Integrand size = 14, antiderivative size = 26 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {a x^2}{2}+\frac {b \text {arctanh}\left (\sin \left (c+d x^2\right )\right )}{2 d} \]
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Time = 0.03 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {14, 4289, 3855} \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {a x^2}{2}+\frac {b \text {arctanh}\left (\sin \left (c+d x^2\right )\right )}{2 d} \]
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Rule 14
Rule 3855
Rule 4289
Rubi steps \begin{align*} \text {integral}& = \int \left (a x+b x \sec \left (c+d x^2\right )\right ) \, dx \\ & = \frac {a x^2}{2}+b \int x \sec \left (c+d x^2\right ) \, dx \\ & = \frac {a x^2}{2}+\frac {1}{2} b \text {Subst}\left (\int \sec (c+d x) \, dx,x,x^2\right ) \\ & = \frac {a x^2}{2}+\frac {b \text {arctanh}\left (\sin \left (c+d x^2\right )\right )}{2 d} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {a x^2}{2}+\frac {b \text {arctanh}\left (\sin \left (c+d x^2\right )\right )}{2 d} \]
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Time = 0.25 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23
method | result | size |
parts | \(\frac {a \,x^{2}}{2}+\frac {b \ln \left (\sec \left (d \,x^{2}+c \right )+\tan \left (d \,x^{2}+c \right )\right )}{2 d}\) | \(32\) |
derivativedivides | \(\frac {\left (d \,x^{2}+c \right ) a +b \ln \left (\sec \left (d \,x^{2}+c \right )+\tan \left (d \,x^{2}+c \right )\right )}{2 d}\) | \(36\) |
default | \(\frac {\left (d \,x^{2}+c \right ) a +b \ln \left (\sec \left (d \,x^{2}+c \right )+\tan \left (d \,x^{2}+c \right )\right )}{2 d}\) | \(36\) |
parallelrisch | \(\frac {a d \,x^{2}-b \ln \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )+b \ln \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(46\) |
norman | \(\frac {a \,x^{2}}{2}-\frac {b \ln \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )-1\right )}{2 d}+\frac {b \ln \left (\tan \left (\frac {d \,x^{2}}{2}+\frac {c}{2}\right )+1\right )}{2 d}\) | \(48\) |
risch | \(\frac {a \,x^{2}}{2}+\frac {b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}+i\right )}{2 d}-\frac {b \ln \left ({\mathrm e}^{i \left (d \,x^{2}+c \right )}-i\right )}{2 d}\) | \(50\) |
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {2 \, a d x^{2} + b \log \left (\sin \left (d x^{2} + c\right ) + 1\right ) - b \log \left (-\sin \left (d x^{2} + c\right ) + 1\right )}{4 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 42 vs. \(2 (20) = 40\).
Time = 2.54 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.62 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\begin {cases} \frac {a \left (c + d x^{2}\right ) + b \log {\left (\tan {\left (c + d x^{2} \right )} + \sec {\left (c + d x^{2} \right )} \right )}}{2 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \sec {\left (c \right )}\right )}{2} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.19 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {1}{2} \, a x^{2} + \frac {b \log \left (\sec \left (d x^{2} + c\right ) + \tan \left (d x^{2} + c\right )\right )}{2 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (22) = 44\).
Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.92 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {{\left (d x^{2} + c\right )} a + b \log \left ({\left | \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - b \log \left ({\left | \tan \left (\frac {1}{2} \, d x^{2} + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{2 \, d} \]
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Time = 0.55 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.58 \[ \int x \left (a+b \sec \left (c+d x^2\right )\right ) \, dx=\frac {a\,x^2}{2}+\frac {b\,\ln \left (-b\,x\,2{}\mathrm {i}-2\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )}{2\,d}-\frac {b\,\ln \left (b\,x\,2{}\mathrm {i}-2\,b\,x\,{\mathrm {e}}^{d\,x^2\,1{}\mathrm {i}}\,{\mathrm {e}}^{c\,1{}\mathrm {i}}\right )}{2\,d} \]
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