Optimal. Leaf size=128 \[ -\frac {5 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d}+\frac {21 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac {7 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 a d}-\frac {5 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.11, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2767, 2748, 2635, 2641, 2639} \[ -\frac {5 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d}+\frac {21 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d}-\frac {\sin (c+d x) \cos ^{\frac {5}{2}}(c+d x)}{d (a \cos (c+d x)+a)}+\frac {7 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 a d}-\frac {5 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2639
Rule 2641
Rule 2748
Rule 2767
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {7}{2}}(c+d x)}{a+a \cos (c+d x)} \, dx &=-\frac {\cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {\int \cos ^{\frac {3}{2}}(c+d x) \left (\frac {5 a}{2}-\frac {7}{2} a \cos (c+d x)\right ) \, dx}{a^2}\\ &=-\frac {\cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {5 \int \cos ^{\frac {3}{2}}(c+d x) \, dx}{2 a}+\frac {7 \int \cos ^{\frac {5}{2}}(c+d x) \, dx}{2 a}\\ &=-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {7 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d}-\frac {\cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}-\frac {5 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 a}+\frac {21 \int \sqrt {\cos (c+d x)} \, dx}{10 a}\\ &=\frac {21 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 a d}-\frac {5 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a d}-\frac {5 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 a d}+\frac {7 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 a d}-\frac {\cos ^{\frac {5}{2}}(c+d x) \sin (c+d x)}{d (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [C] time = 1.81, size = 315, normalized size = 2.46 \[ \frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {2 \csc (c) \sqrt {\cos (c+d x)} \left (5 \sin (2 c) \sin (d x)+10 \sin ^2(c) \cos (d x)-6 \cos (c) \left (\sin ^2(c) \cos (2 d x)-8\right )-3 \sin (c) \cos (2 c) \sin (2 d x)+30 \sin \left (\frac {c}{2}\right ) \sin \left (\frac {d x}{2}\right ) \sec \left (\frac {1}{2} (c+d x)\right )+15\right )}{d}+\frac {2 i \sqrt {2} e^{-i (c+d x)} \left (63 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )+25 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )+63 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{15 a (\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\cos \left (d x + c\right )^{\frac {7}{2}}}{a \cos \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {7}{2}}}{a \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.53, size = 229, normalized size = 1.79 \[ -\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \left (25 \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+63 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )+48 \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-56 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-30 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+23 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{15 a \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{\frac {7}{2}}}{a \cos \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^{7/2}}{a+a\,\cos \left (c+d\,x\right )} \,d x \]
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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