Optimal. Leaf size=425 \[ -\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac{8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right )^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (-114 a^2 b^2+a^4-15 b^4\right )-4 a \left (-6 a^2 b^2+a^4-27 b^4\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^4}+\frac{\left (-21 a^2 b^2+4 a^4-15 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{6 d \left (a^2-b^2\right )^3 \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left (-6 a^2 b^2+a^4-27 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.87617, antiderivative size = 425, normalized size of antiderivative = 1., 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 = {2694, 2864, 2866, 2752, 2663, 2661, 2655, 2653} \[ -\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac{8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right )^2 \sqrt{a+b \sin (c+d x)}}+\frac{2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (-114 a^2 b^2+a^4-15 b^4\right )-4 a \left (-6 a^2 b^2+a^4-27 b^4\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^4}+\frac{\left (-21 a^2 b^2+4 a^4-15 b^4\right ) \sqrt{\frac{a+b \sin (c+d x)}{a+b}} F\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{6 d \left (a^2-b^2\right )^3 \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left (-6 a^2 b^2+a^4-27 b^4\right ) \sqrt{a+b \sin (c+d x)} E\left (\frac{1}{2} \left (c+d x-\frac{\pi }{2}\right )|\frac{2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}} \]
Antiderivative was successfully verified.
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Rule 2694
Rule 2864
Rule 2866
Rule 2752
Rule 2663
Rule 2661
Rule 2655
Rule 2653
Rubi steps
\begin{align*} \int \frac{\sec ^4(c+d x)}{(a+b \sin (c+d x))^{5/2}} \, dx &=\frac{2 b \sec ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}-\frac{2 \int \frac{\sec ^4(c+d x) \left (-\frac{3 a}{2}+\frac{9}{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 ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \sin (c+d x)}}+\frac{4 \int \frac{\sec ^4(c+d x) \left (\frac{3}{4} \left (a^2+3 b^2\right )-21 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 ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{4 \int \frac{\sec ^2(c+d x) \left (-\frac{3}{8} \left (4 a^4-21 a^2 b^2-15 b^4\right )-\frac{9}{8} a b \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{\sqrt{a+b \sin (c+d x)}} \, dx}{9 \left (a^2-b^2\right )^3}\\ &=\frac{2 b \sec ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^4 d}+\frac{4 \int \frac{-\frac{3}{16} b^2 \left (a^4-114 a^2 b^2-15 b^4\right )-\frac{3}{4} a b \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)}{\sqrt{a+b \sin (c+d x)}} \, dx}{9 \left (a^2-b^2\right )^4}\\ &=\frac{2 b \sec ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^4 d}-\frac{\left (a \left (a^4-6 a^2 b^2-27 b^4\right )\right ) \int \sqrt{a+b \sin (c+d x)} \, dx}{3 \left (a^2-b^2\right )^4}+\frac{\left (4 a^4-21 a^2 b^2-15 b^4\right ) \int \frac{1}{\sqrt{a+b \sin (c+d x)}} \, dx}{12 \left (a^2-b^2\right )^3}\\ &=\frac{2 b \sec ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^4 d}-\frac{\left (a \left (a^4-6 a^2 b^2-27 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}{3 \left (a^2-b^2\right )^4 \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (\left (4 a^4-21 a^2 b^2-15 b^4\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}{12 \left (a^2-b^2\right )^3 \sqrt{a+b \sin (c+d x)}}\\ &=\frac{2 b \sec ^3(c+d x)}{3 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^{3/2}}+\frac{8 a b \sec ^3(c+d x)}{\left (a^2-b^2\right )^2 d \sqrt{a+b \sin (c+d x)}}-\frac{2 a \left (a^4-6 a^2 b^2-27 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)}}{3 \left (a^2-b^2\right )^4 d \sqrt{\frac{a+b \sin (c+d x)}{a+b}}}+\frac{\left (4 a^4-21 a^2 b^2-15 b^4\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}}}{6 \left (a^2-b^2\right )^3 d \sqrt{a+b \sin (c+d x)}}-\frac{\sec ^3(c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{\sec (c+d x) \sqrt{a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^4 d}\\ \end{align*}
Mathematica [A] time = 2.4375, size = 341, normalized size = 0.8 \[ \frac{\frac{4 b^5 \left (a^2-b^2\right ) \cos (c+d x)+2 \left (a^2-b^2\right ) \sec ^3(c+d x) (a+b \sin (c+d x))^2 \left (a \left (a^2+3 b^2\right ) \sin (c+d x)-b \left (3 a^2+b^2\right )\right )+\sec (c+d x) (a+b \sin (c+d x))^2 \left (4 a \left (-6 a^2 b^2+a^4-11 b^4\right ) \sin (c+d x)+54 a^2 b^3-a^4 b+11 b^5\right )+64 a b^5 \cos (c+d x) (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}+\frac{\left (\frac{a+b \sin (c+d x)}{a+b}\right )^{3/2} \left (\left (21 a^3 b^2-21 a^2 b^3+4 a^4 b-4 a^5+15 a b^4-15 b^5\right ) F\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )+4 \left (-6 a^3 b^2+a^5-27 a b^4\right ) E\left (\frac{1}{4} (-2 c-2 d x+\pi )|\frac{2 b}{a+b}\right )\right )}{(a-b)^4 (a+b)^2}}{6 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 4.317, size = 2585, normalized size = 6.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4}}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} +{\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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