Optimal. Leaf size=423 \[ -\frac {2 \left (12 a^2 A b+48 A b^3-5 a^3 B-40 a b^2 B\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{15 a^4 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^4 A+24 a^2 A b^2-48 A b^4-25 a^3 b B+40 a b^3 B\right ) \sqrt {\cos (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{15 a^4 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (9 a^2 A b-24 A b^3-5 a^3 B+20 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d} \]
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Rubi [A]
time = 0.91, antiderivative size = 423, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3034, 4115,
4189, 4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} \frac {2 \left (a^2 A+5 a b B-6 A b^2\right ) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}}{5 a^2 d \left (a^2-b^2\right )}+\frac {2 b (A b-a B) \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (-5 a^3 B+9 a^2 A b+20 a b^2 B-24 A b^3\right ) \sin (c+d x) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}{15 a^3 d \left (a^2-b^2\right )}-\frac {2 \left (-5 a^3 B+12 a^2 A b-40 a b^2 B+48 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 )}{15 a^4 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^4 A-25 a^3 b B+24 a^2 A b^2+40 a b^3 B-48 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 )}{15 a^4 d \left (a^2-b^2\right ) \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 3034
Rule 3941
Rule 3943
Rule 4115
Rule 4120
Rule 4189
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {5}{2}}(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {A+B \sec (c+d x)}{\sec ^{\frac {5}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}} \, dx\\ &=\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {\left (2 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{2} \left (-a^2 A+6 A b^2-5 a b B\right )+\frac {1}{2} a (A b-a B) \sec (c+d x)-2 b (A b-a B) \sec ^2(c+d x)}{\sec ^{\frac {5}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}+\frac {\left (4 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{4} \left (-9 a^2 A b+24 A b^3+5 a^3 B-20 a b^2 B\right )+\frac {1}{4} a \left (3 a^2 A+2 A b^2-5 a b B\right ) \sec (c+d x)+\frac {1}{2} b \left (a^2 A-6 A b^2+5 a b B\right ) \sec ^2(c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)}} \, dx}{5 a^2 \left (a^2-b^2\right )}\\ &=\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (9 a^2 A b-24 A b^3-5 a^3 B+20 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac {\left (8 \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\frac {1}{8} \left (-9 a^4 A-24 a^2 A b^2+48 A b^4+25 a^3 b B-40 a b^3 B\right )+\frac {1}{8} a \left (3 a^2 A b+12 A b^3-5 a^3 B-10 a b^2 B\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{15 a^3 \left (a^2-b^2\right )}\\ &=\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (9 a^2 A b-24 A b^3-5 a^3 B+20 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac {\left (\left (12 a^2 A b+48 A b^3-5 a^3 B-40 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}{15 a^4}-\frac {\left (\left (-9 a^4 A-24 a^2 A b^2+48 A b^4+25 a^3 b B-40 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}{15 a^4 \left (a^2-b^2\right )}\\ &=\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (9 a^2 A b-24 A b^3-5 a^3 B+20 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac {\left (\left (12 a^2 A b+48 A b^3-5 a^3 B-40 a b^2 B\right ) \sqrt {b+a \cos (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{15 a^4 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (-9 a^4 A-24 a^2 A b^2+48 A b^4+25 a^3 b B-40 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}{15 a^4 \left (a^2-b^2\right ) \sqrt {b+a \cos (c+d x)}}\\ &=\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (9 a^2 A b-24 A b^3-5 a^3 B+20 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}-\frac {\left (\left (12 a^2 A b+48 A b^3-5 a^3 B-40 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}{15 a^4 \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}-\frac {\left (\left (-9 a^4 A-24 a^2 A b^2+48 A b^4+25 a^3 b B-40 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}{15 a^4 \left (a^2-b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}\\ &=-\frac {2 \left (12 