Optimal. Leaf size=115 \[ \frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 b^4 d \sqrt {b \cos (c+d x)}}+\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac {2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3012, 2636, 2642, 2641} \[ \frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 b^4 d \sqrt {b \cos (c+d x)}}+\frac {2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rule 2642
Rule 3012
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{(b \cos (c+d x))^{9/2}} \, dx &=\frac {2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac {(5 A+7 C) \int \frac {1}{(b \cos (c+d x))^{5/2}} \, dx}{7 b^2}\\ &=\frac {2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac {(5 A+7 C) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx}{21 b^4}\\ &=\frac {2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}+\frac {\left ((5 A+7 C) \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 b^4 \sqrt {b \cos (c+d x)}}\\ &=\frac {2 (5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 b^4 d \sqrt {b \cos (c+d x)}}+\frac {2 A \sin (c+d x)}{7 b d (b \cos (c+d x))^{7/2}}+\frac {2 (5 A+7 C) \sin (c+d x)}{21 b^3 d (b \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 77, normalized size = 0.67 \[ \frac {2 \left ((5 A+7 C) \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\tan (c+d x) \left (3 A \sec ^2(c+d x)+5 A+7 C\right )\right )}{21 b^4 d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (C \cos \left (d x + c\right )^{2} + A\right )} \sqrt {b \cos \left (d x + c\right )}}{b^{5} \cos \left (d x + c\right )^{5}}, x\right ) \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.00, size = 413, normalized size = 3.59 \[ -\frac {2 \sqrt {b \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 (A \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{56 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}-\frac {5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{42 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{21 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right )+C \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{6 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{3 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right )\right )}{b^{4} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, 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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + A}{\left (b \cos \left (d x + c\right )\right )^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{9/2}} \,d x \]
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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