Optimal. Leaf size=444 \[ \frac {2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (539 a^3 B+2 a^2 b (673 A+891 C)+1353 a b^2 B+192 A b^3\right )}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (15 a^4 (9 A+11 C)+660 a^3 b B+9 a^2 b^2 (101 A+143 C)+682 a b^3 B+64 A b^4\right )}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (5 a^4 (9 A+11 C)+220 a^3 b B+66 a^2 b^2 (5 A+7 C)+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (7 a^4 B+4 a^3 b (7 A+9 C)+54 a^2 b^2 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.32, antiderivative size = 444, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {4094, 4074, 4047, 3771, 2639, 4045, 2641} \[ \frac {2 \sin (c+d x) \left (3 a^2 (9 A+11 C)+55 a b B+16 A b^2\right ) (a+b \sec (c+d x))^2}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 a \sin (c+d x) \left (2 a^2 b (673 A+891 C)+539 a^3 B+1353 a b^2 B+192 A b^3\right )}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \sin (c+d x) \left (9 a^2 b^2 (101 A+143 C)+15 a^4 (9 A+11 C)+660 a^3 b B+682 a b^3 B+64 A b^4\right )}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 C)+220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)\right )}{231 d}+\frac {2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (4 a^3 b (7 A+9 C)+54 a^2 b^2 B+7 a^4 B+12 a b^3 (3 A+5 C)+15 b^4 B\right )}{15 d}+\frac {2 (11 a B+8 A b) \sin (c+d x) (a+b \sec (c+d x))^3}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A \sin (c+d x) (a+b \sec (c+d x))^4}{11 d \sec ^{\frac {9}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2639
Rule 2641
Rule 3771
Rule 4045
Rule 4047
Rule 4074
Rule 4094
Rubi steps
\begin {align*} \int \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {2}{11} \int \frac {(a+b \sec (c+d x))^3 \left (\frac {1}{2} (8 A b+11 a B)+\frac {1}{2} (9 a A+11 b B+11 a C) \sec (c+d x)+\frac {1}{2} b (A+11 C) \sec ^2(c+d x)\right )}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {4}{99} \int \frac {(a+b \sec (c+d x))^2 \left (\frac {3}{4} \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right )+\frac {1}{4} \left (146 a A b+77 a^2 B+99 b^2 B+198 a b C\right ) \sec (c+d x)+\frac {1}{4} b (17 A b+11 a B+99 b C) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}+\frac {8}{693} \int \frac {(a+b \sec (c+d x)) \left (\frac {1}{8} \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right )+\frac {1}{8} \left (1441 a^2 b B+693 b^3 B+45 a^3 (9 A+11 C)+a b^2 (1381 A+2079 C)\right ) \sec (c+d x)+\frac {1}{8} b \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)\right )}{\sec ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac {231}{16} \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sec (c+d x)-\frac {5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{3465}\\ &=\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {16 \int \frac {-\frac {15}{16} \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right )-\frac {5}{16} b^2 \left (242 a b B+9 a^2 (9 A+11 C)+b^2 (167 A+693 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x)} \, dx}{3465}-\frac {1}{15} \left (-7 a^4 B-54 a^2 b^2 B-15 b^4 B-12 a b^3 (3 A+5 C)-4 a^3 b (7 A+9 C)\right ) \int \frac {1}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {1}{231} \left (-220 a^3 b B-308 a b^3 B-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 C)\right ) \int \sqrt {\sec (c+d x)} \, dx-\frac {1}{15} \left (\left (-7 a^4 B-54 a^2 b^2 B-15 b^4 B-12 a b^3 (3 A+5 C)-4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 \left (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}-\frac {1}{231} \left (\left (-220 a^3 b B-308 a b^3 B-77 b^4 (A+3 C)-66 a^2 b^2 (5 A+7 C)-5 a^4 (9 A+11 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 (7 a^4 B+54 a^2 b^2 B+15 b^4 B+12 a b^3 (3 A+5 C)+4 a^3 b (7 A+9 C)\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {\sec (c+d x)}}{15 d}+\frac {2 \left (220 a^3 b B+308 a b^3 B+77 b^4 (A+3 C)+66 a^2 b^2 (5 A+7 C)+5 a^4 (9 A+11 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 {2 a \left (192 A b^3+539 a^3 B+1353 a b^2 B+2 a^2 b (673 A+891 C)\right ) \sin (c+d x)}{3465 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (64 A b^4+660 a^3 b B+682 a b^3 B+15 a^4 (9 A+11 C)+9 a^2 b^2 (101 A+143 C)\right ) \sin (c+d x)}{693 d \sqrt {\sec (c+d x)}}+\frac {2 \left (16 A b^2+55 a b B+3 a^2 (9 A+11 C)\right ) (a+b \sec (c+d x))^2 \sin (c+d x)}{231 d \sec ^{\frac {5}{2}}(c+d x)}+\frac {2 (8 A b+11 a B) (a+b \sec (c+d x))^3 \sin (c+d x)}{99 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {2 A (a+b \sec (c+d x))^4 \sin (c+d x)}{11 d \sec ^{\frac {9}{2}}(c+d x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 7.09, size = 580, normalized size = 1.31 \[ \frac {(a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (\frac {1}{88} a^4 A \sin (6 (c+d x))+\frac {1}{36} a^3 (a B+4 A b) \sin (5 (c+d x))+\frac {1}{154} a^2 \sin (4 (c+d x)) \left (16 a^2 A+11 a^2 C+44 a b B+66 A b^2\right )+\frac {1}{90} a \sin (c+d x) \left (19 a^3 B+76 a^2 A b+72 a^2 b C+108 a b^2 B+72 A b^3\right )+\frac {1}{180} a \sin (3 (c+d x)) \left (43 a^3 B+172 a^2 A b+144 a^2 b C+216 a b^2 B+144 A b^3\right )+\frac {\sin (2 (c+d x)) \left (1041 a^4 A+1144 a^4 C+4576 a^3 b B+6864 a^2 A b^2+7392 a^2 b^2 C+4928 a b^3 B+1232 A b^4\right )}{1848}\right )}{d \sec ^{\frac {11}{2}}(c+d x) (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)}+\frac {2 \cos ^6(c+d x) (a+b \sec (c+d x))^4 \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (225 a^4 A+275 a^4 C+1100 a^3 b B+1650 a^2 A b^2+2310 a^2 b^2 C+1540 a b^3 B+385 A b^4+1155 b^4 C\right )+\frac {2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \left (539 a^4 B+2156 a^3 A b+2772 a^3 b C+4158 a^2 b^2 B+2772 a A b^3+4620 a b^3 C+1155 b^4 B\right )}{\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}\right )}{1155 d (a \cos (c+d x)+b)^4 (A \cos (2 c+2 d x)+A+2 B \cos (c+d x)+2 C)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {C b^{4} \sec \left (d x + c\right )^{6} + {\left (4 \, C a b^{3} + B b^{4}\right )} \sec \left (d x + c\right )^{5} + A a^{4} + {\left (6 \, C a^{2} b^{2} + 4 \, B a b^{3} + A b^{4}\right )} \sec \left (d x + c\right )^{4} + 2 \, {\left (2 \, C a^{3} b + 3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} \sec \left (d x + c\right )^{3} + {\left (C a^{4} + 4 \, B a^{3} b + 6 \, A a^{2} b^{2}\right )} \sec \left (d x + c\right )^{2} + {\left (B a^{4} + 4 \, A a^{3} b\right )} \sec \left (d x + c\right )}{\sec \left (d x + c\right )^{\frac {11}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (b \sec \left (d x + c\right ) + a\right )}^{4}}{\sec \left (d x + c\right )^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 5.33, size = 1273, normalized size = 2.87 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [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]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^4\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{11/2}} \,d x \]
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]
________________________________________________________________________________________