Optimal. Leaf size=197 \[ -\frac {6 \sqrt {a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{5 d \left (a^2+b^2\right )^2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}-\frac {6 (b \cos (c+d x)-a \sin (c+d x))}{5 d \left (a^2+b^2\right )^2 \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{5 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3076, 3078, 2639} \[ -\frac {6 \sqrt {a \cos (c+d x)+b \sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right )}{5 d \left (a^2+b^2\right )^2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}-\frac {6 (b \cos (c+d x)-a \sin (c+d x))}{5 d \left (a^2+b^2\right )^2 \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{5 d \left (a^2+b^2\right ) (a \cos (c+d x)+b \sin (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3076
Rule 3078
Rubi steps
\begin {align*} \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^{7/2}} \, dx &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{5/2}}+\frac {3 \int \frac {1}{(a \cos (c+d x)+b \sin (c+d x))^{3/2}} \, dx}{5 \left (a^2+b^2\right )}\\ &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{5/2}}-\frac {6 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right )^2 d \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {3 \int \sqrt {a \cos (c+d x)+b \sin (c+d x)} \, dx}{5 \left (a^2+b^2\right )^2}\\ &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{5/2}}-\frac {6 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right )^2 d \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {\left (3 \sqrt {a \cos (c+d x)+b \sin (c+d x)}\right ) \int \sqrt {\cos \left (c+d x-\tan ^{-1}(a,b)\right )} \, dx}{5 \left (a^2+b^2\right )^2 \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}\\ &=-\frac {2 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right ) d (a \cos (c+d x)+b \sin (c+d x))^{5/2}}-\frac {6 (b \cos (c+d x)-a \sin (c+d x))}{5 \left (a^2+b^2\right )^2 d \sqrt {a \cos (c+d x)+b \sin (c+d x)}}-\frac {6 E\left (\left .\frac {1}{2} \left (c+d x-\tan ^{-1}(a,b)\right )\right |2\right ) \sqrt {a \cos (c+d x)+b \sin (c+d x)}}{5 \left (a^2+b^2\right )^2 d \sqrt {\frac {a \cos (c+d x)+b \sin (c+d x)}{\sqrt {a^2+b^2}}}}\\ \end {align*}
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Mathematica [C] time = 2.44, size = 277, normalized size = 1.41 \[ \frac {\frac {\cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right ) \left (3 b \sin \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (c+d x-\tan ^{-1}\left (\frac {b}{a}\right )\right )\right )-3 \sqrt {\sin ^2\left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )} \left (b \sin \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )-2 a \cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )\right )\right )}{\sqrt {\sin ^2\left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )} \left (a \sqrt {\frac {b^2}{a^2}+1} \cos \left (-\tan ^{-1}\left (\frac {b}{a}\right )+c+d x\right )\right )^{3/2}}-\frac {2 \left (3 a^2 \cos ^3(c+d x)-a b \sin (c+d x)+6 a b \sin (c+d x) \cos ^2(c+d x)+b^2 \left (3 \sin ^2(c+d x)+1\right ) \cos (c+d x)\right )}{(a \cos (c+d x)+b \sin (c+d x))^{5/2}}}{5 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.39, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )}}{{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{4} + b^{4} + 2 \, {\left (3 \, a^{2} b^{2} - b^{4}\right )} \cos \left (d x + c\right )^{2} + 4 \, {\left (a b^{3} \cos \left (d x + c\right ) + {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 309, normalized size = 1.57 \[ \frac {\sqrt {a^{2}+b^{2}}\, \left (6 \sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \sqrt {-2 \sin \left (d x +c -\arctan \left (-a , b\right )\right )+2}\, \sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \left (\sin ^{2}\left (d x +c -\arctan \left (-a , b\right )\right )\right ) \EllipticE \left (\sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \sqrt {-2 \sin \left (d x +c -\arctan \left (-a , b\right )\right )+2}\, \sqrt {-\sin \left (d x +c -\arctan \left (-a , b\right )\right )}\, \left (\sin ^{2}\left (d x +c -\arctan \left (-a , b\right )\right )\right ) \EllipticF \left (\sqrt {1+\sin \left (d x +c -\arctan \left (-a , b\right )\right )}, \frac {\sqrt {2}}{2}\right )+6 \left (\sin ^{4}\left (d x +c -\arctan \left (-a , b\right )\right )\right )-4 \left (\sin ^{2}\left (d x +c -\arctan \left (-a , b\right )\right )\right )-2\right )}{5 \sin \left (d x +c -\arctan \left (-a , b\right )\right )^{2} \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \cos \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {\sin \left (d x +c -\arctan \left (-a , b\right )\right ) \sqrt {a^{2}+b^{2}}}\, d} \]
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 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + b \sin \left (d x + c\right )\right )}^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,\cos \left (c+d\,x\right )+b\,\sin \left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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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