Optimal. Leaf size=156 \[ \frac {a^{5/2} (19 A+20 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{4 d}-\frac {a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (7 A+4 B) \tan (c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}+\frac {a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d} \]
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Rubi [A] time = 0.47, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {2975, 2981, 2773, 206} \[ -\frac {a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt {a \cos (c+d x)+a}}+\frac {a^2 (7 A+4 B) \tan (c+d x) \sqrt {a \cos (c+d x)+a}}{4 d}+\frac {a^{5/2} (19 A+20 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a \cos (c+d x)+a}}\right )}{4 d}+\frac {a A \tan (c+d x) \sec (c+d x) (a \cos (c+d x)+a)^{3/2}}{2 d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2773
Rule 2975
Rule 2981
Rubi steps
\begin {align*} \int (a+a \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \sec ^3(c+d x) \, dx &=\frac {a A (a+a \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int (a+a \cos (c+d x))^{3/2} \left (\frac {1}{2} a (7 A+4 B)-\frac {1}{2} a (A-4 B) \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=\frac {a^2 (7 A+4 B) \sqrt {a+a \cos (c+d x)} \tan (c+d x)}{4 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \sqrt {a+a \cos (c+d x)} \left (\frac {1}{4} a^2 (19 A+20 B)-\frac {1}{4} a^2 (9 A-4 B) \cos (c+d x)\right ) \sec (c+d x) \, dx\\ &=-\frac {a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (7 A+4 B) \sqrt {a+a \cos (c+d x)} \tan (c+d x)}{4 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{8} \left (a^2 (19 A+20 B)\right ) \int \sqrt {a+a \cos (c+d x)} \sec (c+d x) \, dx\\ &=-\frac {a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (7 A+4 B) \sqrt {a+a \cos (c+d x)} \tan (c+d x)}{4 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{2 d}-\frac {\left (a^3 (19 A+20 B)\right ) \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 {a^{5/2} (19 A+20 B) \tanh ^{-1}\left (\frac {\sqrt {a} \sin (c+d x)}{\sqrt {a+a \cos (c+d x)}}\right )}{4 d}-\frac {a^3 (9 A-4 B) \sin (c+d x)}{4 d \sqrt {a+a \cos (c+d x)}}+\frac {a^2 (7 A+4 B) \sqrt {a+a \cos (c+d x)} \tan (c+d x)}{4 d}+\frac {a A (a+a \cos (c+d x))^{3/2} \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.68, size = 126, normalized size = 0.81 \[ \frac {a^2 \sec \left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \sqrt {a (\cos (c+d x)+1)} \left (2 \sin \left (\frac {1}{2} (c+d x)\right ) ((11 A+4 B) \cos (c+d x)+2 (A+2 B \cos (2 (c+d x))+2 B))+\sqrt {2} (19 A+20 B) \cos ^2(c+d x) \tanh ^{-1}\left (\sqrt {2} \sin \left (\frac {1}{2} (c+d x)\right )\right )\right )}{8 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 204, normalized size = 1.31 \[ \frac {{\left ({\left (19 \, A + 20 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} + {\left (19 \, A + 20 \, B\right )} a^{2} \cos \left (d x + c\right )^{2}\right )} \sqrt {a} \log \left (\frac {a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \, \sqrt {a \cos \left (d x + c\right ) + a} \sqrt {a} {\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) + 8 \, a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2}}\right ) + 4 \, {\left (8 \, B a^{2} \cos \left (d x + c\right )^{2} + {\left (11 \, A + 4 \, B\right )} a^{2} \cos \left (d x + c\right ) + 2 \, A a^{2}\right )} \sqrt {a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{16 \, {\left (d \cos \left (d x + c\right )^{3} + d \cos \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 1.19, size = 1016, normalized size = 6.51 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (A+B\,\cos \left (c+d\,x\right )\right )\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^{5/2}}{{\cos \left (c+d\,x\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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