Optimal. Leaf size=178 \[ -\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {78 \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}{35 d e}+\frac {78 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {26 \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}{35 d e}-\frac {2 a (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}{7 d e} \]
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Rubi [A] time = 0.21, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {2678, 2669, 2642, 2641} \[ -\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {26 \left (a^2 \sin (c+d x)+a^2\right )^2 \sqrt {e \cos (c+d x)}}{35 d e}-\frac {78 \left (a^4 \sin (c+d x)+a^4\right ) \sqrt {e \cos (c+d x)}}{35 d e}+\frac {78 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 a (a \sin (c+d x)+a)^3 \sqrt {e \cos (c+d x)}}{7 d e} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rule 2678
Rubi steps
\begin {align*} \int \frac {(a+a \sin (c+d x))^4}{\sqrt {e \cos (c+d x)}} \, dx &=-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}+\frac {1}{7} (13 a) \int \frac {(a+a \sin (c+d x))^3}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}+\frac {1}{35} \left (117 a^2\right ) \int \frac {(a+a \sin (c+d x))^2}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac {1}{7} \left (39 a^3\right ) \int \frac {a+a \sin (c+d x)}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac {1}{7} \left (39 a^4\right ) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}+\frac {\left (39 a^4 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{7 \sqrt {e \cos (c+d x)}}\\ &=-\frac {78 a^4 \sqrt {e \cos (c+d x)}}{7 d e}+\frac {78 a^4 \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{7 d \sqrt {e \cos (c+d x)}}-\frac {2 a \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3}{7 d e}-\frac {26 \sqrt {e \cos (c+d x)} \left (a^2+a^2 \sin (c+d x)\right )^2}{35 d e}-\frac {78 \sqrt {e \cos (c+d x)} \left (a^4+a^4 \sin (c+d x)\right )}{35 d e}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 64, normalized size = 0.36 \[ -\frac {32 \sqrt [4]{2} a^4 \sqrt {e \cos (c+d x)} \, _2F_1\left (-\frac {13}{4},\frac {1}{4};\frac {5}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{d e \sqrt [4]{\sin (c+d x)+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\sqrt {e \cos \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.86, size = 222, normalized size = 1.25 \[ -\frac {2 a^{4} \left (80 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-120 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+224 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-280 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-336 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+195 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}+160 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-392 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+252 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{35 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \frac {{\left (a \sin \left (d x + c\right ) + a\right )}^{4}}{\sqrt {e \cos \left (d x + c\right )}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4}{\sqrt {e\,\cos \left (c+d\,x\right )}} \,d x \]
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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