Optimal. Leaf size=325 \[ \frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {a \left (3 a^2+29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (3 a^2+5 b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d} \]
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Rubi [A]
time = 0.39, antiderivative size = 325, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2773, 2943,
2945, 2831, 2742, 2740, 2734, 2732} \begin {gather*} -\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac {16 a b \sec (c+d x)}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac {\left (3 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}-\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2773
Rule 2831
Rule 2943
Rule 2945
Rubi steps
\begin {align*} \int \frac {\sec ^2(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac {2 \int \frac {\sec ^2(c+d x) \left (-\frac {3 a}{2}+\frac {5}{2} b \sin (c+d x)\right )}{(a+b \sin (c+d x))^{3/2}} \, dx}{3 \left (a^2-b^2\right )}\\ &=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}+\frac {4 \int \frac {\sec ^2(c+d x) \left (\frac {1}{4} \left (3 a^2+5 b^2\right )-6 a b \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac {4 \int \frac {\frac {1}{8} b^2 \left (27 a^2+5 b^2\right )+\frac {1}{8} a b \left (3 a^2+29 b^2\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{3 \left (a^2-b^2\right )^3}\\ &=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac {\left (3 a^2+5 b^2\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx}{6 \left (a^2-b^2\right )^2}-\frac {\left (a \left (3 a^2+29 b^2\right )\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac {\left (a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{6 \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (\left (3 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}}} \, dx}{6 \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {2 b \sec (c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac {16 a b \sec (c+d x)}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {a \left (3 a^2+29 b^2\right ) E\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {a+b \sin (c+d x)}}{3 \left (a^2-b^2\right )^3 d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (3 a^2+5 b^2\right ) F\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )|\frac {2 b}{a+b}\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}{3 \left (a^2-b^2\right )^2 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}\\ \end {align*}
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Mathematica [A]
time = 1.92, size = 241, normalized size = 0.74 \begin {gather*} \frac {\frac {\left (\left (3 a^3+29 a b^2\right ) E\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )+\left (-3 a^3+3 a^2 b-5 a b^2+5 b^3\right ) F\left (\frac {1}{4} (-2 c+\pi -2 d x)|\frac {2 b}{a+b}\right )\right ) \left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2}}{(a-b)^3 (a+b)}-\frac {2 b^3 \left (a^2-b^2\right ) \cos (c+d x)+20 a b^3 \cos (c+d x) (a+b \sin (c+d x))+3 \sec (c+d x) (a+b \sin (c+d x))^2 \left (3 a^2 b+b^3-a \left (a^2+3 b^2\right ) \sin (c+d x)\right )}{\left (a^2-b^2\right )^3}}{3 d (a+b \sin (c+d x))^{3/2}} \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. \(1652\) vs.
\(2(367)=734\).
time = 12.44, size = 1653, normalized size = 5.09
method | result | size |
default | \(\text {Expression too large to display}\) | \(1653\) |
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 = 0.18, size = 993, normalized size = 3.06 \begin {gather*} \frac {{\left (\sqrt {2} {\left (6 \, a^{4} b^{2} - 23 \, a^{2} b^{4} - 15 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, \sqrt {2} {\left (6 \, a^{5} b - 23 \, a^{3} b^{3} - 15 \, a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \sqrt {2} {\left (6 \, a^{6} - 17 \, a^{4} b^{2} - 38 \, a^{2} b^{4} - 15 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right ) + {\left (\sqrt {2} {\left (6 \, a^{4} b^{2} - 23 \, a^{2} b^{4} - 15 \, b^{6}\right )} \cos \left (d x + c\right )^{3} - 2 \, \sqrt {2} {\left (6 \, a^{5} b - 23 \, a^{3} b^{3} - 15 \, a b^{5}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \sqrt {2} {\left (6 \, a^{6} - 17 \, a^{4} b^{2} - 38 \, a^{2} b^{4} - 15 \, b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-i \, b} {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right ) - 3 \, {\left (\sqrt {2} {\left (-3 i \, a^{3} b^{3} - 29 i \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + 2 \, \sqrt {2} {\left (3 i \, a^{4} b^{2} + 29 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (3 i \, a^{5} b + 32 i \, a^{3} b^{3} + 29 i \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 i \, a^{3} - 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) - 3 i \, b \sin \left (d x + c\right ) - 2 i \, a}{3 \, b}\right )\right ) - 3 \, {\left (\sqrt {2} {\left (3 i \, a^{3} b^{3} + 29 i \, a b^{5}\right )} \cos \left (d x + c\right )^{3} + 2 \, \sqrt {2} {\left (-3 i \, a^{4} b^{2} - 29 i \, a^{2} b^{4}\right )} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \sqrt {2} {\left (-3 i \, a^{5} b - 32 i \, a^{3} b^{3} - 29 i \, a b^{5}\right )} \cos \left (d x + c\right )\right )} \sqrt {-i \, b} {\rm weierstrassZeta}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (-\frac {4 \, {\left (4 \, a^{2} - 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (-8 i \, a^{3} + 9 i \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cos \left (d x + c\right ) + 3 i \, b \sin \left (d x + c\right ) + 2 i \, a}{3 \, b}\right )\right ) + 6 \, {\left (3 \, a^{4} b^{2} - 6 \, a^{2} b^{4} + 3 \, b^{6} + {\left (6 \, a^{4} b^{2} + 31 \, a^{2} b^{4} - 5 \, b^{6}\right )} \cos \left (d x + c\right )^{2} - {\left (3 \, a^{5} b - 6 \, a^{3} b^{3} + 3 \, a b^{5} - {\left (3 \, a^{3} b^{3} + 29 \, a b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{18 \, {\left ({\left (a^{6} b^{3} - 3 \, a^{4} b^{5} + 3 \, a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )^{3} - 2 \, {\left (a^{7} b^{2} - 3 \, a^{5} b^{4} + 3 \, a^{3} b^{6} - a b^{8}\right )} d \cos \left (d x + c\right ) \sin \left (d x + c\right ) - {\left (a^{8} b - 2 \, a^{6} b^{3} + 2 \, a^{2} b^{7} - b^{9}\right )} d \cos \left (d x + c\right )\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 {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\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 {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \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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