Optimal. Leaf size=325 \[ -\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac {16 a b \sec (c+d x)}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac {\left (3 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}-\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
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Rubi [A] time = 0.61, 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 = {2694, 2864, 2866, 2752, 2663, 2661, 2655, 2653} \[ -\frac {\sec (c+d x) \sqrt {a+b \sin (c+d x)} \left (b \left (27 a^2+5 b^2\right )-a \left (3 a^2+29 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}+\frac {16 a b \sec (c+d x)}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}+\frac {2 b \sec (c+d x)}{3 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^{3/2}}+\frac {\left (3 a^2+5 b^2\right ) \sqrt {\frac {a+b \sin (c+d x)}{a+b}} F\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^2 \sqrt {a+b \sin (c+d x)}}-\frac {a \left (3 a^2+29 b^2\right ) \sqrt {a+b \sin (c+d x)} E\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )|\frac {2 b}{a+b}\right )}{3 d \left (a^2-b^2\right )^3 \sqrt {\frac {a+b \sin (c+d x)}{a+b}}} \]
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 ^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 = 1.89, size = 241, normalized size = 0.74 \[ \frac {\frac {\left (\frac {a+b \sin (c+d x)}{a+b}\right )^{3/2} \left (\left (3 a^3+29 a b^2\right ) E\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )+\left (-3 a^3+3 a^2 b-5 a b^2+5 b^3\right ) F\left (\frac {1}{4} (-2 c-2 d x+\pi )|\frac {2 b}{a+b}\right )\right )}{(a-b)^3 (a+b)}-\frac {2 b^3 \left (a^2-b^2\right ) \cos (c+d x)+3 \sec (c+d x) (a+b \sin (c+d x))^2 \left (-a \left (a^2+3 b^2\right ) \sin (c+d x)+3 a^2 b+b^3\right )+20 a b^3 \cos (c+d x) (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3}}{3 d (a+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \sin \left (d x + c\right ) + a} \sec \left (d x + c\right )^{2}}{3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.85, size = 1653, normalized size = 5.09 \[ \text {result too large to display} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^2\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sec ^{2}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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