Optimal. Leaf size=119 \[ \frac {32 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {56 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {8 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 119, 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, 2639, 2641, 2635} \[ \frac {32 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {56 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {2 a^4 \sin (c+d x) \cos ^{\frac {3}{2}}(c+d x)}{5 d}+\frac {8 a^4 \sin (c+d x) \sqrt {\cos (c+d x)}}{3 d}+\frac {2 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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 {3}{2}}(c+d x)} \, dx &=\int \left (\frac {a^4}{\cos ^{\frac {3}{2}}(c+d x)}+\frac {4 a^4}{\sqrt {\cos (c+d x)}}+6 a^4 \sqrt {\cos (c+d x)}+4 a^4 \cos ^{\frac {3}{2}}(c+d x)+a^4 \cos ^{\frac {5}{2}}(c+d x)\right ) \, dx\\ &=a^4 \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x)} \, dx+a^4 \int \cos ^{\frac {5}{2}}(c+d x) \, dx+\left (4 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx+\left (4 a^4\right ) \int \cos ^{\frac {3}{2}}(c+d x) \, dx+\left (6 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {12 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {8 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}+\frac {2 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {8 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}+\frac {1}{5} \left (3 a^4\right ) \int \sqrt {\cos (c+d x)} \, dx-a^4 \int \sqrt {\cos (c+d x)} \, dx+\frac {1}{3} \left (4 a^4\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx\\ &=\frac {56 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d}+\frac {32 a^4 F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d}+\frac {2 a^4 \sin (c+d x)}{d \sqrt {\cos (c+d x)}}+\frac {8 a^4 \sqrt {\cos (c+d x)} \sin (c+d x)}{3 d}+\frac {2 a^4 \cos ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{5 d}\\ \end {align*}
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Mathematica [C] time = 6.20, size = 497, normalized size = 4.18 \[ -\frac {7 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4 \left (\frac {\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2\left (d x+\tan ^{-1}(\tan (c))\right )\right )}{\sqrt {\tan ^2(c)+1} \sqrt {1-\cos \left (\tan ^{-1}(\tan (c))+d x\right )} \sqrt {\cos \left (\tan ^{-1}(\tan (c))+d x\right )+1} \sqrt {\cos (c) \sqrt {\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}-\frac {\frac {\tan (c) \sin \left (\tan ^{-1}(\tan (c))+d x\right )}{\sqrt {\tan ^2(c)+1}}+\frac {2 \cos ^2(c) \sqrt {\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}{\sin ^2(c)+\cos ^2(c)}}{\sqrt {\cos (c) \sqrt {\tan ^2(c)+1} \cos \left (\tan ^{-1}(\tan (c))+d x\right )}}\right )}{20 d}-\frac {2 \csc (c) \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4 \sqrt {1-\sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {\sin (c) \left (-\sqrt {\cot ^2(c)+1}\right ) \sin \left (d x-\tan ^{-1}(\cot (c))\right )} \sqrt {\sin \left (d x-\tan ^{-1}(\cot (c))\right )+1} \sec \left (d x-\tan ^{-1}(\cot (c))\right ) \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};\sin ^2\left (d x-\tan ^{-1}(\cot (c))\right )\right )}{3 d \sqrt {\cot ^2(c)+1}}+\sqrt {\cos (c+d x)} \sec ^8\left (\frac {c}{2}+\frac {d x}{2}\right ) (a \cos (c+d x)+a)^4 \left (\frac {\sin (c) \cos (d x)}{6 d}+\frac {\sin (2 c) \cos (2 d x)}{80 d}+\frac {\cos (c) \sin (d x)}{6 d}+\frac {\cos (2 c) \sin (2 d x)}{80 d}+\frac {\sec (c) \sin (d x) \sec (c+d x)}{8 d}-\frac {(33 \cos (2 c)+23) \csc (c) \sec (c)}{80 d}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} \cos \left (d x + c\right )^{4} + 4 \, a^{4} \cos \left (d x + c\right )^{3} + 6 \, a^{4} \cos \left (d x + c\right )^{2} + 4 \, a^{4} \cos \left (d x + c\right ) + a^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}}, x\right ) \]
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 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.52, size = 194, normalized size = 1.63 \[ -\frac {8 a^{4} \left (-6 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+26 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+20 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-21 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-19 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (a \cos \left (d x + c\right ) + a\right )}^{4}}{\cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.79, size = 149, normalized size = 1.25 \[ \frac {12\,a^4\,\mathrm {E}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{d}+\frac {32\,a^4\,\mathrm {F}\left (\frac {c}{2}+\frac {d\,x}{2}\middle |2\right )}{3\,d}+\frac {8\,a^4\,\sqrt {\cos \left (c+d\,x\right )}\,\sin \left (c+d\,x\right )}{3\,d}+\frac {2\,a^4\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}\,\sqrt {{\sin \left (c+d\,x\right )}^2}}-\frac {2\,a^4\,{\cos \left (c+d\,x\right )}^{7/2}\,\sin \left (c+d\,x\right )\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {7}{4};\ \frac {11}{4};\ {\cos \left (c+d\,x\right )}^2\right )}{7\,d\,\sqrt {{\sin \left (c+d\,x\right )}^2}} \]
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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