Optimal. Leaf size=403 \[ \frac {b^2 \left (a^2 B \left (m^2+9 m+26\right )+2 a A b (m+4)^2-b^2 B \left (m^2+7 m+12\right )\right ) \tan ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}-\frac {b \left (-2 a^3 B \left (m^2+8 m+19\right )-a^2 A b \left (5 m^2+37 m+68\right )+4 a b^2 B \left (m^2+7 m+12\right )+A b^3 \left (m^2+7 m+12\right )\right ) \tan ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}+\frac {\left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )}{d (m+1)}+\frac {\left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )}{d (m+2)}+\frac {b (a B (m+7)+A b (m+4)) \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^2}{d (m+3) (m+4)}+\frac {b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^3}{d (m+4)} \]
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Rubi [A] time = 1.33, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3607, 3647, 3637, 3630, 3538, 3476, 364} \[ \frac {\left (-6 a^2 A b^2+a^4 A-4 a^3 b B+4 a b^3 B+A b^4\right ) \tan ^{m+1}(c+d x) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )}{d (m+1)}+\frac {\left (4 a^3 A b-6 a^2 b^2 B+a^4 B-4 a A b^3+b^4 B\right ) \tan ^{m+2}(c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )}{d (m+2)}-\frac {b \left (-a^2 A b \left (5 m^2+37 m+68\right )-2 a^3 B \left (m^2+8 m+19\right )+4 a b^2 B \left (m^2+7 m+12\right )+A b^3 \left (m^2+7 m+12\right )\right ) \tan ^{m+1}(c+d x)}{d (m+1) (m+3) (m+4)}+\frac {b^2 \left (a^2 B \left (m^2+9 m+26\right )+2 a A b (m+4)^2-b^2 B \left (m^2+7 m+12\right )\right ) \tan ^{m+2}(c+d x)}{d (m+2) (m+3) (m+4)}+\frac {b (a B (m+7)+A b (m+4)) \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^2}{d (m+3) (m+4)}+\frac {b B \tan ^{m+1}(c+d x) (a+b \tan (c+d x))^3}{d (m+4)} \]
Antiderivative was successfully verified.
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Rule 364
Rule 3476
Rule 3538
Rule 3607
Rule 3630
Rule 3637
Rule 3647
Rubi steps
\begin {align*} \int \tan ^m(c+d x) (a+b \tan (c+d x))^4 (A+B \tan (c+d x)) \, dx &=\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\frac {\int \tan ^m(c+d x) (a+b \tan (c+d x))^2 \left (-a (b B (1+m)-a A (4+m))+\left (2 a A b+a^2 B-b^2 B\right ) (4+m) \tan (c+d x)+b (A b (4+m)+a B (7+m)) \tan ^2(c+d x)\right ) \, dx}{4+m}\\ &=\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\frac {\int \tan ^m(c+d x) (a+b \tan (c+d x)) \left (-a (a (3+m) (b B (1+m)-a A (4+m))+b (1+m) (A b (4+m)+a B (7+m)))+\left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) (3+m) (4+m) \tan (c+d x)+b \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^2(c+d x)\right ) \, dx}{12+7 m+m^2}\\ &=\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}-\frac {\int \tan ^m(c+d x) \left (a^2 (2+m) (a (3+m) (b B (1+m)-a A (4+m))+b (1+m) (A b (4+m)+a B (7+m)))-\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) (2+m) (3+m) (4+m) \tan (c+d x)+b (2+m) \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^2(c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3}\\ &=-\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}-\frac {\int \tan ^m(c+d x) \left (-\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) (2+m) (3+m) (4+m)-\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) (2+m) (3+m) (4+m) \tan (c+d x)\right ) \, dx}{24+26 m+9 m^2+m^3}\\ &=-\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \int \tan ^m(c+d x) \, dx+\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \int \tan ^{1+m}(c+d x) \, dx\\ &=-\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \operatorname {Subst}\left (\int \frac {x^m}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \operatorname {Subst}\left (\int \frac {x^{1+m}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac {b \left (A b^3 \left (12+7 m+m^2\right )+4 a b^2 B \left (12+7 m+m^2\right )-2 a^3 B \left (19+8 m+m^2\right )-a^2 A b \left (68+37 m+5 m^2\right )\right ) \tan ^{1+m}(c+d x)}{d (1+m) \left (12+7 m+m^2\right )}+\frac {\left (a^4 A-6 a^2 A b^2+A b^4-4 a^3 b B+4 a b^3 B\right ) \, _2F_1\left (1,\frac {1+m}{2};\frac {3+m}{2};-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x)}{d (1+m)}+\frac {b^2 \left (2 a A b (4+m)^2-b^2 B \left (12+7 m+m^2\right )+a^2 B \left (26+9 m+m^2\right )\right ) \tan ^{2+m}(c+d x)}{d (2+m) \left (12+7 m+m^2\right )}+\frac {\left (4 a^3 A b-4 a A b^3+a^4 B-6 a^2 b^2 B+b^4 B\right ) \, _2F_1\left (1,\frac {2+m}{2};\frac {4+m}{2};-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x)}{d (2+m)}+\frac {b (A b (4+m)+a B (7+m)) \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^2}{d (3+m) (4+m)}+\frac {b B \tan ^{1+m}(c+d x) (a+b \tan (c+d x))^3}{d (4+m)}\\ \end {align*}
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Mathematica [A] time = 5.71, size = 355, normalized size = 0.88 \[ \frac {\tan ^{m+1}(c+d x) \left (b^2 (m+1) \tan (c+d x) \left (a^2 B \left (m^2+9 m+26\right )+2 a A b (m+4)^2-b^2 B \left (m^2+7 m+12\right )\right )-b (m+2) \left (-2 a^3 B \left (m^2+8 m+19\right )-a^2 A b \left (5 m^2+37 m+68\right )+4 a b^2 B \left (m^2+7 m+12\right )+A b^3 \left (m^2+7 m+12\right )\right )+(m+2) (m+3) (m+4) \left (a^4 A-4 a^3 b B-6 a^2 A b^2+4 a b^3 B+A b^4\right ) \, _2F_1\left (1,\frac {m+1}{2};\frac {m+3}{2};-\tan ^2(c+d x)\right )+(m+1) (m+3) (m+4) \left (a^4 B+4 a^3 A b-6 a^2 b^2 B-4 a A b^3+b^4 B\right ) \tan (c+d x) \, _2F_1\left (1,\frac {m+2}{2};\frac {m+4}{2};-\tan ^2(c+d x)\right )+b (m+1) (m+2) (a B (m+7)+A b (m+4)) (a+b \tan (c+d x))^2+b B (m+1) (m+2) (m+3) (a+b \tan (c+d x))^3\right )}{d (m+1) (m+2) (m+3) (m+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B b^{4} \tan \left (d x + c\right )^{5} + A a^{4} + {\left (4 \, B a b^{3} + A b^{4}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} \tan \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \tan \left (d x + c\right )\right )} \tan \left (d x + c\right )^{m}, x\right ) \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{4} \tan \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.69, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{m}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{4} \left (A +B \tan \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{4} \tan \left (d x + c\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^m\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^4 \,d x \]
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 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{4} \tan ^{m}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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