Optimal. Leaf size=181 \[ -\frac{b (d \tan (e+f x))^{n+2} \, _2F_1\left (1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right )}{d^2 f (n+2) \left (a^2+b^2\right )}+\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right )}{d f (n+1) \left (a^2+b^2\right )}+\frac{b^2 (d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right )}{a d f (n+1) \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.263712, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3574, 3538, 3476, 364, 3634, 64} \[ -\frac{b (d \tan (e+f x))^{n+2} \, _2F_1\left (1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right )}{d^2 f (n+2) \left (a^2+b^2\right )}+\frac{a (d \tan (e+f x))^{n+1} \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right )}{d f (n+1) \left (a^2+b^2\right )}+\frac{b^2 (d \tan (e+f x))^{n+1} \, _2F_1\left (1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right )}{a d f (n+1) \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 3574
Rule 3538
Rule 3476
Rule 364
Rule 3634
Rule 64
Rubi steps
\begin{align*} \int \frac{(d \tan (e+f x))^n}{a+b \tan (e+f x)} \, dx &=\frac{\int (d \tan (e+f x))^n (a-b \tan (e+f x)) \, dx}{a^2+b^2}+\frac{b^2 \int \frac{(d \tan (e+f x))^n \left (1+\tan ^2(e+f x)\right )}{a+b \tan (e+f x)} \, dx}{a^2+b^2}\\ &=\frac{a \int (d \tan (e+f x))^n \, dx}{a^2+b^2}-\frac{b \int (d \tan (e+f x))^{1+n} \, dx}{\left (a^2+b^2\right ) d}+\frac{b^2 \operatorname{Subst}\left (\int \frac{(d x)^n}{a+b x} \, dx,x,\tan (e+f x)\right )}{\left (a^2+b^2\right ) f}\\ &=\frac{b^2 \, _2F_1\left (1,1+n;2+n;-\frac{b \tan (e+f x)}{a}\right ) (d \tan (e+f x))^{1+n}}{a \left (a^2+b^2\right ) d f (1+n)}-\frac{b \operatorname{Subst}\left (\int \frac{x^{1+n}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{\left (a^2+b^2\right ) f}+\frac{(a d) \operatorname{Subst}\left (\int \frac{x^n}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{\left (a^2+b^2\right ) f}\\ &=\frac{a \, _2F_1\left (1,\frac{1+n}{2};\frac{3+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{\left (a^2+b^2\right ) d f (1+n)}+\frac{b^2 \, _2F_1\left (1,1+n;2+n;-\frac{b \tan (e+f x)}{a}\right ) (d \tan (e+f x))^{1+n}}{a \left (a^2+b^2\right ) d f (1+n)}-\frac{b \, _2F_1\left (1,\frac{2+n}{2};\frac{4+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{\left (a^2+b^2\right ) d^2 f (2+n)}\\ \end{align*}
Mathematica [A] time = 0.460766, size = 142, normalized size = 0.78 \[ \frac{\tan (e+f x) (d \tan (e+f x))^n \left (a^2 (n+2) \, _2F_1\left (1,\frac{n+1}{2};\frac{n+3}{2};-\tan ^2(e+f x)\right )+b \left (b (n+2) \, _2F_1\left (1,n+1;n+2;-\frac{b \tan (e+f x)}{a}\right )-a (n+1) \tan (e+f x) \, _2F_1\left (1,\frac{n+2}{2};\frac{n+4}{2};-\tan ^2(e+f x)\right )\right )\right )}{a f (n+1) (n+2) \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.207, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d\tan \left ( fx+e \right ) \right ) ^{n}}{a+b\tan \left ( fx+e \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 \frac{\left (d \tan \left (f x + e\right )\right )^{n}}{b \tan \left (f x + e\right ) + a}\,{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 (\frac{\left (d \tan \left (f x + e\right )\right )^{n}}{b \tan \left (f x + e\right ) + a}, 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 \frac{\left (d \tan{\left (e + f x \right )}\right )^{n}}{a + b \tan{\left (e + f 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 \frac{\left (d \tan \left (f x + e\right )\right )^{n}}{b \tan \left (f x + e\right ) + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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