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 = 0.05, antiderivative size = 44, normalized size of antiderivative = 1.00, 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 8
Rule 3767
Rule 3770
Rule 3773
Rule 4204
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*}
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Mathematica [A] time = 0.21, 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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fricas [B] time = 0.66, size = 91, normalized size = 2.07 \[ \frac {a^{2} d x^{2} \cos \left (d x^{2} + c\right ) + a b \cos \left (d x^{2} + c\right ) \log \left (\sin \left (d x^{2} + c\right ) + 1\right ) - a b \cos \left (d x^{2} + c\right ) \log \left (-\sin \left (d x^{2} + c\right ) + 1\right ) + b^{2} \sin \left (d x^{2} + c\right )}{2 \, d \cos \left (d x^{2} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.86, size = 88, normalized size = 2.00 \[ \frac {{\left (d x^{2} + c\right )} a^{2} + 2 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 2 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {2 \, b^{2} \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )^{2} - 1}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 59, normalized size = 1.34 \[ \frac {a^{2} x^{2}}{2}+\frac {b^{2} \tan \left (d \,x^{2}+c \right )}{2 d}+\frac {a b \ln \left (\sec \left (d \,x^{2}+c \right )+\tan \left (d \,x^{2}+c \right )\right )}{d}+\frac {a^{2} c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.82, size = 96, normalized size = 2.18 \[ \frac {1}{2} \, a^{2} x^{2} + \frac {a b \log \left (\sec \left (d x^{2} + c\right ) + \tan \left (d x^{2} + c\right )\right )}{d} + \frac {b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} + 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.50, size = 100, normalized size = 2.27 \[ \frac {a^2\,x^2}{2}+\frac {b^2\,1{}\mathrm {i}}{d\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,d\,x^2+c\,2{}\mathrm {i}}+1\right )}+\frac {a\,b\,\ln \left (-a\,b\,x\,4{}\mathrm {i}-4\,a\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )}{d}-\frac {a\,b\,\ln \left (a\,b\,x\,4{}\mathrm {i}-4\,a\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )}{d} \]
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 \[ \int x \left (a + b \sec {\left (c + d x^{2} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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