Integrand size = 21, antiderivative size = 43 \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(3 A+2 C) \tan (c+d x)}{3 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 d} \]
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Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4131, 3852, 8} \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {(3 A+2 C) \tan (c+d x)}{3 d}+\frac {C \tan (c+d x) \sec ^2(c+d x)}{3 d} \]
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Rule 8
Rule 3852
Rule 4131
Rubi steps \begin{align*} \text {integral}& = \frac {C \sec ^2(c+d x) \tan (c+d x)}{3 d}+\frac {1}{3} (3 A+2 C) \int \sec ^2(c+d x) \, dx \\ & = \frac {C \sec ^2(c+d x) \tan (c+d x)}{3 d}-\frac {(3 A+2 C) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d} \\ & = \frac {(3 A+2 C) \tan (c+d x)}{3 d}+\frac {C \sec ^2(c+d x) \tan (c+d x)}{3 d} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.84 \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {A \tan (c+d x)}{d}+\frac {C \left (\tan (c+d x)+\frac {1}{3} \tan ^3(c+d x)\right )}{d} \]
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Time = 0.19 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {A \tan \left (d x +c \right )-C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(35\) |
default | \(\frac {A \tan \left (d x +c \right )-C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(35\) |
parts | \(\frac {A \tan \left (d x +c \right )}{d}-\frac {C \left (-\frac {2}{3}-\frac {\sec \left (d x +c \right )^{2}}{3}\right ) \tan \left (d x +c \right )}{d}\) | \(37\) |
parallelrisch | \(\frac {\left (3 A +2 C \right ) \sin \left (3 d x +3 c \right )+3 \sin \left (d x +c \right ) \left (A +2 C \right )}{3 d \left (\cos \left (3 d x +3 c \right )+3 \cos \left (d x +c \right )\right )}\) | \(57\) |
risch | \(\frac {2 i \left (3 A \,{\mathrm e}^{4 i \left (d x +c \right )}+6 A \,{\mathrm e}^{2 i \left (d x +c \right )}+6 C \,{\mathrm e}^{2 i \left (d x +c \right )}+3 A +2 C \right )}{3 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3}}\) | \(63\) |
norman | \(\frac {-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}-\frac {2 \left (A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{d}+\frac {4 \left (3 A +C \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-1\right )^{3}}\) | \(75\) |
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Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left ({\left (3 \, A + 2 \, C\right )} \cos \left (d x + c\right )^{2} + C\right )} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{3}} \]
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\[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \sec ^{2}{\left (c + d x \right )}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {{\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C + 3 \, A \tan \left (d x + c\right )}{3 \, d} \]
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Time = 0.30 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.79 \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C \tan \left (d x + c\right )^{3} + 3 \, A \tan \left (d x + c\right ) + 3 \, C \tan \left (d x + c\right )}{3 \, d} \]
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Time = 15.74 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.65 \[ \int \sec ^2(c+d x) \left (A+C \sec ^2(c+d x)\right ) \, dx=\frac {C\,{\mathrm {tan}\left (c+d\,x\right )}^3}{3\,d}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (A+C\right )}{d} \]
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