Optimal. Leaf size=100 \[ -\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \]
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Rubi [A] time = 0.06, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2636, 2640, 2639} \[ \frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right ) \sqrt {c \cos (a+b x)}}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rubi steps
\begin {align*} \int \frac {1}{(c \cos (a+b x))^{7/2}} \, dx &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {3 \int \frac {1}{(c \cos (a+b x))^{3/2}} \, dx}{5 c^2}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {3 \int \sqrt {c \cos (a+b x)} \, dx}{5 c^4}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}-\frac {\left (3 \sqrt {c \cos (a+b x)}\right ) \int \sqrt {\cos (a+b x)} \, dx}{5 c^4 \sqrt {\cos (a+b x)}}\\ &=-\frac {6 \sqrt {c \cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^4 \sqrt {\cos (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \cos (a+b x))^{5/2}}+\frac {6 \sin (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 68, normalized size = 0.68 \[ \frac {6 \sin (a+b x)-6 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )+2 \tan (a+b x) \sec (a+b x)}{5 b c^3 \sqrt {c \cos (a+b x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \cos \left (b x + a\right )}}{c^{4} \cos \left (b x + a\right )^{4}}, x\right ) \]
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 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.20, size = 366, normalized size = 3.66 \[ \frac {2 \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}\, \left (12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-24 \cos \left (\frac {b x}{2}+\frac {a}{2}\right ) \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+24 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (b x +a \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {b x}{2}+\frac {a}{2}\right ), \sqrt {2}\right )-8 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \cos \left (\frac {b x}{2}+\frac {a}{2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right ) c +c \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )}}{5 c^{4} \sin \left (\frac {b x}{2}+\frac {a}{2}\right )^{3} \left (8 \left (\sin ^{6}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-12 \left (\sin ^{4}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )+6 \left (\sin ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right ) \sqrt {c \left (2 \left (\cos ^{2}\left (\frac {b x}{2}+\frac {a}{2}\right )\right )-1\right )}\, b} \]
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 \[ \int \frac {1}{\left (c \cos \left (b x + a\right )\right )^{\frac {7}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (c\,\cos \left (a+b\,x\right )\right )}^{7/2}} \,d x \]
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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