Optimal. Leaf size=159 \[ \frac {(7 A+8 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{4 \sqrt {a} d}-\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {A \tan (c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.49, antiderivative size = 159, 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 = {3044, 2984, 2985, 2649, 206, 2773} \[ \frac {(7 A+8 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{4 \sqrt {a} d}-\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a \cos (c+d x)+a}}\right )}{\sqrt {a} d}-\frac {A \tan (c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}+\frac {A \tan (c+d x) \sec (c+d x)}{2 d \sqrt {a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2773
Rule 2984
Rule 2985
Rule 3044
Rubi steps
\begin {align*} \int \frac {\left (A+C \cos ^2(c+d x)\right ) \sec ^3(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx &=\frac {A \sec (c+d x) \tan (c+d x)}{2 d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (-\frac {a A}{2}+\frac {1}{2} a (3 A+4 C) \cos (c+d x)\right ) \sec ^2(c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a}\\ &=-\frac {A \tan (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 d \sqrt {a+a \cos (c+d x)}}+\frac {\int \frac {\left (\frac {1}{4} a^2 (7 A+8 C)-\frac {1}{4} a^2 A \cos (c+d x)\right ) \sec (c+d x)}{\sqrt {a+a \cos (c+d x)}} \, dx}{2 a^2}\\ &=-\frac {A \tan (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 d \sqrt {a+a \cos (c+d x)}}+(-A-C) \int \frac {1}{\sqrt {a+a \cos (c+d x)}} \, dx+\frac {(7 A+8 C) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx}{8 a}\\ &=-\frac {A \tan (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 d \sqrt {a+a \cos (c+d x)}}+\frac {(2 (A+C)) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{d}-\frac {(7 A+8 C) \operatorname {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,-\frac {a \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{4 d}\\ &=\frac {(7 A+8 C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{4 \sqrt {a} d}-\frac {\sqrt {2} (A+C) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {2} \sqrt {a+a \cos (c+d x)}}\right )}{\sqrt {a} d}-\frac {A \tan (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {A \sec (c+d x) \tan (c+d x)}{2 d \sqrt {a+a \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.76, size = 113, normalized size = 0.71 \[ \frac {\cos \left (\frac {1}{2} (c+d x)\right ) \left (-8 (A+C) \tanh ^{-1}\left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+\sqrt {2} (7 A+8 C) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )+A \left (5 \sin \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {3}{2} (c+d x)\right )\right ) \sec ^2(c+d x)\right )}{4 d \sqrt {a (\cos (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 276, normalized size = 1.74 \[ \frac {{\left ({\left (7 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{3} + {\left (7 \, A + 8 \, C\right )} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) - 4 \, {\left (A \cos \left (d x + c\right ) - 2 \, A\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right ) + \frac {8 \, \sqrt {2} {\left ({\left (A + C\right )} a \cos \left (d x + c\right )^{3} + {\left (A + C\right )} a \cos \left (d x + c\right )^{2}\right )} \log \left (-\frac {\cos \left (d x + c\right )^{2} + \frac {2 \, \sqrt {2} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{\sqrt {a}} - 2 \, \cos \left (d x + c\right ) - 3}{\cos \left (d x + c\right )^{2} + 2 \, \cos \left (d x + c\right ) + 1}\right )}{\sqrt {a}}}{16 \, {\left (a d \cos \left (d x + c\right )^{3} + a d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 8.98, size = 397, normalized size = 2.50 \[ \frac {\frac {4 \, \sqrt {2} {\left (A + C\right )} \log \left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2}\right )}{\sqrt {a}} + \frac {{\left (7 \, A \sqrt {a} + 8 \, C \sqrt {a}\right )} \log \left ({\left | {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} - a {\left (2 \, \sqrt {2} + 3\right )} \right |}\right )}{a} - \frac {{\left (7 \, A \sqrt {a} + 8 \, C \sqrt {a}\right )} \log \left ({\left | {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} + a {\left (2 \, \sqrt {2} - 3\right )} \right |}\right )}{a} - \frac {4 \, \sqrt {2} {\left (17 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{6} A \sqrt {a} - 57 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} A a^{\frac {3}{2}} + 19 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} A a^{\frac {5}{2}} - 3 \, A a^{\frac {7}{2}}\right )}}{{\left ({\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{4} - 6 \, {\left (\sqrt {a} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a}\right )}^{2} a + a^{2}\right )}^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.53, size = 1192, normalized size = 7.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+A}{{\cos \left (c+d\,x\right )}^3\,\sqrt {a+a\,\cos \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\left (A + C \cos ^{2}{\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{\sqrt {a \left (\cos {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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