Integrand size = 27, antiderivative size = 231 \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=-\frac {2 c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {c}{c+d},\arcsin \left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right ),\frac {c-d}{c+d}\right ) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (1+\sec (e+f x))}{c+d \sec (e+f x)}} (c+d \sec (e+f x))}{a \sqrt {c+d} f}-\frac {(c-d) E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \]
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Time = 0.30 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4012, 3865, 4053} \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=-\frac {2 c \cot (e+f x) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (\sec (e+f x)+1)}{c+d \sec (e+f x)}} (c+d \sec (e+f x)) \operatorname {EllipticPi}\left (\frac {c}{c+d},\arcsin \left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right ),\frac {c-d}{c+d}\right )}{a f \sqrt {c+d}}-\frac {(c-d) \sqrt {\frac {1}{\sec (e+f x)+1}} \sqrt {c+d \sec (e+f x)} E\left (\arcsin \left (\frac {\tan (e+f x)}{\sec (e+f x)+1}\right )|\frac {c-d}{c+d}\right )}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (\sec (e+f x)+1)}}} \]
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Rule 3865
Rule 4012
Rule 4053
Rubi steps \begin{align*} \text {integral}& = \frac {c \int \sqrt {c+d \sec (e+f x)} \, dx}{a}+(-c+d) \int \frac {\sec (e+f x) \sqrt {c+d \sec (e+f x)}}{a+a \sec (e+f x)} \, dx \\ & = -\frac {2 c \cot (e+f x) \operatorname {EllipticPi}\left (\frac {c}{c+d},\arcsin \left (\frac {\sqrt {c+d}}{\sqrt {c+d \sec (e+f x)}}\right ),\frac {c-d}{c+d}\right ) \sqrt {-\frac {d (1-\sec (e+f x))}{c+d \sec (e+f x)}} \sqrt {\frac {d (1+\sec (e+f x))}{c+d \sec (e+f x)}} (c+d \sec (e+f x))}{a \sqrt {c+d} f}-\frac {(c-d) E\left (\arcsin \left (\frac {\tan (e+f x)}{1+\sec (e+f x)}\right )|\frac {c-d}{c+d}\right ) \sqrt {\frac {1}{1+\sec (e+f x)}} \sqrt {c+d \sec (e+f x)}}{a f \sqrt {\frac {c+d \sec (e+f x)}{(c+d) (1+\sec (e+f x))}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(810\) vs. \(2(231)=462\).
Time = 34.07 (sec) , antiderivative size = 810, normalized size of antiderivative = 3.51 \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=\frac {\cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (c+d \sec (e+f x))^{3/2} \left (2 \sec \left (\frac {1}{2} (e+f x)\right ) \left (-c \sin \left (\frac {1}{2} (e+f x)\right )+d \sin \left (\frac {1}{2} (e+f x)\right )\right )-2 (-c+d) \sin (e+f x)\right )}{f (d+c \cos (e+f x)) (a+a \sec (e+f x))}+\frac {2 \cos ^2\left (\frac {e}{2}+\frac {f x}{2}\right ) (c+d \sec (e+f x))^{3/2} \left (c^2 \tan \left (\frac {1}{2} (e+f x)\right )-d^2 \tan \left (\frac {1}{2} (e+f x)\right )-2 c^2 \tan ^3\left (\frac {1}{2} (e+f x)\right )+2 c d \tan ^3\left (\frac {1}{2} (e+f x)\right )+c^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )-2 c d \tan ^5\left (\frac {1}{2} (e+f x)\right )+d^2 \tan ^5\left (\frac {1}{2} (e+f x)\right )-4 c^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {c+d-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )}{c+d}}-4 c^2 \operatorname {EllipticPi}\left (-1,\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right ) \tan ^2\left (\frac {1}{2} (e+f x)\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \sqrt {\frac {c+d-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )}{c+d}}+\left (c^2-d^2\right ) E\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right )|\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {c+d-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )}{c+d}}+2 c (c-d) \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (e+f x)\right )\right ),\frac {c-d}{c+d}\right ) \sqrt {1-\tan ^2\left (\frac {1}{2} (e+f x)\right )} \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {\frac {c+d-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )}{c+d}}\right )}{f (d+c \cos (e+f x))^{3/2} \sqrt {\sec (e+f x)} (a+a \sec (e+f x)) \sqrt {\frac {1}{1-\tan ^2\left (\frac {1}{2} (e+f x)\right )}} \left (-1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right ) \left (1+\tan ^2\left (\frac {1}{2} (e+f x)\right )\right )^{3/2} \sqrt {\frac {c+d-c \tan ^2\left (\frac {1}{2} (e+f x)\right )+d \tan ^2\left (\frac {1}{2} (e+f x)\right )}{1+\tan ^2\left (\frac {1}{2} (e+f x)\right )}}} \]
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Time = 8.25 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.11
method | result | size |
default | \(\frac {\left (\cos \left (f x +e \right )+1\right ) \left (2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-2 \operatorname {EllipticF}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) c d +\operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) c^{2}-\operatorname {EllipticE}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), \sqrt {\frac {c -d}{c +d}}\right ) d^{2}-4 c^{2} \operatorname {EllipticPi}\left (\cot \left (f x +e \right )-\csc \left (f x +e \right ), -1, \sqrt {\frac {c -d}{c +d}}\right )\right ) \sqrt {\frac {d +c \cos \left (f x +e \right )}{\left (c +d \right ) \left (\cos \left (f x +e \right )+1\right )}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sqrt {c +d \sec \left (f x +e \right )}}{a f \left (d +c \cos \left (f x +e \right )\right )}\) | \(257\) |
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\[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=\int { \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sec \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=\frac {\int \frac {c \sqrt {c + d \sec {\left (e + f x \right )}}}{\sec {\left (e + f x \right )} + 1}\, dx + \int \frac {d \sqrt {c + d \sec {\left (e + f x \right )}} \sec {\left (e + f x \right )}}{\sec {\left (e + f x \right )} + 1}\, dx}{a} \]
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\[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=\int { \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sec \left (f x + e\right ) + a} \,d x } \]
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\[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=\int { \frac {{\left (d \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{a \sec \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(c+d \sec (e+f x))^{3/2}}{a+a \sec (e+f x)} \, dx=\int \frac {{\left (c+\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}}{a+\frac {a}{\cos \left (e+f\,x\right )}} \,d x \]
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