a^2 A b+48 A b^3-5 a^3 B-40 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 )}{15 a^4 d \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)}}+\frac {2 \left (9 a^4 A+24 a^2 A b^2-48 A b^4-25 a^3 b B+40 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)}}{15 a^4 \left (a^2-b^2\right ) d \sqrt {\frac {b+a \cos (c+d x)}{a+b}}}+\frac {2 b (A b-a B) \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}-\frac {2 \left (9 a^2 A b-24 A b^3-5 a^3 B+20 a b^2 B\right ) \sqrt {\cos (c+d x)} \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{15 a^3 \left (a^2-b^2\right ) d}+\frac {2 \left (a^2 A-6 A b^2+5 a b B\right ) \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+b \sec (c+d x)} \sin (c+d x)}{5 a^2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 15.23, size = 533, normalized size = 1.26 \begin {gather*} \frac {(b+a \cos (c+d x))^2 \left (\frac {2 (-9 A b+5 a B) \sin (c+d x)}{15 a^3}+\frac {2 \left (A b^4 \sin (c+d x)-a b^3 B \sin (c+d x)\right )}{a^3 \left (a^2-b^2\right ) (b+a \cos (c+d x))}+\frac {A \sin (2 (c+d x))}{5 a^2}\right )}{d \cos ^{\frac {3}{2}}(c+d x) (a+b \sec (c+d x))^{3/2}}-\frac {2 \cos ^{\frac {3}{2}}(c+d x) (b+a \cos (c+d x)) \sec ^{\frac {3}{2}}(c+d x) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{3/2} \left (-i (a+b) \left (9 a^4 A+24 a^2 A b^2-48 A b^4-25 a^3 b B+40 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 (-48 A b^3-6 a^2 b (2 A+5 B)+a^3 (9 A+5 B)+4 a b^2 (9 A+10 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 (9 a^4 A+24 a^2 A b^2-48 A b^4-25 a^3 b B+40 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 )}{15 a^4 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}} \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. \(2083\) vs.
\(2(449)=898\).
time = 37.36, size = 2084, normalized size = 4.93
method | result | size |
default | \(\text {Expression too large to display}\) | \(2084\) |
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 = 2.01, size = 928, normalized size = 2.19 \begin {gather*} \frac {6 \, {\left (5 \, B a^{5} b - 9 \, A a^{4} b^{2} - 20 \, B a^{3} b^{3} + 24 \, A a^{2} b^{4} + 3 \, {\left (A a^{6} - A a^{4} b^{2}\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, B a^{6} - 6 \, A a^{5} b - 5 \, B a^{4} b^{2} + 6 \, A a^{3} b^{3}\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 ) - {\left (\sqrt {2} {\left (15 i \, B a^{6} - 27 i \, A a^{5} b + 80 i \, B a^{4} b^{2} - 84 i \, A a^{3} b^{3} - 80 i \, B a^{2} b^{4} + 96 i \, A a b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (15 i \, B a^{5} b - 27 i \, A a^{4} b^{2} + 80 i \, B a^{3} b^{3} - 84 i \, A a^{2} b^{4} - 80 i \, B a b^{5} + 96 i \, A b^{6}\right )}\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 ) - {\left (\sqrt {2} {\left (-15 i \, B a^{6} + 27 i \, A a^{5} b - 80 i \, B a^{4} b^{2} + 84 i \, A a^{3} b^{3} + 80 i \, B a^{2} b^{4} - 96 i \, A a b^{5}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-15 i \, B a^{5} b + 27 i \, A a^{4} b^{2} - 80 i \, B a^{3} b^{3} + 84 i \, A a^{2} b^{4} + 80 i \, B a b^{5} - 96 i \, A b^{6}\right )}\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 \, {\left (\sqrt {2} {\left (9 i \, A a^{6} - 25 i \, B a^{5} b + 24 i \, A a^{4} b^{2} + 40 i \, B a^{3} b^{3} - 48 i \, A a^{2} b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (9 i \, A a^{5} b - 25 i \, B a^{4} b^{2} + 24 i \, A a^{3} b^{3} + 40 i \, B a^{2} b^{4} - 48 i \, A a b^{5}\right )}\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 \, {\left (\sqrt {2} {\left (-9 i \, A a^{6} + 25 i \, B a^{5} b - 24 i \, A a^{4} b^{2} - 40 i \, B a^{3} b^{3} + 48 i \, A a^{2} b^{4}\right )} \cos \left (d x + c\right ) + \sqrt {2} {\left (-9 i \, A a^{5} b + 25 i \, B a^{4} b^{2} - 24 i \, A a^{3} b^{3} - 40 i \, B a^{2} b^{4} + 48 i \, A a b^{5}\right )}\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 )}{45 \, {\left ({\left (a^{8} - a^{6} b^{2}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b - a^{5} b^{3}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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 {{\cos \left (c+d\,x\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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