Integrand size = 30, antiderivative size = 93 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a c^2 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a c \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}} \]
[Out]
Time = 0.22 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3991, 3990, 3556} \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a c^2 \tan (e+f x) \log (\cos (e+f x))}{f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a c \tan (e+f x) \sqrt {c-c \sec (e+f x)}}{f \sqrt {a \sec (e+f x)+a}} \]
[In]
[Out]
Rule 3556
Rule 3990
Rule 3991
Rubi steps \begin{align*} \text {integral}& = -\frac {a c \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}+c \int \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)} \, dx \\ & = -\frac {a c \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}}-\frac {\left (a c^2 \tan (e+f x)\right ) \int \tan (e+f x) \, dx}{\sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \\ & = \frac {a c^2 \log (\cos (e+f x)) \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}-\frac {a c \sqrt {c-c \sec (e+f x)} \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)}} \\ \end{align*}
Time = 0.50 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\frac {a c^2 (\log (\cos (e+f x))+\sec (e+f x)) \tan (e+f x)}{f \sqrt {a (1+\sec (e+f x))} \sqrt {c-c \sec (e+f x)}} \]
[In]
[Out]
Time = 2.34 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.42
method | result | size |
default | \(\frac {c \sqrt {a \left (\sec \left (f x +e \right )+1\right )}\, \left (\sec \left (f x +e \right )-1\right ) \sqrt {-c \left (\sec \left (f x +e \right )-1\right )}\, \left (\cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )-1\right )+\cos \left (f x +e \right ) \ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )-\cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+\cos \left (f x +e \right )+1\right ) \cot \left (f x +e \right )}{f \left (\cos \left (f x +e \right )-1\right )}\) | \(132\) |
risch | \(\frac {c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, x}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}-\frac {2 c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (f x +e \right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}-\frac {2 i c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, {\mathrm e}^{i \left (f x +e \right )}}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}-\frac {i c \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{1+{\mathrm e}^{2 i \left (f x +e \right )}}}\, \ln \left (1+{\mathrm e}^{2 i \left (f x +e \right )}\right )}{\left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) f}\) | \(428\) |
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 350, normalized size of antiderivative = 3.76 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\left [-\frac {2 \, c \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - \sqrt {-a c} {\left (c \cos \left (f x + e\right ) + c\right )} \log \left (\frac {a c \cos \left (f x + e\right )^{4} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )\right )} \sqrt {-a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) + a c}{2 \, \cos \left (f x + e\right )^{2}}\right )}{2 \, {\left (f \cos \left (f x + e\right ) + f\right )}}, -\frac {c \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - \sqrt {a c} {\left (c \cos \left (f x + e\right ) + c\right )} \arctan \left (\frac {\sqrt {a c} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sqrt {\frac {c \cos \left (f x + e\right ) - c}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{a c \cos \left (f x + e\right )^{2} + a c}\right )}{f \cos \left (f x + e\right ) + f}\right ] \]
[In]
[Out]
\[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\int \sqrt {a \left (\sec {\left (e + f x \right )} + 1\right )} \left (- c \left (\sec {\left (e + f x \right )} - 1\right )\right )^{\frac {3}{2}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (85) = 170\).
Time = 0.40 (sec) , antiderivative size = 243, normalized size of antiderivative = 2.61 \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=-\frac {{\left ({\left (f x + e\right )} c \cos \left (2 \, f x + 2 \, e\right )^{2} + {\left (f x + e\right )} c \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, {\left (f x + e\right )} c \cos \left (2 \, f x + 2 \, e\right ) + 2 \, c \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) \sin \left (2 \, f x + 2 \, e\right ) + {\left (f x + e\right )} c - {\left (c \cos \left (2 \, f x + 2 \, e\right )^{2} + c \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, c \cos \left (2 \, f x + 2 \, e\right ) + c\right )} \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right ) + 1\right ) - 2 \, {\left (c \cos \left (2 \, f x + 2 \, e\right ) + c\right )} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a} \sqrt {c}}{{\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )} f} \]
[In]
[Out]
\[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\int { \sqrt {a \sec \left (f x + e\right ) + a} {\left (-c \sec \left (f x + e\right ) + c\right )}^{\frac {3}{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2} \, dx=\int \sqrt {a+\frac {a}{\cos \left (e+f\,x\right )}}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{3/2} \,d x \]
[In]
[Out]