Optimal. Leaf size=356 \[ \frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{105 d}-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}-\frac {8 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac {16 A b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.30, antiderivative size = 356, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {4221, 3048, 3047, 3031, 3023, 2748, 2641, 2639} \[ \frac {4 a b \left (a^2 (101 A+175 C)+96 A b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{105 d}-\frac {2 b^2 \left (5 a^2 (5 A+7 C)+b^2 (87 A-35 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {2 \left (5 a^2 (5 A+7 C)+48 A b^2\right ) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a+b \cos (c+d x))^2}{105 d}+\frac {2 \left (42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)+7 b^4 (3 A+C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d}-\frac {8 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 A \sin (c+d x) \sec ^{\frac {7}{2}}(c+d x) (a+b \cos (c+d x))^4}{7 d}+\frac {16 A b \sin (c+d x) \sec ^{\frac {5}{2}}(c+d x) (a+b \cos (c+d x))^3}{35 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3031
Rule 3047
Rule 3048
Rule 4221
Rubi steps
\begin {align*} \int (a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right ) \sec ^{\frac {9}{2}}(c+d x) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^4 \left (A+C \cos ^2(c+d x)\right )}{\cos ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{7} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^3 \left (4 A b+\frac {1}{2} a (5 A+7 C) \cos (c+d x)-\frac {1}{2} b (3 A-7 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{35} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x))^2 \left (\frac {1}{4} \left (48 A b^2+5 a^2 (5 A+7 C)\right )+\frac {1}{2} a b (17 A+35 C) \cos (c+d x)-\frac {1}{4} b^2 (39 A-35 C) \cos ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}+\frac {1}{105} \left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \cos (c+d x)) \left (\frac {1}{4} b \left (96 A b^2+a^2 (101 A+175 C)\right )+\frac {1}{8} a \left (5 a^2 (5 A+7 C)+3 b^2 (11 A+105 C)\right ) \cos (c+d x)-\frac {3}{8} b \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \cos ^2(c+d x)\right )}{\cos ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{105} \left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{16} \left (-192 A b^4-5 a^4 (5 A+7 C)-5 a^2 b^2 (47 A+133 C)\right )+\frac {21}{4} a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \cos (c+d x)+\frac {3}{16} b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \cos ^2(c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{315} \left (32 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {15}{32} \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right )+\frac {63}{8} a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \cos (c+d x)}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}-\frac {1}{5} \left (4 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{21} \left (\left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=-\frac {8 a b \left (5 b^2 (A-C)+a^2 (3 A+5 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{5 d}+\frac {2 \left (7 b^4 (3 A+C)+42 a^2 b^2 (A+3 C)+a^4 (5 A+7 C)\right ) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{21 d}-\frac {2 b^2 \left (b^2 (87 A-35 C)+5 a^2 (5 A+7 C)\right ) \sin (c+d x)}{105 d \sqrt {\sec (c+d x)}}+\frac {4 a b \left (96 A b^2+a^2 (101 A+175 C)\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{105 d}+\frac {2 \left (48 A b^2+5 a^2 (5 A+7 C)\right ) (a+b \cos (c+d x))^2 \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{105 d}+\frac {16 A b (a+b \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{35 d}+\frac {2 A (a+b \cos (c+d x))^4 \sec ^{\frac {7}{2}}(c+d x) \sin (c+d x)}{7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.30, size = 296, normalized size = 0.83 \[ \frac {2 \sec ^{\frac {7}{2}}(c+d x) \left (15 a^4 A \sin (c+d x)+25 a^4 A \sin (c+d x) \cos ^2(c+d x)+35 a^4 C \sin (c+d x) \cos ^2(c+d x)+42 a^3 A b \sin (2 (c+d x))+252 a^3 A b \sin (c+d x) \cos ^3(c+d x)+420 a^3 b C \sin (c+d x) \cos ^3(c+d x)-84 a b \left (a^2 (3 A+5 C)+5 b^2 (A-C)\right ) \cos ^{\frac {7}{2}}(c+d x) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+210 a^2 A b^2 \sin (c+d x) \cos ^2(c+d x)+5 \left (a^4 (5 A+7 C)+42 a^2 b^2 (A+3 C)+7 b^4 (3 A+C)\right ) \cos ^{\frac {7}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+420 a A b^3 \sin (c+d x) \cos ^3(c+d x)+35 b^4 C \sin (c+d x) \cos ^4(c+d x)\right )}{105 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (C b^{4} \cos \left (d x + c\right )^{6} + 4 \, C a b^{3} \cos \left (d x + c\right )^{5} + 4 \, A a^{3} b \cos \left (d x + c\right ) + A a^{4} + {\left (6 \, C a^{2} b^{2} + A b^{4}\right )} \cos \left (d x + c\right )^{4} + 4 \, {\left (C a^{3} b + A a b^{3}\right )} \cos \left (d x + c\right )^{3} + {\left (C a^{4} + 6 \, A a^{2} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \sec \left (d x + c\right )^{\frac {9}{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 12.57, size = 1531, normalized size = 4.30 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (C \cos \left (d x + c\right )^{2} + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{4} \sec \left (d x + c\right )^{\frac {9}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \left (C\,{\cos \left (c+d\,x\right )}^2+A\right )\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{9/2}\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^4 \,d x \]
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]
________________________________________________________________________________________