Optimal. Leaf size=302 \[ \frac {2 \left (3 a^2-2 b^2\right ) \sqrt {\frac {b+a \cos (c+d x)}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {4 b \left (3 a^2-b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^2 \left (a^2-b^2\right )^2 d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}}-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}} \]
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Rubi [A]
time = 0.44, antiderivative size = 302, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3928, 4185,
4120, 3941, 2734, 2732, 3943, 2742, 2740} \begin {gather*} -\frac {2 b \left (5 a^2-b^2\right ) \sin (c+d x) \sqrt {\sec (c+d x)}}{3 a d \left (a^2-b^2\right )^2 \sqrt {a+b \sec (c+d x)}}-\frac {2 b \sin (c+d x) \sqrt {\sec (c+d x)}}{3 d \left (a^2-b^2\right ) (a+b \sec (c+d x))^{3/2}}+\frac {2 \left (3 a^2-2 b^2\right ) \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^2 d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {4 b \left (3 a^2-b^2\right ) \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{3 a^2 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 3928
Rule 3941
Rule 3943
Rule 4120
Rule 4185
Rubi steps
\begin {align*} \int \frac {\sqrt {\sec (c+d x)}}{(a+b \sec (c+d x))^{5/2}} \, dx &=-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 \int \frac {-\frac {b}{2}-\frac {3}{2} a \sec (c+d x)+b \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} (a+b \sec (c+d x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )}\\ &=-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {4 \int \frac {\frac {1}{2} b \left (3 a^2-b^2\right )+\frac {1}{4} a \left (3 a^2+b^2\right ) \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx}{3 a \left (a^2-b^2\right )^2}\\ &=-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (3 a^2-2 b^2\right ) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )}+\frac {\left (2 b \left (3 a^2-b^2\right )\right ) \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{3 a^2 \left (a^2-b^2\right )^2}\\ &=-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (3 a^2-2 b^2\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}{3 a^2 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (2 b \left (3 a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )^2 \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}}\\ &=-\frac {2 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}+\frac {\left (\left (3 a^2-2 b^2\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}{3 a^2 \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}+\frac {\left (2 b \left (3 a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{3 a^2 \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 (3 a^2-2 b^2\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)}}{3 a^2 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}}+\frac {4 b \left (3 a^2-b^2\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{3 a^2 \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 b \sqrt {\sec (c+d x)} \sin (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sec (c+d x))^{3/2}}-\frac {2 b \left (5 a^2-b^2\right ) \sqrt {\sec (c+d x)} \sin (c+d x)}{3 a \left (a^2-b^2\right )^2 d \sqrt {a+b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.84, size = 196, normalized size = 0.65 \begin {gather*} \frac {2 (b+a \cos (c+d x)) \sec ^{\frac {5}{2}}(c+d x) \left (\frac {\left (\frac {b+a \cos (c+d x)}{a+b}\right )^{3/2} \left (\left (6 a^2 b-2 b^3\right ) E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )+\left (3 a^3-3 a^2 b-2 a b^2+2 b^3\right ) F\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )\right )}{(a-b)^2}+\frac {a b \left (-5 a^2 b+b^3+\left (-6 a^3+2 a b^2\right ) \cos (c+d x)\right ) \sin (c+d x)}{\left (a^2-b^2\right )^2}\right )}{3 a^2 d (a+b \sec (c+d x))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(2069\) vs.
\(2(332)=664\).
time = 0.20, size = 2070, normalized size = 6.85
method | result | size |
default | \(\text {Expression too large to display}\) | \(2070\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.86, size = 811, normalized size = 2.69 \begin {gather*} \frac {\sqrt {2} {\left (-9 i \, a^{4} b^{2} + 9 i \, a^{2} b^{4} - 4 i \, b^{6} + {\left (-9 i \, a^{6} + 9 i \, a^{4} b^{2} - 4 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (9 i \, a^{5} b - 9 i \, a^{3} b^{3} + 4 i \, a b^{5}\right )} \cos \left (d x + c\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 ) + \sqrt {2} {\left (9 i \, a^{4} b^{2} - 9 i \, a^{2} b^{4} + 4 i \, b^{6} + {\left (9 i \, a^{6} - 9 i \, a^{4} b^{2} + 4 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left (-9 i \, a^{5} b + 9 i \, a^{3} b^{3} - 4 i \, a b^{5}\right )} \cos \left (d x + c\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 ) - 6 \, \sqrt {2} {\left (-3 i \, a^{3} b^{3} + i \, a b^{5} + {\left (-3 i \, a^{5} b + i \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (-3 i \, a^{4} b^{2} + i \, a^{2} b^{4}\right )} \cos \left (d x + c\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 ) - 6 \, \sqrt {2} {\left (3 i \, a^{3} b^{3} - i \, a b^{5} + {\left (3 i \, a^{5} b - i \, a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 i \, a^{4} b^{2} - i \, a^{2} b^{4}\right )} \cos \left (d x + c\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 ) - \frac {6 \, {\left (2 \, {\left (3 \, a^{5} b - a^{3} b^{3}\right )} \cos \left (d x + c\right )^{2} + {\left (5 \, a^{4} b^{2} - a^{2} b^{4}\right )} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{9 \, {\left ({\left (a^{9} - 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{8} b - 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} d \cos \left (d x + c\right ) + {\left (a^{7} b^{2} - 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {\sec {\left (c + d x \right )}}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \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 {\sqrt {\frac {1}{\cos \left (c+d\,x\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.
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