Optimal. Leaf size=425 \[ -\frac {\sec ^3(c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (29 a^2+3 b^2\right )-a \left (a^2+31 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac {8 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}+\frac {2 b \sec ^3(c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}-\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (a^4-114 a^2 b^2-15 b^4\right )-4 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sin (c+d x)\right )}{6 d \left (a^2-b^2\right )^4}+\frac {\left (4 a^4-21 a^2 b^2-15 b^4\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{6 d \left (a^2-b^2\right )^3 \sqrt {a+b \sin (c+d x)}}-\frac {2 a \left (a^4-6 a^2 b^2-27 b^4\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^4 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.88, antiderivative size = 425, 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 = {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 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2694
Rule 2752
Rule 2864
Rule 2866
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*}
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Mathematica [A] time = 2.50, size = 341, normalized size = 0.80 \[ \frac {\frac {2 \left (a^2-b^2\right ) \sec ^3(c+d x) (a+b \sin (c+d x))^2 \left (a \left (a^2+3 b^2\right ) \sin (c+d x)-b \left (3 a^2+b^2\right )\right )+4 b^5 \left (a^2-b^2\right ) \cos (c+d x)+\sec (c+d x) (a+b \sin (c+d x))^2 \left (-a^4 b+54 a^2 b^3+4 a \left (a^4-6 a^2 b^2-11 b^4\right ) \sin (c+d x)+11 b^5\right )+64 a b^5 \cos (c+d x) (a+b \sin (c+d x))}{\left (a^2-b^2\right )^4}+\frac {\left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2} \left (4 \left (a^5-6 a^3 b^2-27 a b^4\right ) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\left (-4 a^5+4 a^4 b+21 a^3 b^2-21 a^2 b^3+15 a b^4-15 b^5\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )}{(a-b)^4 (a+b)^2}}{6 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{4}}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{4}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 5.84, size = 2585, normalized size = 6.08 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [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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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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