Integrand size = 39, antiderivative size = 149 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d \sqrt {c+d} f} \]
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Time = 0.65 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {4055, 3887, 214, 4050} \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {a \sec (e+f x)+a} \sqrt {g \sec (e+f x)}}\right )}{d f \sqrt {c+d}} \]
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Rule 214
Rule 3887
Rule 4050
Rule 4055
Rubi steps \begin{align*} \text {integral}& = \frac {g \int \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)} \, dx}{d}-\frac {(c g) \int \frac {\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx}{d} \\ & = -\frac {\left (2 a g^2\right ) \text {Subst}\left (\int \frac {1}{a-g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}+\frac {\left (2 a c g^2\right ) \text {Subst}\left (\int \frac {1}{a c+a d-c g x^2} \, dx,x,-\frac {a \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f} \\ & = \frac {2 \sqrt {a} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {g} \tan (e+f x)}{\sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d f}-\frac {2 \sqrt {a} \sqrt {c} g^{3/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {c} \sqrt {g} \tan (e+f x)}{\sqrt {c+d} \sqrt {g \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}\right )}{d \sqrt {c+d} f} \\ \end{align*}
Time = 2.03 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.26 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=-\frac {g^2 \left (\sqrt {c+d} \log \left (\sqrt {2}-2 \sin \left (\frac {1}{2} (e+f x)\right )\right )-\sqrt {c+d} \log \left (\sqrt {2}+2 \sin \left (\frac {1}{2} (e+f x)\right )\right )+\sqrt {c} \left (-\log \left (\sqrt {2} \sqrt {c+d}-2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\sqrt {2} \sqrt {c+d}+2 \sqrt {c} \sin \left (\frac {1}{2} (e+f x)\right )\right )\right )\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {a (1+\sec (e+f x))}}{\sqrt {2} d \sqrt {c+d} f \sqrt {g \sec (e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(684\) vs. \(2(117)=234\).
Time = 22.71 (sec) , antiderivative size = 685, normalized size of antiderivative = 4.60
method | result | size |
default | \(-\frac {g \sqrt {2}\, \left (c -d \right ) \sqrt {-\frac {g \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1\right )}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1\right ) \sqrt {-\frac {2 a}{\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}-1}}\, \left (\sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arctanh}\left (\frac {\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right ) \sqrt {2}}{2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\right ) \sqrt {\frac {c}{c -d}}+\sqrt {\left (c +d \right ) \left (c -d \right )}\, \operatorname {arctanh}\left (\frac {\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right ) \sqrt {2}}{2 \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\right ) \sqrt {\frac {c}{c -d}}-c \ln \left (-\frac {2 \left (\sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, c -\sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, d +\sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+c -d \right )}{-c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )+c \ln \left (\frac {2 \sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, c -2 \sqrt {2}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}\, \sqrt {\frac {c}{c -d}}\, d -2 \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )+2 c -2 d}{c \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right )-\left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )\right ) d +\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )\right )}{f \sqrt {\left (c +d \right ) \left (c -d \right )}\, \left (c -d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \left (-c +d +\sqrt {\left (c +d \right ) \left (c -d \right )}\right ) \sqrt {\frac {c}{c -d}}\, \sqrt {\left (1-\cos \left (f x +e \right )\right )^{2} \csc \left (f x +e \right )^{2}+1}}\) | \(685\) |
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Time = 4.81 (sec) , antiderivative size = 1126, normalized size of antiderivative = 7.56 \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (g \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}{c + d \sec {\left (e + f x \right )}}\, dx \]
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Exception generated. \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int { \frac {\sqrt {a \sec \left (f x + e\right ) + a} \left (g \sec \left (f x + e\right )\right )^{\frac {3}{2}}}{d \sec \left (f x + e\right ) + c} \,d x } \]
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Timed out. \[ \int \frac {(g \sec (e+f x))^{3/2} \sqrt {a+a \sec (e+f x)}}{c+d \sec (e+f x)} \, dx=\int \frac {\sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (\frac {g}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{c+\frac {d}{\cos \left (e+f\,x\right )}} \,d x \]
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