Optimal. Leaf size=295 \[ \frac {32 \sqrt {2} a b \cos (2 c+2 d x)}{3 d \left (4 a^2-b^2\right )^2 \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 d \left (4 a^2-b^2\right ) (2 a+b \sin (2 c+2 d x))^{3/2}}-\frac {4 \sqrt {2} \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{3 d \left (4 a^2-b^2\right ) \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {32 \sqrt {2} a \sqrt {2 a+b \sin (2 c+2 d x)} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{3 d \left (4 a^2-b^2\right )^2 \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}} \]
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Rubi [A] time = 0.30, antiderivative size = 295, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2666, 2664, 2754, 2752, 2663, 2661, 2655, 2653} \[ \frac {32 \sqrt {2} a b \cos (2 c+2 d x)}{3 d \left (4 a^2-b^2\right )^2 \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 d \left (4 a^2-b^2\right ) (2 a+b \sin (2 c+2 d x))^{3/2}}-\frac {4 \sqrt {2} \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}} F\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{3 d \left (4 a^2-b^2\right ) \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {32 \sqrt {2} a \sqrt {2 a+b \sin (2 c+2 d x)} E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{3 d \left (4 a^2-b^2\right )^2 \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2655
Rule 2661
Rule 2663
Rule 2664
Rule 2666
Rule 2752
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cos (c+d x) \sin (c+d x))^{5/2}} \, dx &=\int \frac {1}{\left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )^{5/2}} \, dx\\ &=\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^{3/2}}-\frac {8 \int \frac {-\frac {3 a}{2}+\frac {1}{4} b \sin (2 c+2 d x)}{\left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )^{3/2}} \, dx}{3 \left (4 a^2-b^2\right )}\\ &=\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^{3/2}}+\frac {32 \sqrt {2} a b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right )^2 d \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {64 \int \frac {\frac {1}{16} \left (12 a^2+b^2\right )+\frac {1}{2} a b \sin (2 c+2 d x)}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx}{3 \left (4 a^2-b^2\right )^2}\\ &=\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^{3/2}}+\frac {32 \sqrt {2} a b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right )^2 d \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {(64 a) \int \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)} \, dx}{3 \left (4 a^2-b^2\right )^2}-\frac {4 \int \frac {1}{\sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}} \, dx}{3 \left (4 a^2-b^2\right )}\\ &=\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^{3/2}}+\frac {32 \sqrt {2} a b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right )^2 d \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {\left (64 a \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}\right ) \int \sqrt {\frac {a}{a+\frac {b}{2}}+\frac {b \sin (2 c+2 d x)}{2 \left (a+\frac {b}{2}\right )}} \, dx}{3 \left (4 a^2-b^2\right )^2 \sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}}}-\frac {\left (4 \sqrt {\frac {a+\frac {1}{2} b \sin (2 c+2 d x)}{a+\frac {b}{2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+\frac {b}{2}}+\frac {b \sin (2 c+2 d x)}{2 \left (a+\frac {b}{2}\right )}}} \, dx}{3 \left (4 a^2-b^2\right ) \sqrt {a+\frac {1}{2} b \sin (2 c+2 d x)}}\\ &=\frac {4 \sqrt {2} b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^{3/2}}+\frac {32 \sqrt {2} a b \cos (2 c+2 d x)}{3 \left (4 a^2-b^2\right )^2 d \sqrt {2 a+b \sin (2 c+2 d x)}}+\frac {32 \sqrt {2} a E\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {2 a+b \sin (2 c+2 d x)}}{3 \left (4 a^2-b^2\right )^2 d \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}-\frac {4 \sqrt {2} F\left (c-\frac {\pi }{4}+d x|\frac {2 b}{2 a+b}\right ) \sqrt {\frac {2 a+b \sin (2 c+2 d x)}{2 a+b}}}{3 \left (4 a^2-b^2\right ) d \sqrt {2 a+b \sin (2 c+2 d x)}}\\ \end {align*}
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Mathematica [A] time = 1.52, size = 201, normalized size = 0.68 \[ -\frac {4 \sqrt {2} \left (b \cos (2 (c+d x)) \left (-20 a^2-8 a b \sin (2 (c+d x))+b^2\right )+(2 a-b) (2 a+b)^2 \left (\frac {2 a+b \sin (2 (c+d x))}{2 a+b}\right )^{3/2} F\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )-\frac {8 a (2 a+b \sin (2 (c+d x)))^2 E\left (c+d x-\frac {\pi }{4}|\frac {2 b}{2 a+b}\right )}{\sqrt {\frac {2 a+b \sin (2 (c+d x))}{2 a+b}}}\right )}{3 d \left (b^2-4 a^2\right )^2 (2 a+b \sin (2 (c+d x)))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a}}{3 \, a b^{2} \cos \left (d x + c\right )^{4} - 3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} + {\left (b^{3} \cos \left (d x + c\right )^{5} - b^{3} \cos \left (d x + c\right )^{3} - 3 \, a^{2} b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.52, size = 1554, normalized size = 5.27 \[ \text {result too large to display} \]
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 \[ \int \frac {1}{{\left (b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a+b\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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