Optimal. Leaf size=44 \[ \frac{a^2 x^2}{2}+\frac{a b \tanh ^{-1}\left (\sin \left (c+d x^2\right )\right )}{d}+\frac{b^2 \tan \left (c+d x^2\right )}{2 d} \]
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Rubi [A] time = 0.0515532, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4204, 3773, 3770, 3767, 8} \[ \frac{a^2 x^2}{2}+\frac{a b \tanh ^{-1}\left (\sin \left (c+d x^2\right )\right )}{d}+\frac{b^2 \tan \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 4204
Rule 3773
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x \left (a+b \sec \left (c+d x^2\right )\right )^2 \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int (a+b \sec (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{2}+(a b) \operatorname{Subst}\left (\int \sec (c+d x) \, dx,x,x^2\right )+\frac{1}{2} b^2 \operatorname{Subst}\left (\int \sec ^2(c+d x) \, dx,x,x^2\right )\\ &=\frac{a^2 x^2}{2}+\frac{a b \tanh ^{-1}\left (\sin \left (c+d x^2\right )\right )}{d}-\frac{b^2 \operatorname{Subst}\left (\int 1 \, dx,x,-\tan \left (c+d x^2\right )\right )}{2 d}\\ &=\frac{a^2 x^2}{2}+\frac{a b \tanh ^{-1}\left (\sin \left (c+d x^2\right )\right )}{d}+\frac{b^2 \tan \left (c+d x^2\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.204511, size = 41, normalized size = 0.93 \[ \frac{a^2 d x^2+2 a b \tanh ^{-1}\left (\sin \left (c+d x^2\right )\right )+b^2 \tan \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.023, size = 59, normalized size = 1.3 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}+{\frac{{b}^{2}\tan \left ( d{x}^{2}+c \right ) }{2\,d}}+{\frac{ab\ln \left ( \sec \left ( d{x}^{2}+c \right ) +\tan \left ( d{x}^{2}+c \right ) \right ) }{d}}+{\frac{{a}^{2}c}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.14844, size = 130, normalized size = 2.95 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + \frac{a b \log \left (\sec \left (d x^{2} + c\right ) + \tan \left (d x^{2} + c\right )\right )}{d} + \frac{b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.71378, size = 220, normalized size = 5. \begin{align*} \frac{a^{2} d x^{2} \cos \left (d x^{2} + c\right ) + a b \cos \left (d x^{2} + c\right ) \log \left (\sin \left (d x^{2} + c\right ) + 1\right ) - a b \cos \left (d x^{2} + c\right ) \log \left (-\sin \left (d x^{2} + c\right ) + 1\right ) + b^{2} \sin \left (d x^{2} + c\right )}{2 \, d \cos \left (d x^{2} + c\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \sec{\left (c + d x^{2} \right )}\right )^{2}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.34382, size = 119, normalized size = 2.7 \begin{align*} \frac{{\left (d x^{2} + c\right )} a^{2} + 2 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, a b \log \left ({\left | \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right ) - 1 \right |}\right ) - \frac{2 \, b^{2} \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )}{\tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right )^{2} - 1}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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