Optimal. Leaf size=266 \[ -\frac {(47 A+24 C) \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A+9 C) \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}} \]
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Rubi [A]
time = 0.52, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {4170, 4107,
4005, 3859, 209, 3880} \begin {gather*} -\frac {(47 A+24 C) \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{8 a^{3/2} d}+\frac {(17 A+9 C) \text {ArcTan}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a \sec (c+d x)+a}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt {a \sec (c+d x)+a}}+\frac {(5 A+3 C) \sin (c+d x) \cos ^2(c+d x)}{6 a d \sqrt {a \sec (c+d x)+a}}-\frac {(A+C) \sin (c+d x) \cos ^2(c+d x)}{2 d (a \sec (c+d x)+a)^{3/2}}-\frac {(13 A+6 C) \sin (c+d x) \cos (c+d x)}{12 a d \sqrt {a \sec (c+d x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 3859
Rule 3880
Rule 4005
Rule 4107
Rule 4170
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) \left (A+C \sec ^2(c+d x)\right )}{(a+a \sec (c+d x))^{3/2}} \, dx &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {\int \frac {\cos ^3(c+d x) \left (-a (5 A+3 C)+\frac {1}{2} a (7 A+3 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x) \left (a^2 (13 A+6 C)-\frac {5}{2} a^2 (5 A+3 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{6 a^3}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}-\frac {(13 A+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}-\frac {\int \frac {\cos (c+d x) \left (-\frac {9}{2} a^3 (7 A+4 C)+\frac {3}{2} a^3 (13 A+6 C) \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx}{12 a^4}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}-\frac {\int \frac {\frac {3}{4} a^4 (47 A+24 C)-\frac {9}{4} a^4 (7 A+4 C) \sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{12 a^5}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}+\frac {(17 A+9 C) \int \frac {\sec (c+d x)}{\sqrt {a+a \sec (c+d x)}} \, dx}{4 a}-\frac {(47 A+24 C) \int \sqrt {a+a \sec (c+d x)} \, dx}{16 a^2}\\ &=-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}-\frac {(17 A+9 C) \text {Subst}\left (\int \frac {1}{2 a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{2 a d}+\frac {(47 A+24 C) \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 a d}\\ &=-\frac {(47 A+24 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{8 a^{3/2} d}+\frac {(17 A+9 C) \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2} \sqrt {a+a \sec (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}-\frac {(A+C) \cos ^2(c+d x) \sin (c+d x)}{2 d (a+a \sec (c+d x))^{3/2}}+\frac {3 (7 A+4 C) \sin (c+d x)}{8 a d \sqrt {a+a \sec (c+d x)}}-\frac {(13 A+6 C) \cos (c+d x) \sin (c+d x)}{12 a d \sqrt {a+a \sec (c+d x)}}+\frac {(5 A+3 C) \cos ^2(c+d x) \sin (c+d x)}{6 a d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 3.02, size = 204, normalized size = 0.77 \begin {gather*} -\frac {\left (-3 (47 A+24 C) \text {ArcTan}\left (\sqrt {-1+\sec (c+d x)}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {-1+\sec (c+d x)}+6 \sqrt {2} (17 A+9 C) \text {ArcTan}\left (\frac {\sqrt {-1+\sec (c+d x)}}{\sqrt {2}}\right ) \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sqrt {-1+\sec (c+d x)}+(60 A+36 C+(43 A+24 C) \cos (c+d x)-3 A \cos (2 (c+d x))+2 A \cos (3 (c+d x))) \sin ^2\left (\frac {1}{2} (c+d x)\right )\right ) \tan \left (\frac {1}{2} (c+d x)\right )}{12 a d (-1+\cos (c+d x)) \sqrt {a (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. \(1413\) vs.
\(2(231)=462\).
time = 24.56, size = 1414, normalized size = 5.32
method | result | size |
default | \(\text {Expression too large to display}\) | \(1414\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 8.83, size = 668, normalized size = 2.51 \begin {gather*} \left [-\frac {6 \, \sqrt {2} {\left ({\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right ) + 17 \, A + 9 \, C\right )} \sqrt {-a} \log \left (\frac {2 \, \sqrt {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 ) + 3 \, a \cos \left (d x + c\right )^{2} + 2 \, a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right ) + 3 \, {\left ({\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right ) + 47 \, A + 24 \, C\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 \cos \left (d x + c\right )^{4} - 6 \, A \cos \left (d x + c\right )^{3} + {\left (37 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (7 \, A + 4 \, C\right )} \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 (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}, -\frac {6 \, \sqrt {2} {\left ({\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (17 \, A + 9 \, C\right )} \cos \left (d x + c\right ) + 17 \, A + 9 \, C\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \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 ) - 3 \, {\left ({\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (47 \, A + 24 \, C\right )} \cos \left (d x + c\right ) + 47 \, A + 24 \, C\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 \cos \left (d x + c\right )^{4} - 6 \, A \cos \left (d x + c\right )^{3} + {\left (37 \, A + 24 \, C\right )} \cos \left (d x + c\right )^{2} + 9 \, {\left (7 \, A + 4 \, C\right )} \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 (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \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. 815 vs.
\(2 (231) = 462\).
time = 2.33, size = 815, normalized size = 3.06 \begin {gather*} -\frac {\frac {6 \, \sqrt {2} {\left (17 \, A + 9 \, C\right )} \log \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}\right )}{\sqrt {-a} a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {3 \, {\left (47 \, A + 24 \, C\right )} \log \left (\frac {{\left | -1947111321950560360698936123457536 \, {\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} - 3894222643901120721397872246915072 \, \sqrt {2} {\left | a \right |} + 5841333965851681082096808370372608 \, a \right |}}{{\left | -1947111321950560360698936123457536 \, {\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} + 3894222643901120721397872246915072 \, \sqrt {2} {\left | a \right |} + 5841333965851681082096808370372608 \, a \right |}}\right )}{\sqrt {-a} {\left | a \right |} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} - \frac {12 \, {\left (\sqrt {2} A a \mathrm {sgn}\left (\cos \left (d x + c\right )\right ) + \sqrt {2} C a \mathrm {sgn}\left (\cos \left (d x + c\right )\right )\right )} \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{3}} - \frac {4 \, \sqrt {2} {\left (339 \, {\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} A + 72 \, {\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} C - 3165 \, {\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} A a - 888 \, {\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} C a + 9198 \, {\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} A a^{2} + 3024 \, {\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} C a^{2} - 4938 \, {\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} A a^{3} - 1776 \, {\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} C a^{3} + 975 \, {\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^{4} + 360 \, {\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} C a^{4} - 73 \, A a^{5} - 24 \, C a^{5}\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} \sqrt {-a} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}}{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 = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {C}{{\cos \left (c+d\,x\right )}^2}\right )}{{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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