Optimal. Leaf size=113 \[ -\frac {(A-C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}-\frac {(3 A+C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}+\frac {(3 A+C) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {(A+C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)} \]
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Rubi [A] time = 0.17, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 2748, 2636, 2639, 2641} \[ -\frac {(A-C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}-\frac {(3 A+C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}+\frac {(3 A+C) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {(A+C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} (a \cos (c+d x)+a)} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 3042
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))} \, dx &=-\frac {(A+C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))}+\frac {\int \frac {\frac {1}{2} a (3 A+C)-\frac {1}{2} a (A-C) \cos (c+d x)}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac {(A+C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))}-\frac {(A-C) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{2 a}+\frac {(3 A+C) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{2 a}\\ &=-\frac {(A-C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}+\frac {(3 A+C) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {(A+C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))}-\frac {(3 A+C) \int \sqrt {\cos (c+d x)} \, dx}{2 a}\\ &=-\frac {(3 A+C) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}-\frac {(A-C) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{a d}+\frac {(3 A+C) \sin (c+d x)}{a d \sqrt {\cos (c+d x)}}-\frac {(A+C) \sin (c+d x)}{d \sqrt {\cos (c+d x)} (a+a \cos (c+d x))}\\ \end {align*}
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Mathematica [C] time = 6.68, size = 1128, normalized size = 9.98 \[ \text {result too large to display} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {\cos \left (d x + c\right )}}{a \cos \left (d x + c\right )^{3} + a \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.74, size = 316, normalized size = 2.80 \[ -\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \left (A \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 A \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-C \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-C \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )-2 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \left (3 A +C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \left (5 A +C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}{a \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{{\left (a \cos \left (d x + c\right ) + a\right )} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^{3/2}\,\left (a+a\,\cos \left (c+d\,x\right )\right )} \,d x \]
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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