2) (m+3) (m+4)}+\frac {(c-i d)^3 (A-i B-C) (a+b \tan (e+f x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a)}-\frac {(c+i d)^3 (A+i B-C) (a+b \tan (e+f x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (-b+i a)}-\frac {(c+d \tan (e+f x))^2 (3 a C d-b (B d (m+4)+3 c C)) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+3) (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)} \]
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Rubi [A] time = 2.38, antiderivative size = 551, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3647, 3637, 3630, 3539, 3537, 68} \[ \frac {(a+b \tan (e+f x))^{m+1} \left (d \left (b^3 (m+2) (m+3) (m+4) \left (2 c d (A-C)+B \left (c^2-d^2\right )\right )-a \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )+b c (m+2) \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )\right )}{b^4 f (m+1) (m+2) (m+3) (m+4)}+\frac {d \tan (e+f x) (a+b \tan (e+f x))^{m+1} \left (2 (b c-a d) (-3 a C d+b B d (m+4)+3 b c C)+b^2 d (m+3) (m+4) (d (A-C)+B c)\right )}{b^3 f (m+2) (m+3) (m+4)}+\frac {(c-i d)^3 (A-i B-C) (a+b \tan (e+f x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 f (m+1) (b+i a)}-\frac {(c+i d)^3 (A+i B-C) (a+b \tan (e+f x))^{m+1} \, _2F_1\left (1,m+1;m+2;\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 f (m+1) (-b+i a)}+\frac {(c+d \tan (e+f x))^2 (-3 a C d+b B d (m+4)+3 b c C) (a+b \tan (e+f x))^{m+1}}{b^2 f (m+3) (m+4)}+\frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3537
Rule 3539
Rule 3630
Rule 3637
Rule 3647
Rubi steps
\begin {align*} \int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^3 \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right ) \, dx &=\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}+\frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x))^2 \left (A b c (4+m)-C (3 a d+b c (1+m))+b (B c+(A-C) d) (4+m) \tan (e+f x)+(3 b c C-3 a C d+b B d (4+m)) \tan ^2(e+f x)\right ) \, dx}{b (4+m)}\\ &=\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}+\frac {\int (a+b \tan (e+f x))^m (c+d \tan (e+f x)) \left (-(2 a d+b c (1+m)) (3 b c C-3 a C d+b B d (4+m))+b c (3+m) (A b c (4+m)-C (3 a d+b c (1+m)))+b^2 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (3+m) (4+m) \tan (e+f x)+\left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan ^2(e+f x)\right ) \, dx}{b^2 (3+m) (4+m)}\\ &=\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^3 f (2+m) (3+m) (4+m)}+\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}-\frac {\int (a+b \tan (e+f x))^m \left (a d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+b c (2+m) ((2 a d+b c (1+m)) (3 b c C-3 a C d+b B d (4+m))-b c (3+m) (A b c (4+m)-C (3 a d+b c (1+m))))-b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (2+m) \left (12+7 m+m^2\right ) \tan (e+f x)-\left (b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)-a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )\right ) \tan ^2(e+f x)\right ) \, dx}{b^3 (2+m) (3+m) (4+m)}\\ &=\frac {\left (b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)-a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^4 f (1+m) (2+m) (3+m) (4+m)}+\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^3 f (2+m) (3+m) (4+m)}+\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}-\frac {\int (a+b \tan (e+f x))^m \left (-b^3 \left (A c^3-c^3 C-3 B c^2 d-3 A c d^2+3 c C d^2+B d^3\right ) (2+m) (3+m) (4+m)-b^3 \left ((A-C) d \left (3 c^2-d^2\right )+B \left (c^3-3 c d^2\right )\right ) (2+m) \left (12+7 m+m^2\right ) \tan (e+f x)\right ) \, dx}{b^3 (2+m) (3+m) (4+m)}\\ &=\frac {\left (b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)-a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^4 f (1+m) (2+m) (3+m) (4+m)}+\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^3 f (2+m) (3+m) (4+m)}+\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}+\frac {1}{2} \left ((A-i B-C) (c-i d)^3\right ) \int (1+i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx+\frac {1}{2} \left ((A+i B-C) (c+i d)^3\right ) \int (1-i \tan (e+f x)) (a+b \tan (e+f x))^m \, dx\\ &=\frac {\left (b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)-a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^4 f (1+m) (2+m) (3+m) (4+m)}+\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^3 f (2+m) (3+m) (4+m)}+\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}+\frac {\left ((i A+B-i C) (c-i d)^3\right ) \operatorname {Subst}\left (\int \frac {(a-i b x)^m}{-1+x} \, dx,x,i \tan (e+f x)\right )}{2 f}-\frac {\left (i (A+i B-C) (c+i d)^3\right ) \operatorname {Subst}\left (\int \frac {(a+i b x)^m}{-1+x} \, dx,x,-i \tan (e+f x)\right )}{2 f}\\ &=\frac {\left (b c (2+m) \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )+d \left (b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (2+m) (3+m) (4+m)-a \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right )\right )\right ) (a+b \tan (e+f x))^{1+m}}{b^4 f (1+m) (2+m) (3+m) (4+m)}-\frac {(i A+B-i C) (c-i d)^3 \, _2F_1\left (1,1+m;2+m;\frac {a+b \tan (e+f x)}{a-i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (a-i b) f (1+m)}-\frac {(A+i B-C) (c+i d)^3 \, _2F_1\left (1,1+m;2+m;\frac {a+b \tan (e+f x)}{a+i b}\right ) (a+b \tan (e+f x))^{1+m}}{2 (i a-b) f (1+m)}+\frac {d \left (b^2 d (B c+(A-C) d) (3+m) (4+m)+2 (b c-a d) (3 b c C-3 a C d+b B d (4+m))\right ) \tan (e+f x) (a+b \tan (e+f x))^{1+m}}{b^3 f (2+m) (3+m) (4+m)}+\frac {(3 b c C-3 a C d+b B d (4+m)) (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^2}{b^2 f (3+m) (4+m)}+\frac {C (a+b \tan (e+f x))^{1+m} (c+d \tan (e+f x))^3}{b f (4+m)}\\ \end {align*}
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Mathematica [B] time = 6.41, size = 1390, normalized size = 2.48 \[ \frac {C (c+d \tan (e+f x))^3 (a+b \tan (e+f x))^{m+1}}{b f (m+4)}+\frac {\frac {(3 b c C-3 a d C+b B d (m+4)) (c+d \tan (e+f x))^2 (a+b \tan (e+f x))^{m+1}}{b f (m+3)}+\frac {\frac {d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right ) \tan (e+f x) (a+b \tan (e+f x))^{m+1}}{b f (m+2)}-\frac {\frac {\left (d \left (a \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+2) (m+3) (m+4)\right )-b c (m+2) \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )\right ) (a+b \tan (e+f x))^{m+1}}{b f (m+1)}+\frac {i \left (a d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+b c (m+2) \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )-b c (m+2) (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))-d \left (a \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+2) (m+3) (m+4)\right )-i b (m+2) \left (c \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+3) (m+4) b^2-d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+d (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))\right )\right ) \, _2F_1\left (1,m+1;m+2;\frac {-i a-i b \tan (e+f x)}{b-i a}\right ) (a+b \tan (e+f x))^{m+1}}{2 (a+i b) f (m+1)}-\frac {i \left (a d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+b c (m+2) \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )-b c (m+2) (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))-d \left (a \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )-b^3 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+2) (m+3) (m+4)\right )+i b (m+2) \left (c \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) (m+3) (m+4) b^2-d \left (d (B c+(A-C) d) (m+3) (m+4) b^2+2 (b c-a d) (3 b c C-3 a d C+b B d (m+4))\right )+d (b c (m+3) (A b c (m+4)-C (3 a d+b c (m+1)))-(2 a d+b c (m+1)) (3 b c C-3 a d C+b B d (m+4)))\right )\right ) \, _2F_1\left (1,m+1;m+2;-\frac {i a+i b \tan (e+f x)}{-i a-b}\right ) (a+b \tan (e+f x))^{m+1}}{2 (a-i b) f (m+1)}}{b (m+2)}}{b (m+3)}}{b (m+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.91, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C d^{3} \tan \left (f x + e\right )^{5} + {\left (3 \, C c d^{2} + B d^{3}\right )} \tan \left (f x + e\right )^{4} + A c^{3} + {\left (3 \, C c^{2} d + 3 \, B c d^{2} + A d^{3}\right )} \tan \left (f x + e\right )^{3} + {\left (C c^{3} + 3 \, B c^{2} d + 3 \, A c d^{2}\right )} \tan \left (f x + e\right )^{2} + {\left (B c^{3} + 3 \, A c^{2} d\right )} \tan \left (f x + e\right )\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{m}, x\right ) \]
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 \[ \int {\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{3} {\left (b \tan \left (f x + e\right ) + a\right )}^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 2.62, size = 0, normalized size = 0.00 \[ \int \left (a +b \tan \left (f x +e \right )\right )^{m} \left (c +d \tan \left (f x +e \right )\right )^{3} \left (A +B \tan \left (f x +e \right )+C \left (\tan ^{2}\left (f x +e \right )\right )\right )\, dx \]
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 \[ \int {\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^3\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right ) \,d x \]
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 \[ \int \left (a + b \tan {\left (e + f x \right )}\right )^{m} \left (c + d \tan {\left (e + f x \right )}\right )^{3} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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