Optimal. Leaf size=98 \[ \frac {40 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {8 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2757, 2636, 2641, 2639, 2635} \[ \frac {40 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {2 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {8 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2636
Rule 2639
Rule 2641
Rule 2757
Rubi steps
\begin {align*} \int \frac {(a+a \cos (c+d x))^4}{\cos ^{\frac {5}{2}}(c+d x)} \, dx &=\int \left (\frac {a^4}{\cos ^{\frac {5}{2}}(c+d x)}+\frac {4 a^4}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {6 a^4}{\sqrt {\cos (c+d x)}}+4 a^4 \sqrt {\cos (c+d x)}+a^4 \cos ^{\frac {3}{2}}(c+d x)\right ) \, dx\\ &=a^4 \int \frac {1}{\cos ^{\frac {5}{2}}(c+d x)} \, dx+a^4 \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+\left (4 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx+\left (6 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {8 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {12 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+2 \left (\frac {1}{3} a^4 \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\right )-\left (4 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {40 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a^4 \sin (c+d x)}{3 d \cos ^{\frac {3}{2}}(c+d x)}+\frac {8 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {2 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 70, normalized size = 0.71 \[ \frac {a^4 \left (5 \sin (c+d x)+24 \sin (2 (c+d x))+\sin (3 (c+d x))+80 \cos ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{6 d \cos ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 2.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} \cos \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right ) + a^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.86, size = 292, normalized size = 2.98 \[ \frac {8 \sqrt {-\left (-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, a^{4} \left (2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+10 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-14 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+7 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}}{3 \left (4 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-4 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1\right ) \sin \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 145, normalized size = 1.48 \[ \frac {2\,\left (12\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+19\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )+a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )\right )}{3\,d}+\frac {8\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}+\frac {2\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{4},\frac {1}{2};\ \frac {1}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{3\,d\,{\cos \left (c+d\,x\right )}^{3/2}\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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