Optimal. Leaf size=17 \[ \frac {(b \sec (e+f x))^m}{f m} \]
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Rubi [A]
time = 0.02, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2686, 32}
\begin {gather*} \frac {(b \sec (e+f x))^m}{f m} \end {gather*}
Antiderivative was successfully verified.
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Rule 32
Rule 2686
Rubi steps
\begin {align*} \int (b \sec (e+f x))^m \tan (e+f x) \, dx &=\frac {b \text {Subst}\left (\int (b x)^{-1+m} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac {(b \sec (e+f x))^m}{f m}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 17, normalized size = 1.00 \begin {gather*} \frac {(b \sec (e+f x))^m}{f m} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 18, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {\left (b \sec \left (f x +e \right )\right )^{m}}{f m}\) | \(18\) |
default | \(\frac {\left (b \sec \left (f x +e \right )\right )^{m}}{f m}\) | \(18\) |
risch | \(\frac {{\mathrm e}^{\frac {m \left (-i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{i \left (f x +e \right )}\right )+i \pi \mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )-i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \mathrm {csgn}\left (i {\mathrm e}^{i \left (f x +e \right )}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )+i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \mathrm {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2}-i \pi \,\mathrm {csgn}\left (\frac {i {\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \mathrm {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right ) \mathrm {csgn}\left (i b \right )-i \pi \mathrm {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{3}+i \pi \mathrm {csgn}\left (\frac {i b \,{\mathrm e}^{i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}\right )^{2} \mathrm {csgn}\left (i b \right )+2 \ln \left (b \right )+2 \ln \left (2\right )+2 \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )-2 \ln \left ({\mathrm e}^{2 i \left (f x +e \right )}+1\right )\right )}{2}}}{f m}\) | \(425\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 21, normalized size = 1.24 \begin {gather*} \frac {b^{m} \cos \left (f x + e\right )^{-m}}{f m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 20, normalized size = 1.18 \begin {gather*} \frac {\left (\frac {b}{\cos \left (f x + e\right )}\right )^{m}}{f m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 42 vs.
\(2 (12) = 24\).
time = 0.14, size = 42, normalized size = 2.47 \begin {gather*} \begin {cases} x \tan {\left (e \right )} & \text {for}\: f = 0 \wedge m = 0 \\x \left (b \sec {\left (e \right )}\right )^{m} \tan {\left (e \right )} & \text {for}\: f = 0 \\\frac {\log {\left (\tan ^{2}{\left (e + f x \right )} + 1 \right )}}{2 f} & \text {for}\: m = 0 \\\frac {\left (b \sec {\left (e + f x \right )}\right )^{m}}{f m} & \text {otherwise} \end {cases} \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 [B]
time = 0.12, size = 19, normalized size = 1.12 \begin {gather*} \frac {{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^m}{f\,m} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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