Optimal. Leaf size=26 \[ \frac {a x^2}{2}-\frac {b \log \left (\cos \left (c+d x^2\right )\right )}{2 d} \]
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Rubi [A] time = 0.03, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 3747, 3475} \[ \frac {a x^2}{2}-\frac {b \log \left (\cos \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3475
Rule 3747
Rubi steps
\begin {align*} \int x \left (a+b \tan \left (c+d x^2\right )\right ) \, dx &=\int \left (a x+b x \tan \left (c+d x^2\right )\right ) \, dx\\ &=\frac {a x^2}{2}+b \int x \tan \left (c+d x^2\right ) \, dx\\ &=\frac {a x^2}{2}+\frac {1}{2} b \operatorname {Subst}\left (\int \tan (c+d x) \, dx,x,x^2\right )\\ &=\frac {a x^2}{2}-\frac {b \log \left (\cos \left (c+d x^2\right )\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 26, normalized size = 1.00 \[ \frac {a x^2}{2}-\frac {b \log \left (\cos \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 31, normalized size = 1.19 \[ \frac {2 \, a d x^{2} - b \log \left (\frac {1}{\tan \left (d x^{2} + c\right )^{2} + 1}\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.14, size = 28, normalized size = 1.08 \[ \frac {{\left (d x^{2} + c\right )} a - b \log \left ({\left | \cos \left (d x^{2} + c\right ) \right |}\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 30, normalized size = 1.15 \[ \frac {a \,x^{2}}{2}-\frac {b \ln \left (\cos \left (d \,x^{2}+c \right )\right )}{2 d}+\frac {c a}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 22, normalized size = 0.85 \[ \frac {1}{2} \, a x^{2} + \frac {b \log \left (\sec \left (d x^{2} + c\right )\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 26, normalized size = 1.00 \[ \frac {a\,x^2}{2}+\frac {b\,\ln \left ({\mathrm {tan}\left (d\,x^2+c\right )}^2+1\right )}{4\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.16, size = 36, normalized size = 1.38 \[ \begin {cases} \frac {a x^{2}}{2} + \frac {b \log {\left (\tan ^{2}{\left (c + d x^{2} \right )} + 1 \right )}}{4 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \tan {\relax (c )}\right )}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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