Optimal. Leaf size=233 \[ \frac {a^{5/2} (304 A+200 B+163 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{64 d}+\frac {a^3 (432 A+392 B+299 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{192 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (16 A+24 B+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{32 d}+\frac {a (8 B+5 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d} \]
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Rubi [A] time = 0.75, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {4088, 4018, 4016, 3801, 215} \[ \frac {a^3 (432 A+392 B+299 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{192 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (16 A+24 B+17 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}{32 d}+\frac {a^{5/2} (304 A+200 B+163 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{64 d}+\frac {a (8 B+5 C) \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{3/2}}{24 d}+\frac {C \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) (a \sec (c+d x)+a)^{5/2}}{4 d} \]
Antiderivative was successfully verified.
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Rule 215
Rule 3801
Rule 4016
Rule 4018
Rule 4088
Rubi steps
\begin {align*} \int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (8 A+C)+\frac {1}{2} a (8 B+5 C) \sec (c+d x)\right ) \, dx}{4 a}\\ &=\frac {a (8 B+5 C) \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\int \sqrt {\sec (c+d x)} (a+a \sec (c+d x))^{3/2} \left (\frac {1}{4} a^2 (48 A+8 B+11 C)+\frac {3}{4} a^2 (16 A+24 B+17 C) \sec (c+d x)\right ) \, dx}{12 a}\\ &=\frac {a^2 (16 A+24 B+17 C) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {a (8 B+5 C) \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {\int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (240 A+104 B+95 C)+\frac {1}{8} a^3 (432 A+392 B+299 C) \sec (c+d x)\right ) \, dx}{24 a}\\ &=\frac {a^3 (432 A+392 B+299 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (16 A+24 B+17 C) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {a (8 B+5 C) \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}+\frac {1}{128} \left (a^2 (304 A+200 B+163 C)\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (432 A+392 B+299 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (16 A+24 B+17 C) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {a (8 B+5 C) \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}-\frac {\left (a^2 (304 A+200 B+163 C)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{a}}} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 d}\\ &=\frac {a^{5/2} (304 A+200 B+163 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{64 d}+\frac {a^3 (432 A+392 B+299 C) \sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{192 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (16 A+24 B+17 C) \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{32 d}+\frac {a (8 B+5 C) \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{24 d}+\frac {C \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{4 d}\\ \end {align*}
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Mathematica [A] time = 2.14, size = 179, normalized size = 0.77 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^{\frac {7}{2}}(c+d x) \sqrt {a (\sec (c+d x)+1)} \left (4 \sin \left (\frac {1}{2} (c+d x)\right ) ((1584 A+2056 B+2203 C) \cos (c+d x)+4 (48 A+136 B+163 C) \cos (2 (c+d x))+528 A \cos (3 (c+d x))+192 A+600 B \cos (3 (c+d x))+544 B+489 C \cos (3 (c+d x))+844 C)+24 \sqrt {2} (304 A+200 B+163 C) \cos ^4(c+d x) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{3072 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 542, normalized size = 2.33 \[ \left [\frac {3 \, {\left ({\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - \frac {4 \, {\left (\cos \left (d x + c\right )^{2} - 2 \, \cos \left (d x + c\right )\right )} \sqrt {a} \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 )}} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + \frac {4 \, {\left (3 \, {\left (176 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (48 \, A + 136 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, B + 23 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\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 )}}}{768 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, \frac {3 \, {\left ({\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (304 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3}\right )} \sqrt {-a} \arctan \left (\frac {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 )}{a \cos \left (d x + c\right )^{2} - a \cos \left (d x + c\right ) - 2 \, a}\right ) + \frac {2 \, {\left (3 \, {\left (176 \, A + 200 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (48 \, A + 136 \, B + 163 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, B + 23 \, C\right )} a^{2} \cos \left (d x + c\right ) + 48 \, C a^{2}\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 )}}}{384 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\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 \[ \int {\left (C \sec \left (d x + c\right )^{2} + B \sec \left (d x + c\right ) + A\right )} {\left (a \sec \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \sqrt {\sec \left (d x + c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.96, size = 639, normalized size = 2.74 \[ \frac {\left (-1+\cos \left (d x +c \right )\right ) \left (912 A \left (\cos ^{4}\left (d x +c \right )\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}-912 A \left (\cos ^{4}\left (d x +c \right )\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}+600 B \left (\cos ^{4}\left (d x +c \right )\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}-600 B \left (\cos ^{4}\left (d x +c \right )\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}+489 C \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}\, \left (\cos ^{4}\left (d x +c \right )\right )-489 C \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}\, \left (\cos ^{4}\left (d x +c \right )\right )-1056 A \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-1200 B \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-978 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right )-192 A \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-544 B \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-652 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )-128 B \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}-368 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )-96 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \sqrt {\frac {1}{\cos \left (d x +c \right )}}\, a^{2}}{384 d \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}} \]
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 \[ \int {\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \]
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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