Optimal. Leaf size=142 \[ \frac {2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}} F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{3 d \left (a^2+b^2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{3 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^{3/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 142, 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, 3078, 2641} \[ \frac {2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}} F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{3 d \left (a^2+b^2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{3 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 3076
Rule 3078
Rubi steps
\begin {align*} \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^{5/2}} \, dx &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{3/2}}+\frac {\int \frac {1}{\sqrt {a \cos (c+d x)+b \sin (c+d x)}} \, dx}{3 \left (a^2+b^2\right )}\\ &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{3/2}}+\frac {\sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}} \int \frac {1}{\sqrt {\cos \left (c+d x-\tan ^{-1}(a,b)\right )}} \, dx}{3 \left (a^2+b^2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}\\ &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{3 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{3/2}}+\frac {2 F\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right ) \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}{3 \left (a^2+b^2\right ) d \sqrt {a \cos (c+d x)+b \sin (c+d x)}}\\ \end {align*}
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Mathematica [C] time = 1.68, size = 145, normalized size = 1.02 \[ \frac {2 \left (\frac {\tan \left (\tan ^{-1}\left (\frac {a}{b}\right )+c+d x\right ) \sqrt {\cos ^2\left (\tan ^{-1}\left (\frac {a}{b}\right )+c+d x\right )} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (c+d x+\tan ^{-1}\left (\frac {a}{b}\right )\right )\right )}{\sqrt {b \sqrt {\frac {a^2}{b^2}+1} \sin \left (\tan ^{-1}\left (\frac {a}{b}\right )+c+d x\right )}}+\frac {a \sin (c+d x)-b \cos (c+d x)}{(a \cos (c+d x)+b \sin (c+d x))^{3/2}}\right )}{3 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{3 \, a b^{2} \cos \left (d x + c\right ) + {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} + {\left (b^{3} + {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 178, normalized size = 1.25 \[ \frac {\sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \sqrt {-2 \sin \left (d x +c -\arctan \left (-a , b\right )\right )+2}\, \sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \EllipticF \left (\sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}, \frac {\sqrt {2}}{2}\right ) \sin \left (d x +c -\arctan \left (-a , b\right )\right )-2 \left (\cos ^{2}\left (d x +c -\arctan \left (-a , b\right )\right )\right )}{3 \sin \left (d x +c -\arctan \left (-a , b\right )\right ) \left (a^{2}+b^{2}\right ) \cos \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {a^{2}+b^{2}}}\, d} \]
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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,\cos \left (c+d\,x\right )+b\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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