Optimal. Leaf size=85 \[ \frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}+\frac {\sqrt {2} \sinh ^{-1}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac {\sinh ^{-1}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d} \]
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Rubi [A] time = 0.19, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3822, 4023, 3807, 215, 3801} \[ \frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {\sec (c+d x)+1}}+\frac {\sqrt {2} \sinh ^{-1}\left (\frac {\tan (c+d x)}{\sec (c+d x)+1}\right )}{d}-\frac {\sinh ^{-1}\left (\frac {\tan (c+d x)}{\sqrt {\sec (c+d x)+1}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 215
Rule 3801
Rule 3807
Rule 3822
Rule 4023
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {1+\sec (c+d x)}} \, dx &=\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {1}{2} \int \frac {(1-\sec (c+d x)) \sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx\\ &=\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}-\frac {1}{2} \int \sqrt {\sec (c+d x)} \sqrt {1+\sec (c+d x)} \, dx+\int \frac {\sqrt {\sec (c+d x)}}{\sqrt {1+\sec (c+d x)}} \, dx\\ &=\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,-\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{d}-\frac {\sqrt {2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,-\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}\\ &=\frac {\sqrt {2} \sinh ^{-1}\left (\frac {\tan (c+d x)}{1+\sec (c+d x)}\right )}{d}-\frac {\sinh ^{-1}\left (\frac {\tan (c+d x)}{\sqrt {1+\sec (c+d x)}}\right )}{d}+\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {1+\sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.31, size = 111, normalized size = 1.31 \[ \frac {\tan (c+d x) \left (\sqrt {-((\sec (c+d x)-1) \sec (c+d x))}+\sin ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )+2 \sin ^{-1}\left (\sqrt {\sec (c+d x)}\right )-\sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right )\right )}{d \sqrt {-\tan ^2(c+d x)}} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 0.97, size = 297, normalized size = 3.49 \[ \frac {2 \, {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \log \left (\frac {2 \, \sqrt {2} \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) - {\left (\cos \left (d x + c\right ) + 1\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} - 2 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}{\cos \left (d x + c\right ) + 1}\right ) + \frac {4 \, \sqrt {\frac {\cos \left (d x + c\right ) + 1}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{4 \, {\left (d \cos \left (d x + c\right ) + d\right )}} \]
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 \[ \int \frac {\sec \left (d x + c\right )^{\frac {5}{2}}}{\sqrt {\sec \left (d x + c\right ) + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.42, size = 220, normalized size = 2.59 \[ \frac {\left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1+\cos \left (d x +c \right )}{\cos \left (d x +c \right )}}\, \left (\arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {2}\, \cos \left (d x +c \right )-\arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \sqrt {2}\, \cos \left (d x +c \right )+4 \cos \left (d x +c \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right )+2 \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )-1\right )}{4 d \sin \left (d x +c \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.25, size = 873, normalized size = 10.27 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{\sqrt {\frac {1}{\cos \left (c+d\,x\right )}+1}} \,d x \]
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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