Optimal. Leaf size=73 \[ -\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{13 d \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{13^{3/4} d} \]
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Rubi [A] time = 0.04, antiderivative size = 73, 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, 2639} \[ -\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{13 d \sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{13^{3/4} d} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3076
Rule 3077
Rubi steps
\begin {align*} \int \frac {1}{(2 \cos (c+d x)+3 \sin (c+d x))^{3/2}} \, dx &=-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{13 d \sqrt {2 \cos (c+d x)+3 \sin (c+d x)}}-\frac {1}{13} \int \sqrt {2 \cos (c+d x)+3 \sin (c+d x)} \, dx\\ &=-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{13 d \sqrt {2 \cos (c+d x)+3 \sin (c+d x)}}-\frac {\int \sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )} \, dx}{13^{3/4}}\\ &=-\frac {2 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right |2\right )}{13^{3/4} d}-\frac {2 (3 \cos (c+d x)-2 \sin (c+d x))}{13 d \sqrt {2 \cos (c+d x)+3 \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 1.03, size = 190, normalized size = 2.60 \[ \frac {\frac {3 \sin \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right )}{13^{3/4} \sqrt {-\left (\left (\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )-1\right ) \cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )\right )} \sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )+1}}-\frac {2 \cos (c+d x)}{\sqrt {3 \sin (c+d x)+2 \cos (c+d x)}}+\frac {4 \sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )}}{13^{3/4}}-\frac {3 \sin \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )}{13^{3/4} \sqrt {\cos \left (c+d x-\tan ^{-1}\left (\frac {3}{2}\right )\right )}}}{3 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )}}{5 \, \cos \left (d x + c\right )^{2} - 12 \, \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 9}, x\right ) \]
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 \[ \int \frac {1}{{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 162, normalized size = 2.22 \[ \frac {\sqrt {13}\, \left (2 \sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \sqrt {-2 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+2}\, \sqrt {-\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \EllipticE \left (\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}, \frac {\sqrt {2}}{2}\right )-\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \sqrt {-2 \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )+2}\, \sqrt {-\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \left (\cos ^{2}\left (d x +c +\arctan \left (\frac {2}{3}\right )\right )\right )\right )}{13 \cos \left (d x +c +\arctan \left (\frac {2}{3}\right )\right ) \sqrt {\sqrt {13}\, \sin \left (d x +c +\arctan \left (\frac {2}{3}\right )\right )}\, d} \]
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 \[ \int \frac {1}{{\left (2 \, \cos \left (d x + c\right ) + 3 \, \sin \left (d x + c\right )\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (2\,\cos \left (c+d\,x\right )+3\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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