Optimal. Leaf size=219 \[ \frac{(B+i A) (a+b \tan (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a-i b}\right )}{2 d (n+1) (a-i b)}+\frac{(A+i B) (a+b \tan (c+d x))^{n+1} \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a+i b}\right )}{2 d (n+1) (-b+i a)}-\frac{(a B-A b (n+2)) (a+b \tan (c+d x))^{n+1}}{b^2 d (n+1) (n+2)}+\frac{B \tan (c+d x) (a+b \tan (c+d x))^{n+1}}{b d (n+2)} \]
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Rubi [A] time = 0.353011, antiderivative size = 219, normalized size of antiderivative = 1., 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 3607
Rule 3630
Rule 3539
Rule 3537
Rule 68
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*}
Mathematica [A] time = 1.23716, size = 169, normalized size = 0.77 \[ \frac{(a+b \tan (c+d x))^{n+1} \left (\frac{b (n+2) (B+i A) \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a-i b}\right )}{(n+1) (a-i b)}+\frac{b (n+2) (B-i A) \text{Hypergeometric2F1}\left (1,n+1,n+2,\frac{a+b \tan (c+d x)}{a+i b}\right )}{(n+1) (a+i b)}+\frac{-2 a B+2 A b n+4 A b}{b n+b}+2 B \tan (c+d x)\right )}{2 b d (n+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.368, size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( dx+c \right ) \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n} \left ( A+B\tan \left ( dx+c \right ) \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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