Integrand size = 25, antiderivative size = 68 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \]
[Out]
Time = 0.32 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {4349, 3944, 2886, 2884} \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\frac {2 \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \]
[In]
[Out]
Rule 2884
Rule 2886
Rule 3944
Rule 4349
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\sqrt {b+a \cos (c+d x)} \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{\sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ & = \frac {\sqrt {\frac {b+a \cos (c+d x)}{a+b}} \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{\sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ & = \frac {2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 6.90 (sec) , antiderivative size = 14986, normalized size of antiderivative = 220.38 \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\text {Result too large to show} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 7.57 (sec) , antiderivative size = 195, normalized size of antiderivative = 2.87
method | result | size |
default | \(-\frac {2 \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (\cos \left (d x +c \right )+1\right )}}\, \sqrt {\cos \left (d x +c \right )}\, \left (\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )-2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \frac {a +b}{a -b}, \frac {i}{\sqrt {\frac {a -b}{a +b}}}\right )\right ) \sqrt {a +b \sec \left (d x +c \right )}}{d \sqrt {\frac {a -b}{a +b}}\, \left (b +a \cos \left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(195\) |
[In]
[Out]
Timed out. \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \sec {\left (c + d x \right )}} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \sec \left (d x + c\right ) + a} \cos \left (d x + c\right )^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}} \,d x \]
[In]
[Out]