Optimal. Leaf size=156 \[ -\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{105 d e}+\frac {2 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e} \]
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Rubi [A] time = 0.24, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{105 d e}+\frac {2 a \left (5 a^2+6 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e} \]
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))^3 \, dx &=-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}+\frac {2}{7} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac {7 a^2}{2}+2 b^2+\frac {11}{2} a b \sin (c+d x)\right ) \, dx\\ &=-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}+\frac {4}{35} \int \sqrt {e \cos (c+d x)} \left (\frac {7}{4} a \left (5 a^2+6 b^2\right )+\frac {1}{4} b \left (57 a^2+20 b^2\right ) \sin (c+d x)\right ) \, dx\\ &=-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{105 d e}-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}+\frac {1}{5} \left (a \left (5 a^2+6 b^2\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{105 d e}-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}+\frac {\left (a \left (5 a^2+6 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=-\frac {2 b \left (57 a^2+20 b^2\right ) (e \cos (c+d x))^{3/2}}{105 d e}+\frac {2 a \left (5 a^2+6 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}-\frac {22 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{35 d e}-\frac {2 b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))^2}{7 d e}\\ \end {align*}
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Mathematica [A] time = 0.63, size = 101, normalized size = 0.65 \[ \frac {\sqrt {e \cos (c+d x)} \left (42 \left (5 a^3+6 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+b \cos ^{\frac {3}{2}}(c+d x) \left (-210 a^2-126 a b \sin (c+d x)+15 b^2 \cos (2 (c+d x))-55 b^2\right )\right )}{105 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.28, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 2.12, size = 339, normalized size = 2.17 \[ \frac {2 e \left (240 b^{3} \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-504 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-480 b^{3} \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-420 a^{2} b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+504 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+220 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+105 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}+126 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}+420 a^{2} b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-126 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+20 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-105 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-20 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{105 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}\,{d x} \]
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 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3 \,d x \]
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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