Optimal. Leaf size=181 \[ \frac {a \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{1+n}}{\left (a^2+b^2\right ) d f (1+n)}+\frac {b^2 \, _2F_1\left (1,1+n;2+n;-\frac {b \tan (e+f x)}{a}\right ) (d \tan (e+f x))^{1+n}}{a \left (a^2+b^2\right ) d f (1+n)}-\frac {b \, _2F_1\left (1,\frac {2+n}{2};\frac {4+n}{2};-\tan ^2(e+f x)\right ) (d \tan (e+f x))^{2+n}}{\left (a^2+b^2\right ) d^2 f (2+n)} \]
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Rubi [A]
time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, 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 = {3655, 3619,
3557, 371, 3715, 66} \begin {gather*} -\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 )} \end {gather*}
Antiderivative was successfully verified.
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Rule 66
Rule 371
Rule 3557
Rule 3619
Rule 3655
Rule 3715
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 \text {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 \text {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) \text {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*}
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Mathematica [A]
time = 0.53, size = 142, normalized size = 0.78 \begin {gather*} \frac {\tan (e+f x) (d \tan (e+f x))^n \left (a^2 (2+n) \, _2F_1\left (1,\frac {1+n}{2};\frac {3+n}{2};-\tan ^2(e+f x)\right )+b \left (b (2+n) \, _2F_1\left (1,1+n;2+n;-\frac {b \tan (e+f x)}{a}\right )-a (1+n) \, _2F_1\left (1,\frac {2+n}{2};\frac {4+n}{2};-\tan ^2(e+f x)\right ) \tan (e+f x)\right )\right )}{a \left (a^2+b^2\right ) f (1+n) (2+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.52, size = 0, normalized size = 0.00 \[\int \frac {\left (d \tan \left (f x +e \right )\right )^{n}}{a +b \tan \left (f x +e \right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d \tan {\left (e + f x \right )}\right )^{n}}{a + b \tan {\left (e + f x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^n}{a+b\,\mathrm {tan}\left (e+f\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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