Optimal. Leaf size=439 \[ \frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-b^2\right ) \sin (c+d x)+3 a b\right )}{70 d}+\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 d \sqrt {a+b \sin (c+d x)}}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 d \left (a^2-b^2\right )}-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{280 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 d \left (a^2-b^2\right )^2}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{7 d} \]
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Rubi [A] time = 0.94, antiderivative size = 439, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2691, 2861, 2866, 2752, 2663, 2661, 2655, 2653} \[ \frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (\left (4 a^2-b^2\right ) \sin (c+d x)+3 a b\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (-39 a^2 b^2+32 a^4+7 b^4\right ) \sin (c+d x)\right )}{140 d \left (a^2-b^2\right )}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (-59 a^2 b^2+32 a^4+27 b^4\right )-\left (-272 a^4 b^2+165 a^2 b^4+128 a^6-21 b^6\right ) \sin (c+d x)\right )}{280 d \left (a^2-b^2\right )^2}+\frac {2 a \left (8 a^2-3 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{35 d \sqrt {a+b \sin (c+d x)}}-\frac {\left (-144 a^2 b^2+128 a^4+21 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{280 d \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\sec ^7(c+d x) (a \sin (c+d x)+b) (a+b \sin (c+d x))^{3/2}}{7 d} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2691
Rule 2752
Rule 2861
Rule 2866
Rubi steps
\begin {align*} \int \sec ^8(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}-\frac {1}{7} \int \sec ^6(c+d x) \sqrt {a+b \sin (c+d x)} \left (-6 a^2+\frac {3 b^2}{2}-\frac {9}{2} a b \sin (c+d x)\right ) \, dx\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}+\frac {1}{35} \int \frac {\sec ^4(c+d x) \left (\frac {3}{4} a \left (32 a^2-11 b^2\right )+\frac {21}{4} b \left (4 a^2-b^2\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\int \frac {\sec ^2(c+d x) \left (-6 a \left (8 a^4-11 a^2 b^2+3 b^4\right )-\frac {9}{8} b \left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 \left (a^2-b^2\right )}\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}+\frac {\int \frac {-\frac {3}{16} a b^2 \left (32 a^4-59 a^2 b^2+27 b^4\right )-\frac {3}{16} b \left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)}{\sqrt {a+b \sin (c+d x)}} \, dx}{105 \left (a^2-b^2\right )^2}\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}+\frac {1}{35} \left (a \left (8 a^2-3 b^2\right )\right ) \int \frac {1}{\sqrt {a+b \sin (c+d x)}} \, dx-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\right ) \int \sqrt {a+b \sin (c+d x)} \, dx}{560 \left (a^2-b^2\right )}\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}-\frac {\left (\left (128 a^4-144 a^2 b^2+21 b^4\right ) \sqrt {a+b \sin (c+d x)}\right ) \int \sqrt {\frac {a}{a+b}+\frac {b \sin (c+d x)}{a+b}} \, dx}{560 \left (a^2-b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {\left (a \left (8 a^2-3 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}{35 \sqrt {a+b \sin (c+d x)}}\\ &=\frac {\sec ^7(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))^{3/2}}{7 d}-\frac {\left (128 a^4-144 a^2 b^2+21 b^4\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)}}{280 \left (a^2-b^2\right ) d \sqrt {\frac {a+b \sin (c+d x)}{a+b}}}+\frac {2 a \left (8 a^2-3 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}}}{35 d \sqrt {a+b \sin (c+d x)}}+\frac {3 \sec ^5(c+d x) \sqrt {a+b \sin (c+d x)} \left (3 a b+\left (4 a^2-b^2\right ) \sin (c+d x)\right )}{70 d}-\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (4 a b \left (a^2-b^2\right )-\left (32 a^4-39 a^2 b^2+7 b^4\right ) \sin (c+d x)\right )}{140 \left (a^2-b^2\right ) d}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (a b \left (32 a^4-59 a^2 b^2+27 b^4\right )-\left (128 a^6-272 a^4 b^2+165 a^2 b^4-21 b^6\right ) \sin (c+d x)\right )}{280 \left (a^2-b^2\right )^2 d}\\ \end {align*}
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Mathematica [A] time = 4.44, size = 338, normalized size = 0.77 \[ \frac {\frac {\sec (c+d x) (a+b \sin (c+d x)) \left (128 a^4 \sin (c+d x)-32 a^3 b-144 a^2 b^2 \sin (c+d x)+40 \left (a^2-b^2\right ) \sec ^6(c+d x) \left (\left (a^2+b^2\right ) \sin (c+d x)+2 a b\right )-4 \left (a^2-b^2\right ) \sec ^4(c+d x) \left (3 \left (b^2-4 a^2\right ) \sin (c+d x)+a b\right )+2 \left (a^2-b^2\right ) \sec ^2(c+d x) \left (\left (32 a^2-7 b^2\right ) \sin (c+d x)-4 a b\right )+27 a b^3+21 b^4 \sin (c+d x)\right )}{a^2-b^2}+\frac {\sqrt {\frac {a+b \sin (c+d x)}{a+b}} \left (\left (128 a^4-144 a^2 b^2+21 b^4\right ) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )-16 a \left (8 a^3-8 a^2 b-3 a b^2+3 b^3\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )}{a-b}}{280 d \sqrt {a+b \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (2 \, a b \sec \left (d x + c\right )^{8} \sin \left (d x + c\right ) - {\left (b^{2} \cos \left (d x + c\right )^{2} - a^{2} - b^{2}\right )} \sec \left (d x + c\right )^{8}\right )} \sqrt {b \sin \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 8.63, size = 1888, normalized size = 4.30 \[ \text {result too large to display} \]
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 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{8}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F(-1)] time = 0.00, size = -1, normalized size = -0.00 \[ \text {Hanged} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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