Optimal. Leaf size=45 \[ \frac {a^2 x^2}{2}-\frac {a b \tanh ^{-1}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 d} \]
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Rubi [A] time = 0.05, antiderivative size = 45, 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 = {4205, 3773, 3770, 3767, 8} \[ \frac {a^2 x^2}{2}-\frac {a b \tanh ^{-1}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 8
Rule 3767
Rule 3770
Rule 3773
Rule 4205
Rubi steps
\begin {align*} \int x \left (a+b \csc \left (c+d x^2\right )\right )^2 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int (a+b \csc (c+d x))^2 \, dx,x,x^2\right )\\ &=\frac {a^2 x^2}{2}+(a b) \operatorname {Subst}\left (\int \csc (c+d x) \, dx,x,x^2\right )+\frac {1}{2} b^2 \operatorname {Subst}\left (\int \csc ^2(c+d x) \, dx,x,x^2\right )\\ &=\frac {a^2 x^2}{2}-\frac {a b \tanh ^{-1}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \operatorname {Subst}\left (\int 1 \, dx,x,\cot \left (c+d x^2\right )\right )}{2 d}\\ &=\frac {a^2 x^2}{2}-\frac {a b \tanh ^{-1}\left (\cos \left (c+d x^2\right )\right )}{d}-\frac {b^2 \cot \left (c+d x^2\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 86, normalized size = 1.91 \[ \frac {2 a \left (a c+a d x^2+2 b \log \left (\sin \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )-2 b \log \left (\cos \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )\right )+b^2 \tan \left (\frac {1}{2} \left (c+d x^2\right )\right )+b^2 \left (-\cot \left (\frac {1}{2} \left (c+d x^2\right )\right )\right )}{4 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 94, normalized size = 2.09 \[ \frac {a^{2} d x^{2} \sin \left (d x^{2} + c\right ) - a b \log \left (\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) + a b \log \left (-\frac {1}{2} \, \cos \left (d x^{2} + c\right ) + \frac {1}{2}\right ) \sin \left (d x^{2} + c\right ) - b^{2} \cos \left (d x^{2} + c\right )}{2 \, d \sin \left (d x^{2} + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.34, size = 84, normalized size = 1.87 \[ \frac {2 \, {\left (d x^{2} + c\right )} a^{2} + 4 \, a b \log \left ({\left | \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) \right |}\right ) + b^{2} \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) - \frac {4 \, a b \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + b^{2}}{\tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right )}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.79, size = 61, normalized size = 1.36 \[ \frac {a^{2} x^{2}}{2}-\frac {b^{2} \cot \left (d \,x^{2}+c \right )}{2 d}+\frac {a b \ln \left (\csc \left (d \,x^{2}+c \right )-\cot \left (d \,x^{2}+c \right )\right )}{d}+\frac {a^{2} c}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 98, normalized size = 2.18 \[ \frac {1}{2} \, a^{2} x^{2} - \frac {a b \log \left (\cot \left (d x^{2} + c\right ) + \csc \left (d x^{2} + c\right )\right )}{d} - \frac {b^{2} \sin \left (2 \, d x^{2} + 2 \, c\right )}{d \cos \left (2 \, d x^{2} + 2 \, c\right )^{2} + d \sin \left (2 \, d x^{2} + 2 \, c\right )^{2} - 2 \, d \cos \left (2 \, d x^{2} + 2 \, c\right ) + d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 102, normalized size = 2.27 \[ \frac {a^2\,x^2}{2}-\frac {b^2\,1{}\mathrm {i}}{d\,\left ({\mathrm {e}}^{2{}\mathrm {i}\,d\,x^2+c\,2{}\mathrm {i}}-1\right )}-\frac {a\,b\,\ln \left (-a\,b\,x\,4{}\mathrm {i}-a\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,4{}\mathrm {i}\right )}{d}+\frac {a\,b\,\ln \left (a\,b\,x\,4{}\mathrm {i}-a\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\,4{}\mathrm {i}\right )}{d} \]
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 \[ \int x \left (a + b \csc {\left (c + d x^{2} \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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