Integrand size = 21, antiderivative size = 73 \[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d} \]
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Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3965, 52, 65, 213} \[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=-\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a \sec (c+d x)+a}}{\sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a \sec (c+d x)+a}}{d}+\frac {2 (a \sec (c+d x)+a)^{3/2}}{3 d} \]
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Rule 52
Rule 65
Rule 213
Rule 3965
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+a x)^{3/2}}{x} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {a \text {Subst}\left (\int \frac {\sqrt {a+a x}}{x} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+a x}} \, dx,x,\sec (c+d x)\right )}{d} \\ & = \frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d}+\frac {(2 a) \text {Subst}\left (\int \frac {1}{-1+\frac {x^2}{a}} \, dx,x,\sqrt {a+a \sec (c+d x)}\right )}{d} \\ & = -\frac {2 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+a \sec (c+d x)}}{\sqrt {a}}\right )}{d}+\frac {2 a \sqrt {a+a \sec (c+d x)}}{d}+\frac {2 (a+a \sec (c+d x))^{3/2}}{3 d} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.96 \[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=\frac {2 (a (1+\sec (c+d x)))^{3/2} \left (-3 \text {arctanh}\left (\sqrt {1+\sec (c+d x)}\right )+\sqrt {1+\sec (c+d x)} (4+\sec (c+d x))\right )}{3 d (1+\sec (c+d x))^{3/2}} \]
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Time = 0.75 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(\frac {\frac {2 \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 \sqrt {a +a \sec \left (d x +c \right )}\, a -2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +a \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{d}\) | \(57\) |
default | \(\frac {\frac {2 \left (a +a \sec \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+2 \sqrt {a +a \sec \left (d x +c \right )}\, a -2 a^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {a +a \sec \left (d x +c \right )}}{\sqrt {a}}\right )}{d}\) | \(57\) |
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Time = 0.35 (sec) , antiderivative size = 238, normalized size of antiderivative = 3.26 \[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=\left [\frac {3 \, a^{\frac {3}{2}} \cos \left (d x + c\right ) \log \left (-8 \, a \cos \left (d x + c\right )^{2} + 4 \, {\left (2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )\right )} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} - 8 \, a \cos \left (d x + c\right ) - a\right ) + 4 \, {\left (4 \, a \cos \left (d x + c\right ) + a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{6 \, d \cos \left (d x + c\right )}, \frac {3 \, \sqrt {-a} a \arctan \left (\frac {2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{2 \, a \cos \left (d x + c\right ) + a}\right ) \cos \left (d x + c\right ) + 2 \, {\left (4 \, a \cos \left (d x + c\right ) + a\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}}}{3 \, d \cos \left (d x + c\right )}\right ] \]
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\[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \tan {\left (c + d x \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.18 \[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} - \sqrt {a}}{\sqrt {a + \frac {a}{\cos \left (d x + c\right )}} + \sqrt {a}}\right ) + 2 \, {\left (a + \frac {a}{\cos \left (d x + c\right )}\right )}^{\frac {3}{2}} + 6 \, \sqrt {a + \frac {a}{\cos \left (d x + c\right )}} a}{3 \, d} \]
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\[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}} \tan \left (d x + c\right ) \,d x } \]
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Time = 14.33 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.92 \[ \int (a+a \sec (c+d x))^{3/2} \tan (c+d x) \, dx=\frac {2\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{3\,d}-\frac {2\,a^{3/2}\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{\sqrt {a}}\right )}{d}+\frac {2\,a\,\sqrt {a+\frac {a}{\cos \left (c+d\,x\right )}}}{d} \]
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