Optimal. Leaf size=96 \[ \frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )-1}}\right )}{10 \sqrt {10} e} \]
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Rubi [A] time = 0.05, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {3116, 3115, 2649, 204} \[ \frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )-1}}\right )}{10 \sqrt {10} e} \]
Antiderivative was successfully verified.
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Rule 204
Rule 2649
Rule 3115
Rule 3116
Rubi steps
\begin {align*} \int \frac {1}{(-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}} \, dx &=\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}-\frac {1}{20} \int \frac {1}{\sqrt {-5+4 \cos (d+e x)+3 \sin (d+e x)}} \, dx\\ &=\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}-\frac {1}{20} \int \frac {1}{\sqrt {-5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}} \, dx\\ &=\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{-10-x^2} \, dx,x,-\frac {5 \sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {-5+5 \cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{10 e}\\ &=\frac {\tan ^{-1}\left (\frac {\sin \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}{\sqrt {2} \sqrt {-1+\cos \left (d+e x-\tan ^{-1}\left (\frac {3}{4}\right )\right )}}\right )}{10 \sqrt {10} e}+\frac {3 \cos (d+e x)-4 \sin (d+e x)}{10 e (-5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 0.31, size = 152, normalized size = 1.58 \[ \frac {\left (\frac {1}{250}-\frac {i}{125}\right ) \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right ) \left ((5+10 i) \left (\sin \left (\frac {1}{2} (d+e x)\right )+3 \cos \left (\frac {1}{2} (d+e x)\right )\right )-(1-i) \sqrt {-20-15 i} \left (\cos \left (\frac {1}{2} (d+e x)\right )-3 \sin \left (\frac {1}{2} (d+e x)\right )\right )^2 \tanh ^{-1}\left (\left (\frac {1}{10}+\frac {3 i}{10}\right ) \sqrt {-\frac {4}{5}-\frac {3 i}{5}} \left (\tan \left (\frac {1}{4} (d+e x)\right )+3\right )\right )\right )}{e (3 \sin (d+e x)+4 \cos (d+e x)-5)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.03, size = 210, normalized size = 2.19 \[ -\frac {{\left (13 \, \sqrt {10} \cos \left (e x + d\right )^{2} - 9 \, {\left (\sqrt {10} \cos \left (e x + d\right ) - 2 \, \sqrt {10}\right )} \sin \left (e x + d\right ) - \sqrt {10} \cos \left (e x + d\right ) - 14 \, \sqrt {10}\right )} \arctan \left (-\frac {{\left (3 \, \sqrt {10} \cos \left (e x + d\right ) + \sqrt {10} \sin \left (e x + d\right ) + 3 \, \sqrt {10}\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5}}{10 \, {\left (\cos \left (e x + d\right ) - 3 \, \sin \left (e x + d\right ) + 1\right )}}\right ) + 10 \, \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5} {\left (3 \, \cos \left (e x + d\right ) + \sin \left (e x + d\right ) + 3\right )}}{100 \, {\left (13 \, e \cos \left (e x + d\right )^{2} - e \cos \left (e x + d\right ) - 9 \, {\left (e \cos \left (e x + d\right ) - 2 \, e\right )} \sin \left (e x + d\right ) - 14 \, e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.53, size = 249, normalized size = 2.59 \[ -\frac {1}{450} \, {\left (\frac {9 \, \sqrt {10} \arctan \left (\frac {1}{10} \, \sqrt {10} {\left (-3 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 3 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - i\right )}\right )}{\mathrm {sgn}\left (-3 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )} + \frac {10 \, {\left (33 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{3} - 7 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} + 21 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - 21 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 9 i\right )}}{{\left (-3 i \, {\left (\sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} - \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )\right )}^{2} - 2 i \, \sqrt {\tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 1} + 2 i \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 3 i\right )}^{2} \mathrm {sgn}\left (-3 \, \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 1\right )}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.37, size = 118, normalized size = 1.23 \[ \frac {\left (-\sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right ) \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+\sqrt {10}\, \arctan \left (\frac {\sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\, \sqrt {10}}{10}\right )+2 \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}\right ) \sqrt {-5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-5}}{100 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {-5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e} \]
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 \[ \int \frac {1}{{\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) - 5\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )-5\right )}^{3/2}} \,d x \]
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 \[ \int \frac {1}{\left (3 \sin {\left (d + e x \right )} + 4 \cos {\left (d + e x \right )} - 5\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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