Optimal. Leaf size=253 \[ \frac {a^{5/2} (40 A+38 B+25 C) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{8 d}+\frac {a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+42 B+31 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {a (6 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}} \]
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Rubi [A]
time = 0.60, antiderivative size = 253, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {4350, 4173,
4103, 4100, 3886, 221} \begin {gather*} \frac {a^{5/2} (40 A+38 B+25 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 )}{8 d}+\frac {a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt {\cos (c+d x)} \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (24 A+42 B+31 C) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{24 d \sqrt {\cos (c+d x)}}+\frac {a (6 B+5 C) \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{12 d \sqrt {\cos (c+d x)}}+\frac {C \sin (c+d x) (a \sec (c+d x)+a)^{5/2}}{3 d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 3886
Rule 4100
Rule 4103
Rule 4173
Rule 4350
Rubi steps
\begin {align*} \int \sqrt {\cos (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 &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{5/2} \left (A+B \sec (c+d x)+C \sec ^2(c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx\\ &=\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{5/2} \left (\frac {1}{2} a (6 A-C)+\frac {1}{2} a (6 B+5 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{3 a}\\ &=\frac {a (6 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {(a+a \sec (c+d x))^{3/2} \left (\frac {3}{4} a^2 (8 A-2 B-3 C)+\frac {1}{4} a^2 (24 A+42 B+31 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{6 a}\\ &=\frac {a^2 (24 A+42 B+31 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {a (6 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sqrt {a+a \sec (c+d x)} \left (\frac {1}{8} a^3 (24 A-54 B-49 C)+\frac {3}{8} a^3 (40 A+38 B+25 C) \sec (c+d x)\right )}{\sqrt {\sec (c+d x)}} \, dx}{6 a}\\ &=\frac {a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+42 B+31 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {a (6 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}+\frac {1}{16} \left (a^2 (40 A+38 B+25 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\\ &=\frac {a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+42 B+31 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {a (6 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}-\frac {\left (a^2 (40 A+38 B+25 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 )}{8 d}\\ &=\frac {a^{5/2} (40 A+38 B+25 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)}}{8 d}+\frac {a^3 (24 A-54 B-49 C) \sin (c+d x)}{24 d \sqrt {\cos (c+d x)} \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (24 A+42 B+31 C) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \sqrt {\cos (c+d x)}}+\frac {a (6 B+5 C) (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{12 d \sqrt {\cos (c+d x)}}+\frac {C (a+a \sec (c+d x))^{5/2} \sin (c+d x)}{3 d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 2.66, size = 157, normalized size = 0.62 \begin {gather*} \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sec (c+d x))} \left (3 \sqrt {2} (40 A+38 B+25 C) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^3(c+d x)+(24 A+66 B+91 C+4 (18 A+6 B+17 C) \cos (c+d x)+3 (8 A+22 B+25 C) \cos (2 (c+d x))+24 A \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{48 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(566\) vs.
\(2(217)=434\).
time = 0.18, size = 567, normalized size = 2.24
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 (120 A \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 ) \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+120 A \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 ) \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+96 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 )+114 B \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 ) \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+114 B \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 ) \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+75 C \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 ) \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+75 C \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 ) \sqrt {2}\, \left (\cos ^{3}\left (d x +c \right )\right )+48 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 )+132 B \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 )+150 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 )+24 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+68 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+16 C \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )\right )}{48 d \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{\frac {5}{2}}}\) | \(567\) |
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. 17788 vs.
\(2 (217) = 434\).
time = 3.76, size = 17788, normalized size = 70.31 \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 = 3.17, size = 515, normalized size = 2.04 \begin {gather*} \left [\frac {4 \, {\left (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 22 \, B + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, B + 17 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, 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 (40 \, A + 38 \, B + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (40 \, A + 38 \, B + 25 \, 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} - 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 )}{96 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\right )}}, \frac {2 \, {\left (48 \, A a^{2} \cos \left (d x + c\right )^{3} + 3 \, {\left (8 \, A + 22 \, B + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, B + 17 \, C\right )} a^{2} \cos \left (d x + c\right ) + 8 \, 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 (40 \, A + 38 \, B + 25 \, C\right )} a^{2} \cos \left (d x + c\right )^{4} + {\left (40 \, A + 38 \, B + 25 \, 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 )}{48 \, {\left (d \cos \left (d x + c\right )^{4} + d \cos \left (d x + c\right )^{3}\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 \sqrt {\cos \left (c+d\,x\right )}\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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