Optimal. Leaf size=425 \[ \frac {2 \left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 1.11, antiderivative size = 425, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {3034, 4110,
4179, 4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \left (49 a^2 A+135 a b B+75 A b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{315 d}+\frac {2 \left (75 a^3 B+163 a^2 A b+135 a b^2 B+5 A b^3\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}{315 a d}+\frac {2 \left (a^2-b^2\right ) \left (75 a^3 B+114 a^2 A b+45 a b^2 B-10 A b^3\right ) \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+435 a^3 b B+279 a^2 A b^2+45 a b^3 B-10 A b^4\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 a (3 a B+4 A b) \sin (c+d x) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{21 d}+\frac {2 a A \sin (c+d x) \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}{9 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3034
Rule 3941
Rule 3943
Rule 4110
Rule 4120
Rule 4179
Rule 4189
Rubi steps
\begin {align*} \int \cos ^{\frac {9}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+b \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\sec ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}-\frac {1}{9} \left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+b \sec (c+d x)} \left (-\frac {3}{2} a (4 A b+3 a B)-\frac {1}{2} \left (7 a^2 A+9 A b^2+18 a b B\right ) \sec (c+d x)-\frac {1}{2} b (4 a A+9 b B) \sec ^2(c+d x)\right )}{\sec ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}-\frac {1}{63} \left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {1}{4} a \left (49 a^2 A+75 A b^2+135 a b B\right )-\frac {1}{4} \left (137 a^2 A b+63 A b^3+45 a^3 B+189 a b^2 B\right ) \sec (c+d x)-\frac {1}{4} b \left (76 a A b+36 a^2 B+63 b^2 B\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx\\ &=\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {3}{8} a \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right )+\frac {1}{8} a \left (147 a^3 A+605 a A b^2+585 a^2 b B+315 b^3 B\right ) \sec (c+d x)+\frac {1}{4} a b \left (49 a^2 A+75 A b^2+135 a b B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{315 a}\\ &=\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}-\frac {\left (16 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {-\frac {3}{16} a \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right )-\frac {3}{16} a^2 \left (261 a^2 A b+155 A b^3+75 a^3 B+405 a b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{945 a^2}\\ &=\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\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}{315 a^2}+\frac {\left (\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\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}{315 a^2}\\ &=\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{315 a^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{315 a^2 \sqrt {b+a \cos (c+d x)}}\\ &=\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}+\frac {\left (\left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\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}{315 a^2 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\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}{315 a^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}\\ &=\frac {2 \left (a^2-b^2\right ) \left (114 a^2 A b-10 A b^3+75 a^3 B+45 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{315 a^2 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\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)}}{315 a^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 \left (163 a^2 A b+5 A b^3+75 a^3 B+135 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 a d}+\frac {2 \left (49 a^2 A+75 A b^2+135 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{315 d}+\frac {2 a (4 A b+3 a B) \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{21 d}+\frac {2 a A \cos ^{\frac {7}{2}}(c+d x) (a+b \sec (c+d x))^{3/2} \sin (c+d x)}{9 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 19.00, size = 542, normalized size = 1.28 \begin {gather*} \frac {\cos ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2} \left (\frac {\left (747 a^2 A b+20 A b^3+345 a^3 B+540 a b^2 B\right ) \sin (c+d x)}{630 a}+\frac {1}{630} \left (133 a^2 A+150 A b^2+270 a b B\right ) \sin (2 (c+d x))+\frac {1}{126} a (19 A b+9 a B) \sin (3 (c+d x))+\frac {1}{36} a^2 A \sin (4 (c+d x))\right )}{d (b+a \cos (c+d x))^2}-\frac {2 \cos ^{\frac {3}{2}}(c+d x) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} (a+b \sec (c+d x))^{5/2} \left (-i (a+b) \left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) E\left (i \sinh ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}+i a (a+b) \left (-10 A b^3+15 a b^2 (11 A+3 B)+3 a^3 (49 A+25 B)+6 a^2 b (19 A+60 B)\right ) F\left (i \sinh ^{-1}\left (\tan \left (\frac {1}{2} (c+d x)\right )\right )|\frac {-a+b}{a+b}\right ) \sec ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {\frac {(b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )}{a+b}}-\left (147 a^4 A+279 a^2 A b^2-10 A b^4+435 a^3 b B+45 a b^3 B\right ) (b+a \cos (c+d x)) \sec ^2\left (\frac {1}{2} (c+d x)\right )^{3/2} \tan \left (\frac {1}{2} (c+d x)\right )\right )}{315 a^2 d (b+a \cos (c+d x))^3 \sec ^{\frac {5}{2}}(c+d x)} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(3068\) vs.
\(2(443)=886\).
time = 21.03, size = 3069, normalized size = 7.22
method | result | size |
default | \(\text {Expression too large to display}\) | \(3069\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.88, size = 656, normalized size = 1.54 \begin {gather*} \frac {6 \, {\left (35 \, A a^{5} \cos \left (d x + c\right )^{3} + 75 \, B a^{5} + 163 \, A a^{4} b + 135 \, B a^{3} b^{2} + 5 \, A a^{2} b^{3} + 5 \, {\left (9 \, B a^{5} + 19 \, A a^{4} b\right )} \cos \left (d x + c\right )^{2} + {\left (49 \, A a^{5} + 135 \, B a^{4} b + 75 \, A a^{3} b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) + \sqrt {2} {\left (-225 i \, B a^{5} - 489 i \, A a^{4} b - 345 i \, B a^{3} b^{2} + 93 i \, A a^{2} b^{3} + 90 i \, B a b^{4} - 20 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) + \sqrt {2} {\left (225 i \, B a^{5} + 489 i \, A a^{4} b + 345 i \, B a^{3} b^{2} - 93 i \, A a^{2} b^{3} - 90 i \, B a b^{4} + 20 i \, A b^{5}\right )} \sqrt {a} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right ) - 3 \, \sqrt {2} {\left (-147 i \, A a^{5} - 435 i \, B a^{4} b - 279 i \, A a^{3} b^{2} - 45 i \, B a^{2} b^{3} + 10 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) + 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right ) - 3 \, \sqrt {2} {\left (147 i \, A a^{5} + 435 i \, B a^{4} b + 279 i \, A a^{3} b^{2} + 45 i \, B a^{2} b^{3} - 10 i \, A a b^{4}\right )} \sqrt {a} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (3 \, a^{2} - 4 \, b^{2}\right )}}{3 \, a^{2}}, \frac {8 \, {\left (9 \, a^{2} b - 8 \, b^{3}\right )}}{27 \, a^{3}}, \frac {3 \, a \cos \left (d x + c\right ) - 3 i \, a \sin \left (d x + c\right ) + 2 \, b}{3 \, a}\right )\right )}{945 \, a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^{9/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________