Optimal. Leaf size=422 \[ \frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (192 a^4 C+11 a^2 b^2 (637 A+491 C)+77 b^4 (13 A+11 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac {3}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.32, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {4221, 3050, 3049, 3033, 3023, 2748, 2639, 2635, 2641} \[ \frac {2 \left (11 a^2 b^2 (637 A+491 C)+192 a^4 C+77 b^4 (13 A+11 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {4 a b \left (96 a^2 C+1573 A b^2+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \sin (c+d x) (a+b \cos (c+d x))^2}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{231 d}+\frac {2 \left (78 a^2 b^2 (9 A+7 C)+39 a^4 (5 A+3 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{195 d}+\frac {2 C \sin (c+d x) (a+b \cos (c+d x))^4}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C \sin (c+d x) (a+b \cos (c+d x))^3}{143 d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3033
Rule 3049
Rule 3050
Rule 4221
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{13} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^3 \left (\frac {1}{2} a (13 A+3 C)+\frac {1}{2} b (13 A+11 C) \cos (c+d x)+4 a C \cos ^2(c+d x)\right ) \, dx\\ &=\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{143} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x))^2 \left (\frac {1}{4} a^2 (143 A+57 C)+\frac {1}{2} a b (143 A+113 C) \cos (c+d x)+\frac {1}{4} \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) \cos ^2(c+d x)\right ) \, dx\\ &=\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} (a+b \cos (c+d x)) \left (\frac {3}{8} a \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {1}{8} b \left (77 b^2 (13 A+11 C)+3 a^2 (1287 A+961 C)\right ) \cos (c+d x)+\frac {1}{4} a \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \cos ^2(c+d x)\right ) \, dx}{1287}\\ &=\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {21}{16} a^2 \left (11 b^2 (13 A+11 C)+3 a^2 (143 A+73 C)\right )+\frac {117}{4} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)+\frac {7}{16} \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \cos ^2(c+d x)\right ) \, dx}{9009}\\ &=\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \left (\frac {231}{32} \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right )+\frac {585}{8} a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \cos (c+d x)\right ) \, dx}{45045}\\ &=\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {1}{77} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\frac {1}{195} \left (\left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}+\frac {1}{231} \left (4 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {2 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{195 d}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{231 d}+\frac {4 a b \left (1573 A b^2+96 a^2 C+1259 b^2 C\right ) \sin (c+d x)}{9009 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (192 a^4 C+77 b^4 (13 A+11 C)+11 a^2 b^2 (637 A+491 C)\right ) \sin (c+d x)}{6435 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (48 a^2 C+11 b^2 (13 A+11 C)\right ) (a+b \cos (c+d x))^2 \sin (c+d x)}{1287 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {16 a C (a+b \cos (c+d x))^3 \sin (c+d x)}{143 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 C (a+b \cos (c+d x))^4 \sin (c+d x)}{13 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {8 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sin (c+d x)}{231 d \sqrt {\sec (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.78, size = 303, normalized size = 0.72 \[ \frac {\sqrt {\sec (c+d x)} \left (49920 a b \left (11 a^2 (7 A+5 C)+5 b^2 (11 A+9 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+14784 \left (39 a^4 (5 A+3 C)+78 a^2 b^2 (9 A+7 C)+7 b^4 (13 A+11 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sin (2 (c+d x)) \left (5 b \left (77 \left (312 a^2 b C+52 A b^3+89 b^3 C\right ) \cos (3 (c+d x))+3744 a \left (11 a^2 C+11 A b^2+16 b^2 C\right ) \cos (2 (c+d x))+312 a \left (44 a^2 (14 A+13 C)+b^2 (572 A+531 C)\right )+6552 a b^2 C \cos (4 (c+d x))+693 b^3 C \cos (5 (c+d x))\right )+154 \left (936 a^4 C+156 a^2 b^2 (36 A+43 C)+b^4 (1118 A+1171 C)\right ) \cos (c+d x)\right )\right )}{1441440 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}}{\sqrt {\sec \left (d x + c\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 3.16, size = 1017, normalized size = 2.41 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4}}{\sqrt {\sec \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________