Optimal. Leaf size=19 \[ \frac {(a+b \tan (x))^{n+1}}{b (n+1)} \]
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Rubi [A] time = 0.04, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3506, 32} \[ \frac {(a+b \tan (x))^{n+1}}{b (n+1)} \]
Antiderivative was successfully verified.
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Rule 32
Rule 3506
Rubi steps
\begin {align*} \int \sec ^2(x) (a+b \tan (x))^n \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^n \, dx,x,b \tan (x)\right )}{b}\\ &=\frac {(a+b \tan (x))^{1+n}}{b (1+n)}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 18, normalized size = 0.95 \[ \frac {(a+b \tan (x))^{n+1}}{b n+b} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.06, size = 37, normalized size = 1.95 \[ \frac {{\left (a \cos \relax (x) + b \sin \relax (x)\right )} \left (\frac {a \cos \relax (x) + b \sin \relax (x)}{\cos \relax (x)}\right )^{n}}{{\left (b n + b\right )} \cos \relax (x)} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 20, normalized size = 1.05 \[ \frac {\left (a +b \tan \relax (x )\right )^{n +1}}{b \left (n +1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 19, normalized size = 1.00 \[ \frac {{\left (b \tan \relax (x) + a\right )}^{n + 1}}{b {\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.56, size = 37, normalized size = 1.95 \[ \left \{\begin {array}{cl} \frac {\ln \left (a+b\,\mathrm {tan}\relax (x)\right )}{b} & \text {\ if\ \ }n=-1\\ \frac {{\left (a+b\,\mathrm {tan}\relax (x)\right )}^{n+1}}{b\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \]
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 \[ \int \left (a + b \tan {\relax (x )}\right )^{n} \sec ^{2}{\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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