Optimal. Leaf size=207 \[ \frac {11 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}+\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]
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Rubi [A] time = 0.36, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2766, 2978, 2748, 2636, 2641, 2639} \[ \frac {11 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3 \cos (c+d x)+a^3\right )}+\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^2}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a \cos (c+d x)+a)^3} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2641
Rule 2748
Rule 2766
Rule 2978
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^3} \, dx &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3}+\frac {\int \frac {\frac {13 a}{2}-\frac {7}{2} a \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))^2} \, dx}{5 a^2}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2}+\frac {\int \frac {\frac {69 a^2}{2}-25 a^2 \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (a+a \cos (c+d x))} \, dx}{15 a^4}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right )}+\frac {\int \frac {\frac {495 a^3}{4}-\frac {357}{4} a^3 \cos (c+d x)}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{15 a^6}\\ &=-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right )}-\frac {119 \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx}{20 a^3}+\frac {33 \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx}{4 a^3}\\ &=\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right )}+\frac {11 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{4 a^3}+\frac {119 \int \sqrt {\cos (c+d x)} \, dx}{20 a^3}\\ &=\frac {119 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{10 a^3 d}+\frac {11 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{2 a^3 d}+\frac {11 \sin (c+d x)}{2 a^3 d \cos ^{\frac {3}{2}}(c+d x)}-\frac {119 \sin (c+d x)}{10 a^3 d \sqrt {\cos (c+d x)}}-\frac {\sin (c+d x)}{5 d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^3}-\frac {2 \sin (c+d x)}{3 a d \cos ^{\frac {3}{2}}(c+d x) (a+a \cos (c+d x))^2}-\frac {119 \sin (c+d x)}{30 d \cos ^{\frac {3}{2}}(c+d x) \left (a^3+a^3 \cos (c+d x)\right )}\\ \end {align*}
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Mathematica [C] time = 2.64, size = 394, normalized size = 1.90 \[ \frac {\cos ^6\left (\frac {1}{2} (c+d x)\right ) \left (-\frac {\csc \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}\right ) \left (5134 \cos \left (\frac {1}{2} (c-d x)\right )+4148 \cos \left (\frac {1}{2} (3 c+d x)\right )+4664 \cos \left (\frac {1}{2} (c+3 d x)\right )+2476 \cos \left (\frac {1}{2} (5 c+3 d x)\right )+3340 \cos \left (\frac {1}{2} (3 c+5 d x)\right )+944 \cos \left (\frac {1}{2} (7 c+5 d x)\right )+1620 \cos \left (\frac {1}{2} (5 c+7 d x)\right )+165 \cos \left (\frac {1}{2} (9 c+7 d x)\right )+357 \cos \left (\frac {1}{2} (7 c+9 d x)\right )\right ) \sec ^5\left (\frac {1}{2} (c+d x)\right )}{96 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {4 i \sqrt {2} e^{-i (c+d x)} \left (119 \left (-1+e^{2 i c}\right ) \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};-e^{2 i (c+d x)}\right )-55 \left (-1+e^{2 i c}\right ) e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-e^{2 i (c+d x)}\right )+119 \left (1+e^{2 i (c+d x)}\right )\right )}{\left (-1+e^{2 i c}\right ) d \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )}}\right )}{5 a^3 (\cos (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.88, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {\cos \left (d x + c\right )}}{a^{3} \cos \left (d x + c\right )^{6} + 3 \, a^{3} \cos \left (d x + c\right )^{5} + 3 \, a^{3} \cos \left (d x + c\right )^{4} + a^{3} \cos \left (d x + c\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \cos \left (d x + c\right ) + a\right )}^{3} \cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 453, normalized size = 2.19 \[ -\frac {\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (\frac {32 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{15 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {118 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {128 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{5 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}+\frac {238 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}+\frac {48 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {\sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{5 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}-\frac {4 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{3 \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}\right )}{4 a^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 = -0.00 \[ \int \frac {1}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (a+a\,\cos \left (c+d\,x\right )\right )}^3} \,d x \]
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 \[ \frac {\int \frac {1}{\cos ^{\frac {11}{2}}{\left (c + d x \right )} + 3 \cos ^{\frac {9}{2}}{\left (c + d x \right )} + 3 \cos ^{\frac {7}{2}}{\left (c + d x \right )} + \cos ^{\frac {5}{2}}{\left (c + d x \right )}}\, dx}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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