Optimal. Leaf size=132 \[ \frac {(A-C) (b \tan (c+d x))^{n+3} \, _2F_1\left (1,\frac {n+3}{2};\frac {n+5}{2};-\tan ^2(c+d x)\right )}{b^3 d (n+3)}+\frac {B (b \tan (c+d x))^{n+4} \, _2F_1\left (1,\frac {n+4}{2};\frac {n+6}{2};-\tan ^2(c+d x)\right )}{b^4 d (n+4)}+\frac {C (b \tan (c+d x))^{n+3}}{b^3 d (n+3)} \]
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Rubi [A] time = 0.16, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {16, 3630, 3538, 3476, 364} \[ \frac {(A-C) (b \tan (c+d x))^{n+3} \, _2F_1\left (1,\frac {n+3}{2};\frac {n+5}{2};-\tan ^2(c+d x)\right )}{b^3 d (n+3)}+\frac {B (b \tan (c+d x))^{n+4} \, _2F_1\left (1,\frac {n+4}{2};\frac {n+6}{2};-\tan ^2(c+d x)\right )}{b^4 d (n+4)}+\frac {C (b \tan (c+d x))^{n+3}}{b^3 d (n+3)} \]
Antiderivative was successfully verified.
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Rule 16
Rule 364
Rule 3476
Rule 3538
Rule 3630
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\frac {\int (b \tan (c+d x))^{2+n} \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx}{b^2}\\ &=\frac {C (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {\int (b \tan (c+d x))^{2+n} (A-C+B \tan (c+d x)) \, dx}{b^2}\\ &=\frac {C (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {B \int (b \tan (c+d x))^{3+n} \, dx}{b^3}+\frac {(A-C) \int (b \tan (c+d x))^{2+n} \, dx}{b^2}\\ &=\frac {C (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {B \operatorname {Subst}\left (\int \frac {x^{3+n}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b^2 d}+\frac {(A-C) \operatorname {Subst}\left (\int \frac {x^{2+n}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {C (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {(A-C) \, _2F_1\left (1,\frac {3+n}{2};\frac {5+n}{2};-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{3+n}}{b^3 d (3+n)}+\frac {B \, _2F_1\left (1,\frac {4+n}{2};\frac {6+n}{2};-\tan ^2(c+d x)\right ) (b \tan (c+d x))^{4+n}}{b^4 d (4+n)}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 110, normalized size = 0.83 \[ \frac {\tan ^3(c+d x) (b \tan (c+d x))^n \left ((n+4) (A-C) \, _2F_1\left (1,\frac {n+3}{2};\frac {n+5}{2};-\tan ^2(c+d x)\right )+B (n+3) \tan (c+d x) \, _2F_1\left (1,\frac {n+4}{2};\frac {n+6}{2};-\tan ^2(c+d x)\right )+C (n+4)\right )}{d (n+3) (n+4)} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C \tan \left (d x + c\right )^{4} + B \tan \left (d x + c\right )^{3} + A \tan \left (d x + c\right )^{2}\right )} \left (b \tan \left (d x + c\right )\right )^{n}, x\right ) \]
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 \[ \int {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.11, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{2}\left (d x +c \right )\right ) \left (b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )+C \left (\tan ^{2}\left (d x +c \right )\right )\right )\, dx \]
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 \[ \int {\left (C \tan \left (d x + c\right )^{2} + B \tan \left (d x + c\right ) + A\right )} \left (b \tan \left (d x + c\right )\right )^{n} \tan \left (d x + c\right )^{2}\,{d x} \]
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 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^2\,{\left (b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n\,\left (C\,{\mathrm {tan}\left (c+d\,x\right )}^2+B\,\mathrm {tan}\left (c+d\,x\right )+A\right ) \,d x \]
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 \[ \int \left (b \tan {\left (c + d x \right )}\right )^{n} \left (A + B \tan {\left (c + d x \right )} + C \tan ^{2}{\left (c + d x \right )}\right ) \tan ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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