Optimal. Leaf size=72 \[ \frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 b^3 d}+\frac {2 \sin (c+d x)}{3 b^2 d \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.04, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {16, 3769, 3771, 2641} \[ \frac {2 \sin (c+d x)}{3 b^2 d \sqrt {b \sec (c+d x)}}+\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 b^3 d} \]
Antiderivative was successfully verified.
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Rule 16
Rule 2641
Rule 3769
Rule 3771
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac {\int \frac {1}{(b \sec (c+d x))^{3/2}} \, dx}{b}\\ &=\frac {2 \sin (c+d x)}{3 b^2 d \sqrt {b \sec (c+d x)}}+\frac {\int \sqrt {b \sec (c+d x)} \, dx}{3 b^3}\\ &=\frac {2 \sin (c+d x)}{3 b^2 d \sqrt {b \sec (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{3 b^3}\\ &=\frac {2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \sec (c+d x)}}{3 b^3 d}+\frac {2 \sin (c+d x)}{3 b^2 d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 62, normalized size = 0.86 \[ \frac {\sec ^2(c+d x) \left (\sin (2 (c+d x))+2 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{3 b d (b \sec (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.63, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \sec \left (d x + c\right )}}{b^{3} \sec \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {\sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.74, size = 131, normalized size = 1.82 \[ -\frac {2 \left (-1+\cos \left (d x +c \right )\right ) \left (i \sin \left (d x +c \right ) \EllipticF \left (\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \sqrt {\frac {\cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}-\left (\cos ^{2}\left (d x +c \right )\right )+\cos \left (d x +c \right )\right ) \left (1+\cos \left (d x +c \right )\right )^{2}}{3 d \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{3} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}} \]
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 \[ \int \frac {\sec \left (d x + c\right )}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\cos \left (c+d\,x\right )\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\sec {\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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