Integrand size = 45, antiderivative size = 586 \[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=-\frac {(a-b) \sqrt {a+b} \left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 a b^2 d}+\frac {\sqrt {a+b} \left (24 A b^2+(a+2 b) (6 b B-3 a C+8 b C)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{24 b^2 d}+\frac {\sqrt {a+b} \left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (1+\sec (c+d x))}{a-b}}}{8 b^3 d}+\frac {\left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B-a C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d} \]
[Out]
Time = 2.33 (sec) , antiderivative size = 586, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3128, 3140, 3132, 2888, 3077, 2895, 3073} \[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {\sqrt {a+b} \cot (c+d x) \left (a^3 (-C)+2 a^2 b B-4 a b^2 (2 A+C)-8 b^3 B\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{b},\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{8 b^3 d}+\frac {\sqrt {a+b} \cot (c+d x) \left ((a+2 b) (-3 a C+6 b B+8 b C)+24 A b^2\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right ),-\frac {a+b}{a-b}\right )}{24 b^2 d}-\frac {(a-b) \sqrt {a+b} \cot (c+d x) \left (3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right ) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{24 a b^2 d}+\frac {\sin (c+d x) \left (3 a (2 b B-a C)+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \cos (c+d x)}}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B-a C) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}{4 b d}+\frac {C \sin (c+d x) \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2}}{3 b d} \]
[In]
[Out]
Rule 2888
Rule 2895
Rule 3073
Rule 3077
Rule 3128
Rule 3132
Rule 3140
Rubi steps \begin{align*} \text {integral}& = \frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d}+\frac {\int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {a C}{2}+b (3 A+2 C) \cos (c+d x)+\frac {3}{2} (2 b B-a C) \cos ^2(c+d x)\right )}{\sqrt {\cos (c+d x)}} \, dx}{3 b} \\ & = \frac {(2 b B-a C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d}+\frac {\int \frac {\frac {1}{4} a (6 b B+a C)+\frac {1}{2} b (12 a A+6 b B+7 a C) \cos (c+d x)+\frac {1}{4} \left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{6 b} \\ & = \frac {\left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B-a C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d}+\frac {\int \frac {-\frac {1}{4} a \left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right )+\frac {1}{2} a b (6 b B+a C) \cos (c+d x)-\frac {3}{4} \left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{12 b^2} \\ & = \frac {\left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B-a C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d}+\frac {\int \frac {-\frac {1}{4} a \left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right )+\frac {1}{2} a b (6 b B+a C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{12 b^2}-\frac {\left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \int \frac {\sqrt {\cos (c+d x)}}{\sqrt {a+b \cos (c+d x)}} \, dx}{16 b^2} \\ & = \frac {\sqrt {a+b} \left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\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}}}{8 b^3 d}+\frac {\left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B-a C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d}-\frac {\left (a \left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right )\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2}+\frac {\left (a \left (24 A b^2+(a+2 b) (6 b B-3 a C+8 b C)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{48 b^2} \\ & = -\frac {(a-b) \sqrt {a+b} \left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \cot (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}}}{24 a b^2 d}+\frac {\sqrt {a+b} \left (24 A b^2+(a+2 b) (6 b B-3 a C+8 b C)\right ) \cot (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}}}{24 b^2 d}+\frac {\sqrt {a+b} \left (2 a^2 b B-8 b^3 B-a^3 C-4 a b^2 (2 A+C)\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{b},\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}}}{8 b^3 d}+\frac {\left (8 b^2 (3 A+2 C)+3 a (2 b B-a C)\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{24 b^2 d \sqrt {\cos (c+d x)}}+\frac {(2 b B-a C) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{4 b d}+\frac {C \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{3 b d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 7.42 (sec) , antiderivative size = 1242, normalized size of antiderivative = 2.12 \[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\frac {-\frac {4 a \left (24 A b^2+18 a b B-a^2 C+16 b^2 C\right ) \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-4 a \left (48 a A b+24 b^2 B+28 a b C\right ) \left (\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {\sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )+2 \left (24 A b^2+6 a b B-3 a^2 C+16 b^2 C\right ) \left (\frac {i \cos \left (\frac {1}{2} (c+d x)\right ) \sqrt {a+b \cos (c+d x)} E\left (i \text {arcsinh}\left (\frac {\sin \left (\frac {1}{2} (c+d x)\right )}{\sqrt {\cos (c+d x)}}\right )|-\frac {2 a}{-a-b}\right ) \sec (c+d x)}{b \sqrt {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)} \sqrt {\frac {(a+b \cos (c+d x)) \sec (c+d x)}{a+b}}}+\frac {2 a \left (\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{(a+b) \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}-\frac {a \sqrt {\frac {(a+b) \cot ^2\left (\frac {1}{2} (c+d x)\right )}{-a+b}} \sqrt {-\frac {(a+b) \cos (c+d x) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}} \csc (c+d x) \operatorname {EllipticPi}\left (-\frac {a}{b},\arcsin \left (\frac {\sqrt {\frac {(a+b \cos (c+d x)) \csc ^2\left (\frac {1}{2} (c+d x)\right )}{a}}}{\sqrt {2}}\right ),-\frac {2 a}{-a+b}\right ) \sin ^4\left (\frac {1}{2} (c+d x)\right )}{b \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}}\right )}{b}+\frac {\sqrt {a+b \cos (c+d x)} \sin (c+d x)}{b \sqrt {\cos (c+d x)}}\right )}{48 b d}+\frac {\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (\frac {(6 b B+a C) \sin (c+d x)}{12 b}+\frac {1}{6} C \sin (2 (c+d x))\right )}{d} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(5084\) vs. \(2(538)=1076\).
Time = 11.00 (sec) , antiderivative size = 5085, normalized size of antiderivative = 8.68
method | result | size |
parts | \(\text {Expression too large to display}\) | \(5085\) |
default | \(\text {Expression too large to display}\) | \(5142\) |
[In]
[Out]
\[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \,d x } \]
[In]
[Out]
\[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int { {\left (C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A\right )} \sqrt {b \cos \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)} \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,\sqrt {a+b\,\cos \left (c+d\,x\right )}\,\left (C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A\right ) \,d x \]
[In]
[Out]