Optimal. Leaf size=15 \[ \frac {\tan ^4(a+b x)}{4 b} \]
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Rubi [A]
time = 0.02, antiderivative size = 15, 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 = {2687, 30}
\begin {gather*} \frac {\tan ^4(a+b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 2687
Rubi steps
\begin {align*} \int \sec ^2(a+b x) \tan ^3(a+b x) \, dx &=\frac {\text {Subst}\left (\int x^3 \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\tan ^4(a+b x)}{4 b}\\ \end {align*}
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Mathematica [A]
time = 0.00, size = 15, normalized size = 1.00 \begin {gather*} \frac {\tan ^4(a+b x)}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.06, size = 22, normalized size = 1.47
method | result | size |
derivativedivides | \(\frac {\sin ^{4}\left (b x +a \right )}{4 b \cos \left (b x +a \right )^{4}}\) | \(22\) |
default | \(\frac {\sin ^{4}\left (b x +a \right )}{4 b \cos \left (b x +a \right )^{4}}\) | \(22\) |
norman | \(\frac {4 \left (\tan ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}{b \left (\tan ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )-1\right )^{4}}\) | \(32\) |
risch | \(-\frac {2 \left ({\mathrm e}^{6 i \left (b x +a \right )}+{\mathrm e}^{2 i \left (b x +a \right )}\right )}{b \left ({\mathrm e}^{2 i \left (b x +a \right )}+1\right )^{4}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs.
\(2 (13) = 26\).
time = 0.29, size = 39, normalized size = 2.60 \begin {gather*} \frac {2 \, \sin \left (b x + a\right )^{2} - 1}{4 \, {\left (\sin \left (b x + a\right )^{4} - 2 \, \sin \left (b x + a\right )^{2} + 1\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 25, normalized size = 1.67 \begin {gather*} -\frac {2 \, \cos \left (b x + a\right )^{2} - 1}{4 \, b \cos \left (b x + a\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.14, size = 25, normalized size = 1.67 \begin {gather*} -\frac {2 \, \cos \left (b x + a\right )^{2} - 1}{4 \, b \cos \left (b x + a\right )^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.38, size = 13, normalized size = 0.87 \begin {gather*} \frac {{\mathrm {tan}\left (a+b\,x\right )}^4}{4\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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