Optimal. Leaf size=217 \[ \frac {163 \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x) \sqrt {\sec (c+d x)}}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {299 \sin (c+d x) \sqrt {\sec (c+d x)}}{48 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {95 \sin (c+d x)}{48 a^2 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.51, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3817, 4020, 4022, 4013, 3808, 206} \[ -\frac {299 \sin (c+d x) \sqrt {\sec (c+d x)}}{48 a^2 d \sqrt {a \sec (c+d x)+a}}+\frac {95 \sin (c+d x)}{48 a^2 d \sqrt {\sec (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {163 \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x) \sqrt {\sec (c+d x)}}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{3/2}}-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a \sec (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 3808
Rule 3817
Rule 4013
Rule 4020
Rule 4022
Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2}} \, dx &=-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac {\int \frac {-\frac {11 a}{2}+3 a \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2}} \, dx}{4 a^2}\\ &=-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}-\frac {\int \frac {-\frac {95 a^2}{4}+17 a^2 \sec (c+d x)}{\sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac {95 \sin (c+d x)}{48 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}-\frac {\int \frac {\frac {299 a^3}{8}-\frac {95}{4} a^3 \sec (c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}} \, dx}{12 a^5}\\ &=-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac {95 \sin (c+d x)}{48 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}-\frac {299 \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt {a+a \sec (c+d x)}}+\frac {163 \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+a \sec (c+d x)}} \, dx}{32 a^2}\\ &=-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac {95 \sin (c+d x)}{48 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}-\frac {299 \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt {a+a \sec (c+d x)}}-\frac {163 \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{16 a^2 d}\\ &=\frac {163 \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {\sec (c+d x)} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{16 \sqrt {2} a^{5/2} d}-\frac {\sin (c+d x)}{4 d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2}}-\frac {17 \sin (c+d x)}{16 a d \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2}}+\frac {95 \sin (c+d x)}{48 a^2 d \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)}}-\frac {299 \sqrt {\sec (c+d x)} \sin (c+d x)}{48 a^2 d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.49, size = 165, normalized size = 0.76 \[ -\frac {\sec (c+d x) \left (\tan (c+d x) \sqrt {1-\sec (c+d x)} (379 \sec (c+d x)+16 \cos (2 (c+d x)) (5 \sec (c+d x)-1)+487)+978 \sqrt {2} \sin (c+d x) \cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {5}{2}}(c+d x) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sec (c+d x)}}{\sqrt {1-\sec (c+d x)}}\right )\right )}{48 d \sqrt {-((\sec (c+d x)-1) \sec (c+d x))} (a (\sec (c+d x)+1))^{5/2}} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 0.97, size = 466, normalized size = 2.15 \[ \left [\frac {489 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {a} \log \left (-\frac {a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {2} \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, a \cos \left (d x + c\right ) - 3 \, a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + \frac {4 \, {\left (32 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 503 \, \cos \left (d x + c\right )^{2} - 299 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{192 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}, -\frac {489 \, \sqrt {2} {\left (\cos \left (d x + c\right )^{3} + 3 \, \cos \left (d x + c\right )^{2} + 3 \, \cos \left (d x + c\right ) + 1\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )}}{a \sin \left (d x + c\right )}\right ) - \frac {2 \, {\left (32 \, \cos \left (d x + c\right )^{4} - 160 \, \cos \left (d x + c\right )^{3} - 503 \, \cos \left (d x + c\right )^{2} - 299 \, \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{96 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )}}\right ] \]
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 \[ \int \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sec \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.70, size = 254, normalized size = 1.17 \[ -\frac {\sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (-1+\cos \left (d x +c \right )\right )^{2} \left (489 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+64 \left (\cos ^{4}\left (d x +c \right )\right )+978 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-384 \left (\cos ^{3}\left (d x +c \right )\right )+489 \arctan \left (\frac {\sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}{2}\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )-686 \left (\cos ^{2}\left (d x +c \right )\right )+408 \cos \left (d x +c \right )+598\right ) \left (\cos ^{2}\left (d x +c \right )\right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{96 d \sin \left (d x +c \right )^{5} a^{3}} \]
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 = -0.00 \[ \int \frac {1}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
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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