Optimal. Leaf size=50 \[ \frac{(d \tan (a+b x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(a+b x)\right )}{b d (n+1)} \]
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Rubi [A] time = 0.0288274, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3476, 364} \[ \frac{(d \tan (a+b x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(a+b x)\right )}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 364
Rubi steps
\begin{align*} \int (d \tan (a+b x))^n \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{x^n}{d^2+x^2} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{\, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};-\tan ^2(a+b x)\right ) (d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end{align*}
Mathematica [A] time = 0.0475396, size = 53, normalized size = 1.06 \[ \frac{\tan (a+b x) (d \tan (a+b x))^n \, _2F_1\left (1,\frac{n+1}{2};\frac{n+1}{2}+1;-\tan ^2(a+b x)\right )}{b (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.347, size = 0, normalized size = 0. \begin{align*} \int \left ( d\tan \left ( bx+a \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 (d \tan \left (b x + a\right )\right )^{n}\,{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 (d \tan \left (b x + a\right )\right )^{n}, 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 (d \tan{\left (a + b x \right )}\right )^{n}\, 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 (d \tan \left (b x + a\right )\right )^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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