Optimal. Leaf size=332 \[ \frac {a^{5/2} (1304 A+1015 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{512 d}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A]
time = 0.69, antiderivative size = 332, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {4350, 4174,
4103, 4101, 3888, 3886, 221} \begin {gather*} \frac {a^{5/2} (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)} \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{512 d}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (24 A+23 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{6 d \cos ^{\frac {7}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 3886
Rule 3888
Rule 4101
Rule 4103
Rule 4174
Rule 4350
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right )}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (A+C \sec ^2(c+d x)\right ) \, dx\\ &=\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (12 A+5 C)+\frac {5}{2} a C \sec (c+d x)\right ) \, dx}{6 a}\\ &=\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {15}{4} a^2 (8 A+5 C)+\frac {5}{4} a^2 (24 A+23 C) \sec (c+d x)\right ) \, dx}{30 a}\\ &=\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {5}{8} a^3 (312 A+235 C)+\frac {15}{8} a^3 (136 A+109 C) \sec (c+d x)\right ) \, dx}{120 a}\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{384} \left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{512} \left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {\left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sqrt {\sec (c+d x)} \sqrt {a+a \sec (c+d x)} \, dx}{1024}\\ &=\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\left (a^2 (1304 A+1015 C) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \text {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 )}{512 d}\\ &=\frac {a^{5/2} (1304 A+1015 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{512 d}+\frac {a^3 (136 A+109 C) \sin (c+d x)}{192 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{768 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (1304 A+1015 C) \sin (c+d x)}{512 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+23 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{96 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {a C (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{6 d \cos ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 4.77, size = 200, normalized size = 0.60 \begin {gather*} \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sec (c+d x))} \left (24 \sqrt {2} (1304 A+1015 C) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^6(c+d x)+(18720 A+27412 C+14 (4056 A+4591 C) \cos (c+d x)+16 (1496 A+1711 C) \cos (2 (c+d x))+25448 A \cos (3 (c+d x))+21721 C \cos (3 (c+d x))+5216 A \cos (4 (c+d x))+4060 C \cos (4 (c+d x))+3912 A \cos (5 (c+d x))+3045 C \cos (5 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{24576 d \cos ^{\frac {11}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 564, normalized size = 1.70
method | result | size |
default | \(-\frac {a^{2} \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 ) \left (3912 A \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{6}\left (d x +c \right )\right )+3912 A \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{6}\left (d x +c \right )\right )+3045 C \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (-1-\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{6}\left (d x +c \right )\right )+3045 C \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right ) \left (\cos ^{6}\left (d x +c \right )\right )+7824 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+6090 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )+5216 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+4060 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \left (\cos ^{4}\left (d x +c \right )\right )+2944 A \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 )+3248 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 )+768 A \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 )+2784 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 )+1792 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+512 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )\right )}{3072 d \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{\frac {11}{2}} \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}}\) | \(564\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 11081 vs.
\(2 (284) = 568\).
time = 2.58, size = 11081, normalized size = 33.38 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.18, size = 577, normalized size = 1.74 \begin {gather*} \left [\frac {4 \, {\left (3 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 2 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 896 \, C a^{2} \cos \left (d x + c\right ) + 256 \, C a^{2}\right )} \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 ) + 3 \, {\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 4 \, \sqrt {a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 7 \, a \cos \left (d x + c\right )^{2} + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right )}{6144 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}, \frac {2 \, {\left (3 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{5} + 2 \, {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + 8 \, {\left (184 \, A + 203 \, C\right )} a^{2} \cos \left (d x + c\right )^{3} + 48 \, {\left (8 \, A + 29 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 896 \, C a^{2} \cos \left (d x + c\right ) + 256 \, C a^{2}\right )} \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 ) + 3 \, {\left ({\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{7} + {\left (1304 \, A + 1015 \, C\right )} a^{2} \cos \left (d x + c\right )^{6}\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 )}{3072 \, {\left (d \cos \left (d x + c\right )^{7} + d \cos \left (d x + c\right )^{6}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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