Optimal. Leaf size=15 \[ \frac {\tan ^3(a+b x)}{3 b} \]
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Rubi [A] time = 0.03, 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 = {2607, 30} \[ \frac {\tan ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2607
Rubi steps
\begin {align*} \int \sec ^2(a+b x) \tan ^2(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^2 \, dx,x,\tan (a+b x)\right )}{b}\\ &=\frac {\tan ^3(a+b x)}{3 b}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 15, normalized size = 1.00 \[ \frac {\tan ^3(a+b x)}{3 b} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 29, normalized size = 1.93 \[ -\frac {{\left (\cos \left (b x + a\right )^{2} - 1\right )} \sin \left (b x + a\right )}{3 \, b \cos \left (b x + a\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 13, normalized size = 0.87 \[ \frac {\tan \left (b x + a\right )^{3}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 22, normalized size = 1.47 \[ \frac {\sin ^{3}\left (b x +a \right )}{3 b \cos \left (b x +a \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 13, normalized size = 0.87 \[ \frac {\tan \left (b x + a\right )^{3}}{3 \, b} \]
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 \[ \frac {{\mathrm {tan}\left (a+b\,x\right )}^3}{3\,b} \]
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 \[ \int \sin ^{2}{\left (a + b x \right )} \sec ^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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