Optimal. Leaf size=75 \[ \frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{39 \sqrt [4]{13} d}-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3076, 3077, 2641} \[ \frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{39 \sqrt [4]{13} d}-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3076
Rule 3077
Rubi steps
\begin {align*} \int \frac {1}{(2 \cos (c+d x)+3 \sin (c+d x))^{5/2}} \, dx &=-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (2 \cos (c+d x)+3 \sin (c+d x))^{3/2}}+\frac {1}{39} \int \frac {1}{\sqrt {2 \cos (c+d x)+3 \sin (c+d x)}} \, dx\\ &=-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (2 \cos (c+d x)+3 \sin (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )}} \, dx}{39 \sqrt [4]{13}}\\ &=\frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{39 \sqrt [4]{13} d}-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{39 d (2 \cos (c+d x)+3 \sin (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.71, size = 157, normalized size = 2.09 \[ \frac {\sqrt {2} 13^{3/4} \sqrt {\sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )+1} (3 \sin (c+d x)+2 \cos (c+d x))^{3/2} \sec \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right ) \sqrt {2 \sin \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )+\cos \left (2 \left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )\right )-1} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (c+d x+\tan ^{-1}\left (\frac {2}{3}\right )\right )\right )+52 \sin (c+d x)-78 \cos (c+d x)}{507 d (3 \sin (c+d x)+2 \cos (c+d x))^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{46 \, \cos \left (d x + c\right )^{3} - 9 \, {\left (\cos \left (d x + c\right )^{2} + 3\right )} \sin \left (d x + c\right ) - 54 \, \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 118, normalized size = 1.57 \[ \frac {\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \sqrt {-2 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+2}\, \sqrt {-\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )-2 \left (\cos ^{2}\left (d x +c +\arctan \left (\frac {2}{3}\right )\right )\right )}{39 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right ) \cos \left (d x +c +\arctan \left (\frac {2}{3}\right )\right ) \sqrt {\sqrt {13}\, \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, d} \]
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 \[ \int \frac {1}{{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (2\,\cos \left (c+d\,x\right )+3\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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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