Optimal. Leaf size=65 \[ \frac {3 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d}+\frac {a^2 \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}} \]
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Rubi [A] time = 0.12, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3814, 21, 3805, 3774, 203} \[ \frac {a^2 \sin (c+d x)}{d \sqrt {a \sec (c+d x)+a}}+\frac {3 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a \sec (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 203
Rule 3774
Rule 3805
Rule 3814
Rubi steps
\begin {align*} \int \cos (c+d x) (a+a \sec (c+d x))^{3/2} \, dx &=-\frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-(2 a) \int \frac {\cos (c+d x) \left (-\frac {3 a}{2}-\frac {3}{2} a \sec (c+d x)\right )}{\sqrt {a+a \sec (c+d x)}} \, dx\\ &=-\frac {2 a^2 \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}+(3 a) \int \cos (c+d x) \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^2 \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}+\frac {1}{2} (3 a) \int \sqrt {a+a \sec (c+d x)} \, dx\\ &=\frac {a^2 \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+x^2} \, dx,x,-\frac {a \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}\\ &=\frac {3 a^{3/2} \tan ^{-1}\left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {a+a \sec (c+d x)}}\right )}{d}+\frac {a^2 \sin (c+d x)}{d \sqrt {a+a \sec (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 89, normalized size = 1.37 \[ \frac {a \sqrt {\cos (c+d x)} \sec \left (\frac {1}{2} (c+d x)\right ) \sqrt {a (\sec (c+d x)+1)} \left (3 \sqrt {2} \sin ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \sin \left (\frac {1}{2} (c+d x)\right ) \sqrt {\cos (c+d x)}\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 248, normalized size = 3.82 \[ \left [\frac {2 \, a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + 3 \, {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {-a} \log \left (\frac {2 \, a \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a} \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \cos \left (d x + c\right ) - a}{\cos \left (d x + c\right ) + 1}\right )}{2 \, {\left (d \cos \left (d x + c\right ) + d\right )}}, \frac {a \sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 3 \, {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {\frac {a \cos \left (d x + c\right ) + a}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )}{\sqrt {a} \sin \left (d x + c\right )}\right )}{d \cos \left (d x + c\right ) + d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 7.16, size = 278, normalized size = 4.28 \[ -\frac {\sqrt {2} \sqrt {-a} a^{3} {\left (\frac {3 \, \sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}{{\left | 2 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + 4 \, \sqrt {2} {\left | a \right |} - 6 \, a \right |}}\right )}{a {\left | a \right |}} + \frac {8 \, {\left (3 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a\right )}}{{\left ({\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 6 \, {\left (\sqrt {-a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {-a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + a^{2}\right )} a}\right )} \mathrm {sgn}\left (\cos \left (d x + c\right )\right )}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.99, size = 125, normalized size = 1.92 \[ -\frac {\left (3 \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sqrt {2}\, \sin \left (d x +c \right )+2 \left (\cos ^{2}\left (d x +c \right )\right )-2 \cos \left (d x +c \right )\right ) \sqrt {\frac {a \left (1+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, a}{2 d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.19, size = 803, normalized size = 12.35 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \cos \left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \cos {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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