Optimal. Leaf size=663 \[ \frac {2 \left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}} \]
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Rubi [A]
time = 1.51, antiderivative size = 663, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 9, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4185, 4189,
4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \sin (c+d x) \left (A b^2-a (b B-a C)\right )}{3 a d \left (a^2-b^2\right ) \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}+\frac {2 \sin (c+d x) \left (a^4 (3 A-35 C)+50 a^3 b B-a^2 b^2 (71 A-15 C)-30 a b^3 B+48 A b^4\right ) \sqrt {a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \sin (c+d x) \left (-6 a^4 C+9 a^3 b B-2 a^2 b^2 (6 A-C)-5 a b^3 B+8 A b^4\right )}{3 a^2 d \left (a^2-b^2\right )^2 \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}-\frac {2 \sin (c+d x) \left (-5 a^5 B+2 a^4 b (7 A-20 C)+65 a^3 b^2 B-2 a^2 b^3 (49 A-10 C)-40 a b^4 B+64 A b^5\right ) \sqrt {a+b \sec (c+d x)}}{15 a^4 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)}}+\frac {2 \sqrt {\sec (c+d x)} \left (5 a^5 B-a^4 b (17 A+45 C)+80 a^3 b^2 B-4 a^2 b^3 (29 A-10 C)-80 a b^4 B+128 A b^5\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 )}{15 a^5 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (3 a^6 (3 A+5 C)-40 a^5 b B+5 a^4 b^2 (11 A-15 C)+140 a^3 b^3 B-4 a^2 b^4 (53 A-10 C)-80 a b^5 B+128 A b^6\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{15 a^5 d \left (a^2-b^2\right )^2 \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 3941
Rule 3943
Rule 4120
Rule 4185
Rule 4189
Rubi steps
\begin {align*} \int \frac {A+B \sec (c+d x)+C \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{5/2}} \, dx &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {\frac {1}{2} \left (8 A b^2-5 a b B-a^2 (3 A-5 C)\right )+\frac {3}{2} a (A b-a B+b C) \sec (c+d x)-3 \left (A b^2-a (b B-a C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx}{3 a \left (a^2-b^2\right )}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {4 \int \frac {\frac {1}{4} \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right )+\frac {1}{4} a \left (2 A b^3+3 a^3 B+a b^2 B-2 a^2 b (3 A+2 C)\right ) \sec (c+d x)-\left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {8 \int \frac {\frac {3}{8} \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right )+\frac {1}{8} a \left (16 A b^4+30 a^3 b B-10 a b^3 B-3 a^4 (3 A+5 C)-a^2 b^2 (27 A+5 C)\right ) \sec (c+d x)-\frac {1}{4} b \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )^2}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {16 \int \frac {\frac {3}{16} \left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right )+\frac {3}{16} a \left (32 A b^5+5 a^5 B+35 a^3 b^2 B-20 a b^4 B-2 a^2 b^3 (22 A-5 C)-2 a^4 b (4 A+15 C)\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{45 a^4 \left (a^2-b^2\right )^2}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )^2}+\frac {\left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right )}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}+\frac {\left (\left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{15 a^5 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{15 a^5 \left (a^2-b^2\right )^2 \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}\\ &=\frac {2 \left (128 A b^5+5 a^5 B+80 a^3 b^2 B-80 a b^4 B-4 a^2 b^3 (29 A-10 C)-a^4 b (17 A+45 C)\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 ) \sqrt {\sec (c+d x)}}{15 a^5 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (128 A b^6-40 a^5 b B+140 a^3 b^3 B-80 a b^5 B+5 a^4 b^2 (11 A-15 C)-4 a^2 b^4 (53 A-10 C)+3 a^6 (3 A+5 C)\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^5 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}+\frac {2 \left (A b^2-a (b B-a C)\right ) \sin (c+d x)}{3 a \left (a^2-b^2\right ) d \sec ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \left (8 A b^4+9 a^3 b B-5 a b^3 B-2 a^2 b^2 (6 A-C)-6 a^4 C\right ) \sin (c+d x)}{3 a^2 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (48 A b^4+50 a^3 b B-30 a b^3 B+a^4 (3 A-35 C)-a^2 b^2 (71 A-15 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right )^2 d \sec ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (64 A b^5-5 a^5 B+65 a^3 b^2 B-40 a b^4 B+2 a^4 b (7 A-20 C)-2 a^2 b^3 (49 A-10 C)\right ) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^4 \left (a^2-b^2\right )^2 d \sqrt {\sec (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 10.65, size = 9192, normalized size = 13.86 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(11336\) vs.
\(2(681)=1362\).
time = 0.52, size = 11337, normalized size = 17.10
method | result | size |
default | \(\text {Expression too large to display}\) | \(11337\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.87, size = 1771, normalized size = 2.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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