Optimal. Leaf size=622 \[ \frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (539 a^3 B+1145 a^2 A b+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1793 a^3 b B+1025 a^2 A b^2+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (1617 a^5 B+3705 a^4 A b+3069 a^3 b^2 B+255 a^2 A b^3-110 a b^4 B+40 A b^5\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^4 d}+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 2.62, antiderivative size = 622, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {2989, 3047, 3055, 2998, 2816, 2994} \[ \frac {2 \left (1025 a^2 A b^2+675 a^4 A+1793 a^3 b B+55 a b^3 B-20 A b^4\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+539 a^3 B+825 a b^2 B+15 A b^3\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+209 a b B+113 A b^2\right ) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 (a-b) \sqrt {a+b} \left (15 a^2 b^2 (19 A-121 B)-6 a^3 b (505 A-209 B)+3 a^4 (225 A-539 B)+10 a b^3 (3 A-11 B)+40 A b^4\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^3 d}+\frac {2 (a-b) \sqrt {a+b} \left (255 a^2 A b^3+3705 a^4 A b+3069 a^3 b^2 B+1617 a^5 B-110 a b^4 B+40 A b^5\right ) \cot (c+d x) \sqrt {\frac {a (1-\sec (c+d x))}{a+b}} \sqrt {\frac {a (\sec (c+d x)+1)}{a-b}} E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \cos (c+d x)}}{\sqrt {a+b} \sqrt {\cos (c+d x)}}\right )|-\frac {a+b}{a-b}\right )}{3465 a^4 d}+\frac {2 a (11 a B+14 A b) \sin (c+d x) \sqrt {a+b \cos (c+d x)}}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a A \sin (c+d x) (a+b \cos (c+d x))^{3/2}}{11 d \cos ^{\frac {11}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2816
Rule 2989
Rule 2994
Rule 2998
Rule 3047
Rule 3055
Rubi steps
\begin {align*} \int \frac {(a+b \cos (c+d x))^{5/2} (A+B \cos (c+d x))}{\cos ^{\frac {13}{2}}(c+d x)} \, dx &=\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {2}{11} \int \frac {\sqrt {a+b \cos (c+d x)} \left (\frac {1}{2} a (14 A b+11 a B)+\frac {1}{2} \left (9 a^2 A+11 A b^2+22 a b B\right ) \cos (c+d x)+\frac {1}{2} b (6 a A+11 b B) \cos ^2(c+d x)\right )}{\cos ^{\frac {11}{2}}(c+d x)} \, dx\\ &=\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {4}{99} \int \frac {\frac {1}{4} a \left (81 a^2 A+113 A b^2+209 a b B\right )+\frac {1}{4} \left (233 a^2 A b+99 A b^3+77 a^3 B+297 a b^2 B\right ) \cos (c+d x)+\frac {3}{4} b \left (46 a A b+22 a^2 B+33 b^2 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {9}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx\\ &=\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {8 \int \frac {\frac {1}{8} a \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right )+\frac {1}{8} a \left (405 a^3 A+1531 a A b^2+1507 a^2 b B+693 b^3 B\right ) \cos (c+d x)+\frac {1}{2} a b \left (81 a^2 A+113 A b^2+209 a b B\right ) \cos ^2(c+d x)}{\cos ^{\frac {7}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{693 a}\\ &=\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {16 \int \frac {\frac {3}{16} a \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right )+\frac {1}{16} a^2 \left (5055 a^2 A b+2305 A b^3+1617 a^3 B+6655 a b^2 B\right ) \cos (c+d x)+\frac {1}{8} a b \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}\\ &=\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {32 \int \frac {\frac {3}{32} a \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right )+\frac {3}{32} a^2 \left (675 a^4 A+3315 a^2 A b^2+10 A b^4+2871 a^3 b B+1705 a b^3 B\right ) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{10395 a^3}\\ &=\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}+\frac {\left ((a-b) \left (40 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right )\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}+\frac {\left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \int \frac {1+\cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \cos (c+d x)}} \, dx}{3465 a^2}\\ &=\frac {2 (a-b) \sqrt {a+b} \left (3705 a^4 A b+255 a^2 A b^3+40 A b^5+1617 a^5 B+3069 a^3 b^2 B-110 a b^4 B\right ) \cot (c+d x) E\left (\sin ^{-1}\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}}}{3465 a^4 d}+\frac {2 (a-b) \sqrt {a+b} \left (40 A b^4+3 a^4 (225 A-539 B)-6 a^3 b (505 A-209 B)+15 a^2 b^2 (19 A-121 B)+10 a b^3 (3 A-11 B)\right ) \cot (c+d x) F\left (\sin ^{-1}\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}}}{3465 a^3 d}+\frac {2 a (14 A b+11 a B) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{99 d \cos ^{\frac {9}{2}}(c+d x)}+\frac {2 \left (81 a^2 A+113 A b^2+209 a b B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{693 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {2 \left (1145 a^2 A b+15 A b^3+539 a^3 B+825 a b^2 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a d \cos ^{\frac {5}{2}}(c+d x)}+\frac {2 \left (675 a^4 A+1025 a^2 A b^2-20 A b^4+1793 a^3 b B+55 a b^3 B\right ) \sqrt {a+b \cos (c+d x)} \sin (c+d x)}{3465 a^2 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a A (a+b \cos (c+d x))^{3/2} \sin (c+d x)}{11 d \cos ^{\frac {11}{2}}(c+d x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 6.85, size = 1640, normalized size = 2.64 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (B b^{2} \cos \left (d x + c\right )^{3} + A a^{2} + {\left (2 \, B a b + A b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (B a^{2} + 2 \, A a b\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right ) + a}}{\cos \left (d x + c\right )^{\frac {13}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [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]
________________________________________________________________________________________
maple [B] time = 1.43, size = 5373, normalized size = 8.64 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \cos \left (d x + c\right ) + A\right )} {\left (b \cos \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}{\cos \left (d x + c\right )^{\frac {13}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+b\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{13/2}} \,d x \]
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]
________________________________________________________________________________________