Optimal. Leaf size=247 \[ \frac {a^{5/2} (200 A+163 B) \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 d}+\frac {a^3 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (200 A+163 B) \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (8 A+11 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)} \]
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Rubi [A]
time = 0.48, antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3034, 4103,
4101, 3888, 3886, 221} \begin {gather*} \frac {a^{5/2} (200 A+163 B) \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 )}{64 d}+\frac {a^3 (200 A+163 B) \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^3 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 (8 A+11 B) \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B \sin (c+d x) (a \sec (c+d x)+a)^{3/2}}{4 d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 3034
Rule 3886
Rule 3888
Rule 4101
Rule 4103
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{5/2} (A+B \sec (c+d x)) \, dx\\ &=\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{4} \left (\sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \sec ^{\frac {3}{2}}(c+d x) (a+a \sec (c+d x))^{3/2} \left (\frac {1}{2} a (8 A+3 B)+\frac {1}{2} a (8 A+11 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^2 (8 A+11 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{12} \left (\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)} \left (\frac {3}{4} a^2 (24 A+17 B)+\frac {1}{4} a^2 (104 A+95 B) \sec (c+d x)\right ) \, dx\\ &=\frac {a^3 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (8 A+11 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{64} \left (a^2 (200 A+163 B) \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 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (200 A+163 B) \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (8 A+11 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {1}{128} \left (a^2 (200 A+163 B) \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 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (200 A+163 B) \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (8 A+11 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)}-\frac {\left (a^2 (200 A+163 B) \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 )}{64 d}\\ &=\frac {a^{5/2} (200 A+163 B) \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)}}{64 d}+\frac {a^3 (104 A+95 B) \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^3 (200 A+163 B) \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 (8 A+11 B) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{24 d \cos ^{\frac {5}{2}}(c+d x)}+\frac {a B (a+a \sec (c+d x))^{3/2} \sin (c+d x)}{4 d \cos ^{\frac {5}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 1.90, size = 154, 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 (6 \sqrt {2} (200 A+163 B) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right ) \cos ^4(c+d x)+(544 A+844 B+(2056 A+2203 B) \cos (c+d x)+(544 A+652 B) \cos (2 (c+d x))+600 A \cos (3 (c+d x))+489 B \cos (3 (c+d x))) \sin \left (\frac {1}{2} (c+d x)\right )\right )}{768 d \cos ^{\frac {7}{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. \(468\) vs.
\(2(211)=422\).
time = 12.58, size = 469, normalized size = 1.90
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 (600 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 ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}-600 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 ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}+489 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 ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}-489 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 ) \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}+1200 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 )+978 B \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 )+544 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 )}}+652 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 )}}+128 A \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \cos \left (d x +c \right )+368 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \sin \left (d x +c \right )+96 B \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right )\right )}{384 d \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )^{\frac {7}{2}} \sin \left (d x +c \right )^{2}}\) | \(469\) |
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. 7331 vs.
\(2 (211) = 422\).
time = 1.65, size = 7331, normalized size = 29.68 \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 = 2.86, size = 509, normalized size = 2.06 \begin {gather*} \left [\frac {4 \, {\left (3 \, {\left (200 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (136 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, A + 23 \, B\right )} a^{2} \cos \left (d x + c\right ) + 48 \, B 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 (200 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (200 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{4}\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 )}{768 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\right )}}, \frac {2 \, {\left (3 \, {\left (200 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + 2 \, {\left (136 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{2} + 8 \, {\left (8 \, A + 23 \, B\right )} a^{2} \cos \left (d x + c\right ) + 48 \, B 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 (200 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} + {\left (200 \, A + 163 \, B\right )} a^{2} \cos \left (d x + c\right )^{4}\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 )}{384 \, {\left (d \cos \left (d x + c\right )^{5} + d \cos \left (d x + c\right )^{4}\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 {B}{\cos \left (c+d\,x\right )}\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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