Optimal. Leaf size=178 \[ -\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {22 \left (a^4 \sin (c+d x)+a^4\right ) (e \cos (c+d x))^{3/2}}{21 d e}+\frac {22 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}-\frac {10 \left (a^2 \sin (c+d x)+a^2\right )^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e} \]
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Rubi [A] time = 0.20, 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, 2640, 2639} \[ -\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {10 \left (a^2 \sin (c+d x)+a^2\right )^2 (e \cos (c+d x))^{3/2}}{21 d e}-\frac {22 \left (a^4 \sin (c+d x)+a^4\right ) (e \cos (c+d x))^{3/2}}{21 d e}+\frac {22 a^4 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{3 d \sqrt {\cos (c+d x)}}-\frac {2 a (a \sin (c+d x)+a)^3 (e \cos (c+d x))^{3/2}}{9 d e} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rule 2678
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^4 \, dx &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}+\frac {1}{3} (5 a) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^3 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}+\frac {1}{21} \left (55 a^2\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^2 \, dx\\ &=-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac {1}{3} \left (11 a^3\right ) \int \sqrt {e \cos (c+d x)} (a+a \sin (c+d x)) \, dx\\ &=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac {1}{3} \left (11 a^4\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}+\frac {\left (11 a^4 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{3 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a^4 (e \cos (c+d x))^{3/2}}{9 d e}+\frac {22 a^4 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{3 d \sqrt {\cos (c+d x)}}-\frac {2 a (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^3}{9 d e}-\frac {10 (e \cos (c+d x))^{3/2} \left (a^2+a^2 \sin (c+d x)\right )^2}{21 d e}-\frac {22 (e \cos (c+d x))^{3/2} \left (a^4+a^4 \sin (c+d x)\right )}{21 d e}\\ \end {align*}
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Mathematica [C] time = 0.06, size = 66, normalized size = 0.37 \[ -\frac {32\ 2^{3/4} a^4 (e \cos (c+d x))^{3/2} \, _2F_1\left (-\frac {15}{4},\frac {3}{4};\frac {7}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{3 d e (\sin (c+d x)+1)^{3/4}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.86, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{4} \cos \left (d x + c\right )^{4} - 8 \, a^{4} \cos \left (d x + c\right )^{2} + 8 \, a^{4} - 4 \, {\left (a^{4} \cos \left (d x + c\right )^{2} - 2 \, a^{4}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.98, size = 258, normalized size = 1.45 \[ \frac {2 a^{4} e \left (224 \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-448 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+576 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-392 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-1152 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+616 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+192 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+231 \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}-168 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+384 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-132 \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{63 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (a \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
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 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]
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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