Optimal. Leaf size=144 \[ \frac {25 a^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}+\frac {25 a^3 \sin (c+d x)}{8 d \sqrt {a+a \sec (c+d x)}}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{3 d} \]
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Rubi [A]
time = 0.18, antiderivative size = 144, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3898, 4100,
3890, 3859, 209} \begin {gather*} \frac {25 a^{5/2} \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 d}+\frac {25 a^3 \sin (c+d x)}{8 d \sqrt {a \sec (c+d x)+a}}+\frac {13 a^3 \sin (c+d x) \cos (c+d x)}{12 d \sqrt {a \sec (c+d x)+a}}+\frac {a^2 \sin (c+d x) \cos ^2(c+d x) \sqrt {a \sec (c+d x)+a}}{3 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3859
Rule 3890
Rule 3898
Rule 4100
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+a \sec (c+d x))^{5/2} \, dx &=\frac {a^2 \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{3} a \int \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \left (\frac {13 a}{2}+\frac {9}{2} a \sec (c+d x)\right ) \, dx\\ &=\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{8} \left (25 a^2\right ) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {25 a^3 \sin (c+d x)}{8 d \sqrt {a+a \sec (c+d x)}}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{3 d}+\frac {1}{16} \left (25 a^2\right ) \int \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {25 a^3 \sin (c+d x)}{8 d \sqrt {a+a \sec (c+d x)}}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{3 d}-\frac {\left (25 a^3\right ) \text {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}\\ &=\frac {25 a^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 d}+\frac {25 a^3 \sin (c+d x)}{8 d \sqrt {a+a \sec (c+d x)}}+\frac {13 a^3 \cos (c+d x) \sin (c+d x)}{12 d \sqrt {a+a \sec (c+d x)}}+\frac {a^2 \cos ^2(c+d x) \sqrt {a+a \sec (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 0.82, size = 151, normalized size = 1.05 \begin {gather*} \frac {a^2 \left (165 \tanh ^{-1}\left (\sqrt {1-\sec (c+d x)}\right )+(31+159 \cos (c+d x)+31 \cos (2 (c+d x))-2 \cos (3 (c+d x))) \sqrt {1-\sec (c+d x)}+192 \, _2F_1\left (\frac {1}{2},4;\frac {3}{2};1-\sec (c+d x)\right ) \sqrt {1-\sec (c+d x)}\right ) \sqrt {a (1+\sec (c+d x))} \sin (c+d x)}{72 d (1+\cos (c+d x)) \sqrt {1-\sec (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. \(312\) vs.
\(2(124)=248\).
time = 0.16, size = 313, normalized size = 2.17
method | result | size |
default | \(-\frac {\left (75 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {2}+150 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sqrt {2}+75 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \left (-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \sin \left (d x +c \right )+64 \left (\cos ^{6}\left (d x +c \right )\right )+208 \left (\cos ^{5}\left (d x +c \right )\right )+328 \left (\cos ^{4}\left (d x +c \right )\right )-600 \left (\cos ^{3}\left (d x +c \right )\right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a^{2}}{192 d \sin \left (d x +c \right ) \cos \left (d x +c \right )^{2}}\) | \(313\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [A]
time = 3.81, size = 320, normalized size = 2.22 \begin {gather*} \left [\frac {75 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right ) + 2 \, {\left (8 \, a^{2} \cos \left (d x + c\right )^{3} + 34 \, a^{2} \cos \left (d x + c\right )^{2} + 75 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{48 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, -\frac {75 \, {\left (a^{2} \cos \left (d x + c\right ) + a^{2}\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right ) - {\left (8 \, a^{2} \cos \left (d x + c\right )^{3} + 34 \, a^{2} \cos \left (d x + c\right )^{2} + 75 \, a^{2} \cos \left (d x + c\right )\right )} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{24 \, {\left (d \cos \left (d x + c\right ) + d\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 539 vs.
\(2 (124) = 248\).
time = 1.87, size = 539, normalized size = 3.74 \begin {gather*} -\frac {75 \, \sqrt {-a} a^{2} \log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 75 \, \sqrt {-a} a^{2} \log \left ({\left | {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right ) \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + \frac {4 \, {\left (75 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{10} \sqrt {-a} a^{3} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 1125 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{8} \sqrt {-a} a^{4} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 6174 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{6} \sqrt {-a} a^{5} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 4314 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} \sqrt {-a} a^{6} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + 807 \, \sqrt {2} {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} \sqrt {-a} a^{7} \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) - 49 \, \sqrt {2} \sqrt {-a} a^{8} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 6 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + a^{2}\right )}^{3}}}{48 \, d} \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 = -0.01 \begin {gather*} \int {\cos \left (c+d\,x\right )}^3\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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