Optimal. Leaf size=220 \[ \frac {163 a^{5/2} \sinh ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right ) \sqrt {\cos (c+d x)} \sqrt {\sec (c+d x)}}{64 d}+\frac {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)} \]
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Rubi [A]
time = 0.29, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4349, 3899,
4101, 3888, 3886, 221} \begin {gather*} \frac {163 a^{5/2} \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 {163 a^3 \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {163 a^3 \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a \sec (c+d x)+a}}+\frac {a^2 \sin (c+d x) \sqrt {a \sec (c+d x)+a}}{4 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 3899
Rule 4101
Rule 4349
Rubi steps
\begin {align*} \int \frac {(a+a \sec (c+d x))^{5/2}}{\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} \, dx\\ &=\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{4} \left (a \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 {13 a}{2}+\frac {17}{2} a \sec (c+d x)\right ) \, dx\\ &=\frac {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{48} \left (163 a^2 \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 {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{64} \left (163 a^2 \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 {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)}+\frac {1}{128} \left (163 a^2 \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 {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)}-\frac {\left (163 a^2 \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 {163 a^{5/2} \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 {17 a^3 \sin (c+d x)}{24 d \cos ^{\frac {7}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{96 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {163 a^3 \sin (c+d x)}{64 d \cos ^{\frac {3}{2}}(c+d x) \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{4 d \cos ^{\frac {7}{2}}(c+d x)}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 5.56, size = 190, normalized size = 0.86 \begin {gather*} \frac {a^2 (1+\cos (c+d x))^2 \sec ^5\left (\frac {1}{2} (c+d x)\right ) \sqrt {a (1+\sec (c+d x))} \left (-489 i e^{\frac {1}{2} i (c+d x)} \cos ^4(c+d x) \, _2F_1\left (\frac {1}{4},1;\frac {5}{4};-e^{2 i (c+d x)}\right )-163 i e^{\frac {3}{2} i (c+d x)} \cos ^4(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-e^{2 i (c+d x)}\right )+\left (48+184 \cos (c+d x)+326 \cos ^2(c+d x)+489 \cos ^3(c+d x)\right ) \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 [A]
time = 0.12, size = 276, normalized size = 1.25
method | result | size |
default | \(\frac {\left (-489 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )-\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right )+489 \left (\cos ^{4}\left (d x +c \right )\right ) \sqrt {2}\, \arctan \left (\frac {\sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {2}}{4}\right )+978 \sin \left (d x +c \right ) \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}+652 \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}+368 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}+96 \sin \left (d x +c \right ) \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \sqrt {-\frac {2}{1+\cos \left (d x +c \right )}}\, \left (\cos ^{2}\left (d x +c \right )-1\right ) a^{2}}{768 d \cos \left (d x +c \right )^{\frac {7}{2}} \sin \left (d x +c \right )^{2}}\) | \(276\) |
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. 3860 vs.
\(2 (184) = 368\).
time = 0.84, size = 3860, normalized size = 17.55 \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 = 2.84, size = 437, normalized size = 1.99 \begin {gather*} \left [\frac {4 \, {\left (489 \, a^{2} \cos \left (d x + c\right )^{3} + 326 \, a^{2} \cos \left (d x + c\right )^{2} + 184 \, a^{2} \cos \left (d x + c\right ) + 48 \, 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 ) + 489 \, {\left (a^{2} \cos \left (d x + c\right )^{5} + 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 (489 \, a^{2} \cos \left (d x + c\right )^{3} + 326 \, a^{2} \cos \left (d x + c\right )^{2} + 184 \, a^{2} \cos \left (d x + c\right ) + 48 \, 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 ) + 489 \, {\left (a^{2} \cos \left (d x + c\right )^{5} + 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 {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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