Optimal. Leaf size=219 \[ -\frac {(a B-A b (n+2)) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac {(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}+\frac {(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (-b+i a)}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)} \]
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Rubi [A] time = 0.35, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3607, 3630, 3539, 3537, 68} \[ -\frac {(a B-A b (n+2)) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac {(B+i A) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}+\frac {(A+i B) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (-b+i a)}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)} \]
Antiderivative was successfully verified.
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Rule 68
Rule 3537
Rule 3539
Rule 3607
Rule 3630
Rubi steps
\begin {align*} \int \tan ^2(c+d x) (a+b \tan (c+d x))^n (A+B \tan (c+d x)) \, dx &=\frac {B \tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}+\frac {\int (a+b \tan (c+d x))^n \left (-a B-b B (2+n) \tan (c+d x)-(a B-A b (2+n)) \tan ^2(c+d x)\right ) \, dx}{b (2+n)}\\ &=-\frac {(a B-A b (2+n)) (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}+\frac {\int (a+b \tan (c+d x))^n (-A b (2+n)-b B (2+n) \tan (c+d x)) \, dx}{b (2+n)}\\ &=-\frac {(a B-A b (2+n)) (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}+\frac {1}{2} (-A-i B) \int (1-i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx+\frac {1}{2} (-A+i B) \int (1+i \tan (c+d x)) (a+b \tan (c+d x))^n \, dx\\ &=-\frac {(a B-A b (2+n)) (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}+\frac {(i A-B) \operatorname {Subst}\left (\int \frac {(a+i b x)^n}{-1+x} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac {(i A+B) \operatorname {Subst}\left (\int \frac {(a-i b x)^n}{-1+x} \, dx,x,i \tan (c+d x)\right )}{2 d}\\ &=-\frac {(a B-A b (2+n)) (a+b \tan (c+d x))^{1+n}}{b^2 d (1+n) (2+n)}+\frac {(i A+B) \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a-i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (a-i b) d (1+n)}+\frac {(A+i B) \, _2F_1\left (1,1+n;2+n;\frac {a+b \tan (c+d x)}{a+i b}\right ) (a+b \tan (c+d x))^{1+n}}{2 (i a-b) d (1+n)}+\frac {B \tan (c+d x) (a+b \tan (c+d x))^{1+n}}{b d (2+n)}\\ \end {align*}
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Mathematica [A] time = 1.48, size = 169, normalized size = 0.77 \[ \frac {(a+b \tan (c+d x))^{n+1} \left (\frac {b (n+2) (B+i A) \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a-i b}\right )}{(n+1) (a-i b)}+\frac {b (n+2) (B-i A) \, _2F_1\left (1,n+1;n+2;\frac {a+b \tan (c+d x)}{a+i b}\right )}{(n+1) (a+i b)}+\frac {-2 a B+2 A b n+4 A b}{b n+b}+2 B \tan (c+d x)\right )}{2 b d (n+2)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \tan \left (d x + c\right )^{3} + A \tan \left (d x + c\right )^{2}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n}, x\right ) \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.06, size = 0, normalized size = 0.00 \[ \int \left (\tan ^{2}\left (d x +c \right )\right ) \left (a +b \tan \left (d x +c \right )\right )^{n} \left (A +B \tan \left (d x +c \right )\right )\, dx \]
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 \[ \int {\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{n} \tan \left (d x + c\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {tan}\left (c+d\,x\right )}^2\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^n \,d x \]
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 \[ \int \left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{n} \tan ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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