Integrand size = 45, antiderivative size = 453 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}} \]
[Out]
Time = 2.19 (sec) , antiderivative size = 453, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {4350, 4181, 4193, 3944, 2886, 2884, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\frac {\sin (c+d x) \left (15 a^2 C+42 a b B+24 A b^2+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\cos (c+d x)}}-\frac {\sqrt {\cos (c+d x)} \left (-\left (a^2 (48 A-33 C)\right )+54 a b B+8 b^2 (3 A+2 C)\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{24 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {\left (48 a^3 B+a^2 b (96 A+59 C)+66 a b^2 B+8 b^3 (3 A+2 C)\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (5 a^3 C+30 a^2 b B+20 a b^2 (2 A+C)+8 b^3 B\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{8 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {(5 a C+6 b B) \sin (c+d x) (a+b \sec (c+d x))^{3/2}}{12 d \sqrt {\cos (c+d x)}}+\frac {C \sin (c+d x) (a+b \sec (c+d x))^{5/2}}{3 d \sqrt {\cos (c+d x)}} \]
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 3941
Rule 3943
Rule 3944
Rule 4120
Rule 4181
Rule 4193
Rule 4350
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {1}{3} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{3/2} \left (\frac {1}{2} a (6 A-C)+(3 A b+3 a B+2 b C) \sec (c+d x)+\frac {1}{2} (6 b B+5 a C) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {1}{6} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)} \left (\frac {3}{4} a (8 a A-2 b B-3 a C)+\frac {1}{2} \left (12 a^2 B+6 b^2 B+a b (24 A+11 C)\right ) \sec (c+d x)+\frac {1}{4} \left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx \\ & = \frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {1}{6} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )+\frac {1}{4} a \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right ) \sec (c+d x)+\frac {3}{8} \left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {1}{6} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{8} a \left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right )+\frac {1}{4} a \left (24 a^2 B+6 b^2 B+a b (72 A+13 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx+\frac {1}{16} \left (\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \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 {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {1}{48} \left (\left (-54 a b B+a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx+\frac {1}{48} \left (\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx+\frac {\left (\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{16 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}} \\ & = \frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {\left (\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{48 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{16 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-54 a b B+a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{48 \sqrt {b+a \cos (c+d x)}} \\ & = \frac {\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \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 )}{8 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {\left (\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{48 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (-54 a b B+a^2 (48 A-33 C)-8 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{48 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}} \\ & = \frac {\left (48 a^3 B+66 a b^2 B+8 b^3 (3 A+2 C)+a^2 b (96 A+59 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{24 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (30 a^2 b B+8 b^3 B+5 a^3 C+20 a b^2 (2 A+C)\right ) \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 )}{8 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (54 a b B-a^2 (48 A-33 C)+8 b^2 (3 A+2 C)\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{24 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {\left (24 A b^2+42 a b B+15 a^2 C+16 b^2 C\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {(6 b B+5 a C) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+b \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 37.58 (sec) , antiderivative size = 243913, normalized size of antiderivative = 538.44 \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Result too large to show} \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 18.69 (sec) , antiderivative size = 5340, normalized size of antiderivative = 11.79
[In]
[Out]
\[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \]
[In]
[Out]
\[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int { {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\cos \left (d x + c\right )} \,d x } \]
[In]
[Out]
Timed out. \[ \int \sqrt {\cos (c+d x)} (a+b \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx=\int \sqrt {\cos \left (c+d\,x\right )}\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
[In]
[Out]