Optimal. Leaf size=36 \[ \frac {\sqrt {\sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt {b \sec (c+d x)}} \]
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Rubi [A] time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 3770} \[ \frac {\sqrt {\sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt {b \sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3770
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {7}{2}}(c+d x)}{(b \sec (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \sec (c+d x) \, dx}{b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {\tanh ^{-1}(\sin (c+d x)) \sqrt {\sec (c+d x)}}{b^2 d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.92 \[ \frac {\sec ^{\frac {5}{2}}(c+d x) \tanh ^{-1}(\sin (c+d x))}{d (b \sec (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.97, size = 114, normalized size = 3.17 \[ \left [\frac {\log \left (-\frac {b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b}{\cos \left (d x + c\right )^{2}}\right )}{2 \, b^{\frac {5}{2}} d}, -\frac {\sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b}\right )}{b^{3} d}\right ] \]
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 \[ \int \frac {\sec \left (d x + c\right )^{\frac {7}{2}}}{\left (b \sec \left (d x + c\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.78, size = 52, normalized size = 1.44 \[ -\frac {2 \cos \left (d x +c \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \arctanh \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )}{d \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 65, normalized size = 1.81 \[ \frac {\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )}{2 \, b^{\frac {5}{2}} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{7/2}}{{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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