Integrand size = 35, antiderivative size = 270 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 A (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{a^2 d \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {a+b} (A-B) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{a d \sqrt {\sec (c+d x)}} \]
[Out]
Time = 0.54 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {3040, 3077, 2895, 3073} \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 A (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{a^2 d \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {a+b} (A-B) \sqrt {\cos (c+d x)} \csc (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{a d \sqrt {\sec (c+d x)}} \]
[In]
[Out]
Rule 2895
Rule 3040
Rule 3073
Rule 3077
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \left (A \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx+\left ((-A+B) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx \\ & = \frac {2 A (a-b) \sqrt {a+b} \sqrt {\cos (c+d x)} \csc (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{a^2 d \sqrt {\sec (c+d x)}}-\frac {2 \sqrt {a+b} (A-B) \sqrt {\cos (c+d x)} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{a d \sqrt {\sec (c+d x)}} \\ \end{align*}
Time = 9.70 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.03 \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\frac {2 \left (A (a+b \cos (c+d x)) \sqrt {\sec (c+d x)} \sin (c+d x)-\frac {\sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \left (2 A (a+b) \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} E\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}}-2 a (A+B) \sqrt {\frac {a+b \cos (c+d x)}{(a+b) (1+\cos (c+d x))}} \operatorname {EllipticF}\left (\arcsin \left (\tan \left (\frac {1}{2} (c+d x)\right )\right ),\frac {-a+b}{a+b}\right ) \sqrt {\frac {1}{1+\sec (c+d x)}}+A \cos (c+d x) (a+b \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \tan \left (\frac {1}{2} (c+d x)\right )\right )}{\sqrt {\sec ^2\left (\frac {1}{2} (c+d x)\right )}}\right )}{a d \sqrt {a+b \cos (c+d x)}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(777\) vs. \(2(246)=492\).
Time = 18.68 (sec) , antiderivative size = 778, normalized size of antiderivative = 2.88
method | result | size |
parts | \(-\frac {2 A \left (-\sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a +\sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a +\sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) b +\left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3} a -\left (\csc ^{3}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{3} b +a \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+b \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right ) \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}}\, \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right ) {\left (-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}\right )}^{\frac {3}{2}}}{d \left (\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b \right ) \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right ) a}-\frac {2 B \sqrt {\frac {a +\cos \left (d x +c \right ) b}{\left (1+\cos \left (d x +c \right )\right ) \left (a +b \right )}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \left (\sec ^{\frac {3}{2}}\left (d x +c \right )\right ) \left (\cos ^{2}\left (d x +c \right )+\cos \left (d x +c \right )\right )}{d \sqrt {a +\cos \left (d x +c \right ) b}}\) | \(778\) |
default | \(-\frac {2 {\left (-\frac {\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1}\right )}^{\frac {3}{2}} \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}-1\right ) \left (-A \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a +A \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a +A \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, E\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) b +\left (\csc ^{3}\left (d x +c \right )\right ) A a \left (1-\cos \left (d x +c \right )\right )^{3}-\left (\csc ^{3}\left (d x +c \right )\right ) A b \left (1-\cos \left (d x +c \right )\right )^{3}-B \sqrt {-\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}\, \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{a +b}}\, F\left (\cot \left (d x +c \right )-\csc \left (d x +c \right ), \sqrt {-\frac {a -b}{a +b}}\right ) a +a A \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )+A b \left (\csc \left (d x +c \right )-\cot \left (d x +c \right )\right )\right ) \sqrt {\frac {\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b}{\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1}}}{d \left (\left (\csc ^{2}\left (d x +c \right )\right ) \left (1-\cos \left (d x +c \right )\right )^{2}+1\right ) \left (\left (\csc ^{2}\left (d x +c \right )\right ) a \left (1-\cos \left (d x +c \right )\right )^{2}-\left (\csc ^{2}\left (d x +c \right )\right ) b \left (1-\cos \left (d x +c \right )\right )^{2}+a +b \right ) a}\) | \(781\) |
[In]
[Out]
\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
\[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int { \frac {{\left (B \cos \left (d x + c\right ) + A\right )} \sec \left (d x + c\right )^{\frac {3}{2}}}{\sqrt {b \cos \left (d x + c\right ) + a}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(A+B \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \cos (c+d x)}} \, dx=\int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}}{\sqrt {a+b\,\cos \left (c+d\,x\right )}} \,d x \]
[In]
[Out]