Optimal. Leaf size=57 \[ \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac {\sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4264, 3815, 21, 3771, 2641} \[ \frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac {\sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2641
Rule 3771
Rule 3815
Rule 4264
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{(a+a \sec (c+d x))^2} \, dx\\ &=\frac {\sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {\sec (c+d x)} \left (\frac {a}{2}+\frac {1}{2} a \sec (c+d x)\right )}{a+a \sec (c+d x)} \, dx}{3 a^2}\\ &=\frac {\sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \, dx}{6 a^2}\\ &=\frac {\sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}+\frac {\int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{6 a^2}\\ &=\frac {F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 a^2 d}+\frac {\sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^2}\\ \end {align*}
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Mathematica [A] time = 0.24, size = 63, normalized size = 1.11 \[ \frac {4 \cos ^4\left (\frac {1}{2} (c+d x)\right ) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\sin (c+d x) \sqrt {\cos (c+d x)}}{3 a^2 d (\cos (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \left (d x + c\right )}}{a^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \cos \left (d x + c\right )^{2} \sec \left (d x + c\right ) + a^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 3.72, size = 188, normalized size = 3.30 \[ -\frac {\sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (2 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \left (\cos ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right )}{6 a^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^{3/2}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^2} \,d x \]
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 \[ \frac {\int \frac {1}{\cos ^{\frac {3}{2}}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )} + 2 \cos ^{\frac {3}{2}}{\left (c + d x \right )} \sec {\left (c + d x \right )} + \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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