Optimal. Leaf size=272 \[ -\frac{\left (\sqrt{-b^2} \left (\frac{a^2}{b^2}-n+1\right )-a n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right )}{4 b d (n+1) \left (\frac{a^2}{b^2}+1\right ) \left (a-\sqrt{-b^2}\right )}+\frac{b \left (\sqrt{-b^2} \left (\frac{a^2}{b^2}-n+1\right )+a n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )}{4 d (n+1) \left (a^2+b^2\right ) \left (a+\sqrt{-b^2}\right )}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.457396, antiderivative size = 272, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3506, 741, 831, 68} \[ -\frac{\left (\sqrt{-b^2} \left (\frac{a^2}{b^2}-n+1\right )-a n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right )}{4 b d (n+1) \left (\frac{a^2}{b^2}+1\right ) \left (a-\sqrt{-b^2}\right )}+\frac{b \left (\sqrt{-b^2} \left (\frac{a^2}{b^2}-n+1\right )+a n\right ) (a+b \tan (c+d x))^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )}{4 d (n+1) \left (a^2+b^2\right ) \left (a+\sqrt{-b^2}\right )}+\frac{\cos ^2(c+d x) (a \tan (c+d x)+b) (a+b \tan (c+d x))^{n+1}}{2 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 3506
Rule 741
Rule 831
Rule 68
Rubi steps
\begin{align*} \int \cos ^2(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+x)^n}{\left (1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac{\cos ^2(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{2 \left (a^2+b^2\right ) d}-\frac{b \operatorname{Subst}\left (\int \frac{(a+x)^n \left (-1-\frac{a^2}{b^2}+n+\frac{a n x}{b^2}\right )}{1+\frac{x^2}{b^2}} \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac{\cos ^2(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{2 \left (a^2+b^2\right ) d}-\frac{b \operatorname{Subst}\left (\int \left (\frac{\left (-a n+\sqrt{-b^2} \left (-1-\frac{a^2}{b^2}+n\right )\right ) (a+x)^n}{2 \left (\sqrt{-b^2}-x\right )}+\frac{\left (a n+\sqrt{-b^2} \left (-1-\frac{a^2}{b^2}+n\right )\right ) (a+x)^n}{2 \left (\sqrt{-b^2}+x\right )}\right ) \, dx,x,b \tan (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac{\cos ^2(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{2 \left (a^2+b^2\right ) d}+\frac{\left (b \left (\sqrt{-b^2} \left (1+\frac{a^2}{b^2}-n\right )-a n\right )\right ) \operatorname{Subst}\left (\int \frac{(a+x)^n}{\sqrt{-b^2}+x} \, dx,x,b \tan (c+d x)\right )}{4 \left (a^2+b^2\right ) d}+\frac{\left (b \left (\sqrt{-b^2} \left (1+\frac{a^2}{b^2}-n\right )+a n\right )\right ) \operatorname{Subst}\left (\int \frac{(a+x)^n}{\sqrt{-b^2}-x} \, dx,x,b \tan (c+d x)\right )}{4 \left (a^2+b^2\right ) d}\\ &=-\frac{b \left (\sqrt{-b^2} \left (1+\frac{a^2}{b^2}-n\right )-a n\right ) \, _2F_1\left (1,1+n;2+n;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) \left (a-\sqrt{-b^2}\right ) d (1+n)}+\frac{b \left (\sqrt{-b^2} \left (1+\frac{a^2}{b^2}-n\right )+a n\right ) \, _2F_1\left (1,1+n;2+n;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right ) (a+b \tan (c+d x))^{1+n}}{4 \left (a^2+b^2\right ) \left (a+\sqrt{-b^2}\right ) d (1+n)}+\frac{\cos ^2(c+d x) (b+a \tan (c+d x)) (a+b \tan (c+d x))^{1+n}}{2 \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [A] time = 1.24, size = 225, normalized size = 0.83 \[ \frac{(a+b \tan (c+d x))^{n+1} \left (-\frac{\left (\sqrt{-b^2} \left (a^2-b^2 (n-1)\right )-a b^2 n\right ) \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a-\sqrt{-b^2}}\right )}{(n+1) \left (a-\sqrt{-b^2}\right )}+\frac{\left (a^2 \sqrt{-b^2}+a b^2 n+\left (-b^2\right )^{3/2} (n-1)\right ) \, _2F_1\left (1,n+1;n+2;\frac{a+b \tan (c+d x)}{a+\sqrt{-b^2}}\right )}{(n+1) \left (a+\sqrt{-b^2}\right )}+2 b \cos ^2(c+d x) (a \tan (c+d x)+b)\right )}{4 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.377, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) ^{n}\, 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 )}^{n} \cos \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 ) + a\right )}^{n} \cos \left (d x + c\right )^{2}, 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 )}^{n} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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