Integrand size = 19, antiderivative size = 42 \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {(2 a-b) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {3757, 393, 212} \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {(2 a-b) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b \tan (c+d x) \sec (c+d x)}{2 d} \]
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Rule 212
Rule 393
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {a-(a-b) x^2}{\left (1-x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = \frac {b \sec (c+d x) \tan (c+d x)}{2 d}+\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (c+d x)\right )}{2 d} \\ & = \frac {(2 a-b) \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b \sec (c+d x) \tan (c+d x)}{2 d} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.14 \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {a \text {arctanh}(\sin (c+d x))}{d}-\frac {b \text {arctanh}(\sin (c+d x))}{2 d}+\frac {b \sec (c+d x) \tan (c+d x)}{2 d} \]
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Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.60
method | result | size |
derivativedivides | \(\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+b \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(67\) |
default | \(\frac {a \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )+b \left (\frac {\sin \left (d x +c \right )^{3}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{2}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{2}\right )}{d}\) | \(67\) |
risch | \(-\frac {i b \left ({\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) b}{2 d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) b}{2 d}\) | \(118\) |
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Time = 0.28 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.81 \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, a - b\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, a - b\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, b \sin \left (d x + c\right )}{4 \, d \cos \left (d x + c\right )^{2}} \]
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\[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\int \left (a + b \tan ^{2}{\left (c + d x \right )}\right ) \sec {\left (c + d x \right )}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.48 \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, a - b\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (2 \, a - b\right )} \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, b \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
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Time = 0.42 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.52 \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {{\left (2 \, a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - {\left (2 \, a - b\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, b \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1}}{4 \, d} \]
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Time = 12.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.88 \[ \int \sec (c+d x) \left (a+b \tan ^2(c+d x)\right ) \, dx=\frac {\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )\,\left (2\,a-b\right )}{d}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]
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