Optimal. Leaf size=154 \[ \frac{(A-C) \tan ^{m+1}(c+d x) (b \tan (c+d x))^n \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+n+1),\frac{1}{2} (m+n+3),-\tan ^2(c+d x)\right )}{d (m+n+1)}+\frac{B \tan ^{m+2}(c+d x) (b \tan (c+d x))^n \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+n+2),\frac{1}{2} (m+n+4),-\tan ^2(c+d x)\right )}{d (m+n+2)}+\frac{C \tan ^{m+1}(c+d x) (b \tan (c+d x))^n}{d (m+n+1)} \]
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Rubi [A] time = 0.137775, antiderivative size = 154, normalized size of antiderivative = 1., 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 = {20, 3630, 3538, 3476, 364} \[ \frac{(A-C) \tan ^{m+1}(c+d x) (b \tan (c+d x))^n \, _2F_1\left (1,\frac{1}{2} (m+n+1);\frac{1}{2} (m+n+3);-\tan ^2(c+d x)\right )}{d (m+n+1)}+\frac{B \tan ^{m+2}(c+d x) (b \tan (c+d x))^n \, _2F_1\left (1,\frac{1}{2} (m+n+2);\frac{1}{2} (m+n+4);-\tan ^2(c+d x)\right )}{d (m+n+2)}+\frac{C \tan ^{m+1}(c+d x) (b \tan (c+d x))^n}{d (m+n+1)} \]
Antiderivative was successfully verified.
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Rule 20
Rule 3630
Rule 3538
Rule 3476
Rule 364
Rubi steps
\begin{align*} \int \tan ^m(c+d x) (b \tan (c+d x))^n \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx &=\left (\tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{m+n}(c+d x) \left (A+B \tan (c+d x)+C \tan ^2(c+d x)\right ) \, dx\\ &=\frac{C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\left (\tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{m+n}(c+d x) (A-C+B \tan (c+d x)) \, dx\\ &=\frac{C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\left (B \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{1+m+n}(c+d x) \, dx+\left ((A-C) \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \int \tan ^{m+n}(c+d x) \, dx\\ &=\frac{C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac{\left (B \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \operatorname{Subst}\left (\int \frac{x^{1+m+n}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}+\frac{\left ((A-C) \tan ^{-n}(c+d x) (b \tan (c+d x))^n\right ) \operatorname{Subst}\left (\int \frac{x^{m+n}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{C \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac{(A-C) \, _2F_1\left (1,\frac{1}{2} (1+m+n);\frac{1}{2} (3+m+n);-\tan ^2(c+d x)\right ) \tan ^{1+m}(c+d x) (b \tan (c+d x))^n}{d (1+m+n)}+\frac{B \, _2F_1\left (1,\frac{1}{2} (2+m+n);\frac{1}{2} (4+m+n);-\tan ^2(c+d x)\right ) \tan ^{2+m}(c+d x) (b \tan (c+d x))^n}{d (2+m+n)}\\ \end{align*}
Mathematica [A] time = 0.376412, size = 115, normalized size = 0.75 \[ \frac{\tan ^{m+1}(c+d x) (b \tan (c+d x))^n \left (\frac{(A-C) \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+n+1),\frac{1}{2} (m+n+3),-\tan ^2(c+d x)\right )}{m+n+1}+\frac{B \tan (c+d x) \text{Hypergeometric2F1}\left (1,\frac{1}{2} (m+n+2),\frac{1}{2} (m+n+4),-\tan ^2(c+d x)\right )}{m+n+2}+\frac{C}{m+n+1}\right )}{d} \]
Antiderivative was successfully verified.
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Maple [F] time = 180., size = 0, normalized size = 0. \begin{align*} \int \left ( \tan \left ( dx+c \right ) \right ) ^{m} \left ( b\tan \left ( dx+c \right ) \right ) ^{n} \left ( A+B\tan \left ( dx+c \right ) +C \left ( \tan \left ( dx+c \right ) \right ) ^{2} \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 (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 )^{m}\,{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 (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 )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \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 ^{m}{\left (c + d x \right )}\, dx \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 (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 )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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