Optimal. Leaf size=24 \[ \frac {(d \tan (a+b x))^{n+1}}{b d (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2607, 32} \[ \frac {(d \tan (a+b x))^{n+1}}{b d (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 2607
Rubi steps
\begin {align*} \int \sec ^2(a+b x) (d \tan (a+b x))^n \, dx &=\frac {\operatorname {Subst}\left (\int (d x)^n \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {(d \tan (a+b x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.04 \[ \frac {\tan (a+b x) (d \tan (a+b x))^n}{b (n+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 40, normalized size = 1.67 \[ \frac {\left (\frac {d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}\right )^{n} \sin \left (b x + a\right )}{{\left (b n + b\right )} \cos \left (b x + a\right )} \]
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 \[ \int \left (d \tan \left (b x + a\right )\right )^{n} \sec \left (b x + a\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 25, normalized size = 1.04 \[ \frac {\left (d \tan \left (b x +a \right )\right )^{1+n}}{b d \left (1+n \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 24, normalized size = 1.00 \[ \frac {\left (d \tan \left (b x + a\right )\right )^{n + 1}}{b d {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 49, normalized size = 2.04 \[ \frac {\sin \left (2\,a+2\,b\,x\right )\,{\left (\frac {d\,\sin \left (2\,a+2\,b\,x\right )}{2\,{\cos \left (a+b\,x\right )}^2}\right )}^n}{2\,b\,{\cos \left (a+b\,x\right )}^2\,\left (n+1\right )} \]
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 \[ \int \left (d \tan {\left (a + b x \right )}\right )^{n} \sec ^{2}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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