Optimal. Leaf size=26 \[ \frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3587, 32}
\begin {gather*} \frac {(a+b \tan (c+d x))^{n+1}}{b d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 3587
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \tan (c+d x))^n \, dx &=\frac {\text {Subst}\left (\int (a+x)^n \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.19, size = 26, normalized size = 1.00 \begin {gather*} \frac {(a+b \tan (c+d x))^{1+n}}{b d (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 27, normalized size = 1.04
method | result | size |
derivativedivides | \(\frac {\left (a +b \tan \left (d x +c \right )\right )^{1+n}}{b d \left (1+n \right )}\) | \(27\) |
default | \(\frac {\left (a +b \tan \left (d x +c \right )\right )^{1+n}}{b d \left (1+n \right )}\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 26, normalized size = 1.00 \begin {gather*} \frac {{\left (b \tan \left (d x + c\right ) + a\right )}^{n + 1}}{b d {\left (n + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 64 vs.
\(2 (26) = 52\).
time = 0.34, size = 64, normalized size = 2.46 \begin {gather*} \frac {{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )} \left (\frac {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}{\cos \left (d x + c\right )}\right )^{n}}{{\left (b d n + b d\right )} \cos \left (d x + c\right )} \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 \left (a + b \tan {\left (c + d x \right )}\right )^{n} \sec ^{2}{\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.31, size = 51, normalized size = 1.96 \begin {gather*} \left \{\begin {array}{cl} \frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}{b\,d} & \text {\ if\ \ }n=-1\\ \frac {{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^{n+1}}{b\,d\,\left (n+1\right )} & \text {\ if\ \ }n\neq -1 \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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