Optimal. Leaf size=72 \[ \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3769, 3771, 2639} \[ \frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {1}{(c \sec (a+b x))^{5/2}} \, dx &=\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}}+\frac {3 \int \frac {1}{\sqrt {c \sec (a+b x)}} \, dx}{5 c^2}\\ &=\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}}+\frac {3 \int \sqrt {\cos (a+b x)} \, dx}{5 c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}\\ &=\frac {6 E\left (\left .\frac {1}{2} (a+b x)\right |2\right )}{5 b c^2 \sqrt {\cos (a+b x)} \sqrt {c \sec (a+b x)}}+\frac {2 \sin (a+b x)}{5 b c (c \sec (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 60, normalized size = 0.83 \[ \frac {\sqrt {c \sec (a+b x)} \left (\sin (a+b x)+\sin (3 (a+b x))+12 \sqrt {\cos (a+b x)} E\left (\left .\frac {1}{2} (a+b x)\right |2\right )\right )}{10 b c^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c \sec \left (b x + a\right )}}{c^{3} \sec \left (b x + a\right )^{3}}, x\right ) \]
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 \[ \int \frac {1}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.97, size = 321, normalized size = 4.46 \[ -\frac {2 \left (-3 i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}\, \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right )+3 i \cos \left (b x +a \right ) \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}-3 i \sin \left (b x +a \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+3 i \sin \left (b x +a \right ) \EllipticE \left (\frac {i \left (-1+\cos \left (b x +a \right )\right )}{\sin \left (b x +a \right )}, i\right ) \sqrt {\frac {1}{\cos \left (b x +a \right )+1}}\, \sqrt {\frac {\cos \left (b x +a \right )}{\cos \left (b x +a \right )+1}}+\cos ^{4}\left (b x +a \right )+2 \left (\cos ^{2}\left (b x +a \right )\right )-3 \cos \left (b x +a \right )\right )}{5 b \left (\frac {c}{\cos \left (b x +a \right )}\right )^{\frac {5}{2}} \cos \left (b x +a \right )^{3} \sin \left (b x +a \right )} \]
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 \[ \int \frac {1}{\left (c \sec \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (\frac {c}{\cos \left (a+b\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c \sec {\left (a + b x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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