Optimal. Leaf size=210 \[ -\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac {2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e} \]
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Rubi [A] time = 0.44, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2692, 2862, 2669, 2640, 2639} \[ -\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}+\frac {2 \left (36 a^2 b^2+15 a^4+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rule 2692
Rule 2862
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^4 \, dx &=-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {2}{9} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2 \left (\frac {9 a^2}{2}+3 b^2+\frac {15}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {4}{63} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac {3}{4} a \left (21 a^2+34 b^2\right )+\frac {3}{4} b \left (41 a^2+14 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {8}{315} \int \sqrt {e \cos (c+d x)} \left (\frac {21}{8} \left (15 a^4+36 a^2 b^2+4 b^4\right )+\frac {33}{8} a b \left (17 a^2+18 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {1}{15} \left (15 a^4+36 a^2 b^2+4 b^4\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}+\frac {\left (\left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)}}\\ &=-\frac {22 a b \left (17 a^2+18 b^2\right ) (e \cos (c+d x))^{3/2}}{315 d e}+\frac {2 \left (15 a^4+36 a^2 b^2+4 b^4\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (41 a^2+14 b^2\right ) (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{105 d e}-\frac {10 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{21 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^3}{9 d e}\\ \end {align*}
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Mathematica [A] time = 1.10, size = 137, normalized size = 0.65 \[ \frac {\sqrt {e \cos (c+d x)} \left (84 \left (15 a^4+36 a^2 b^2+4 b^4\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )-b \cos ^{\frac {3}{2}}(c+d x) \left (5 \left (336 a^3+264 a b^2-7 b^3 \sin (3 (c+d x))\right )+21 b \left (72 a^2+13 b^2\right ) \sin (c+d x)-360 a b^2 \cos (2 (c+d x))\right )\right )}{630 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{4} \cos \left (d x + c\right )^{4} + a^{4} + 6 \, a^{2} b^{2} + b^{4} - 2 \, {\left (3 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2} - 4 \, {\left (a b^{3} \cos \left (d x + c\right )^{2} - a^{3} b - a b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.59, size = 525, normalized size = 2.50 \[ \frac {2 e \left (1120 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2880 a \,b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2240 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-3024 a^{2} b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-5760 a \,b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1064 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1680 a^{3} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3024 a^{2} b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2640 a \,b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+56 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+315 \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 ) a^{4}+756 \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 ) a^{2} b^{2}+84 \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 ) b^{4}+1680 a^{3} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-756 a^{2} b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+240 a \,b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-84 b^{4} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 a^{3} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-240 a \,b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{315 \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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^4 \,d x \]
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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