Optimal. Leaf size=160 \[ \frac {2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac {2 a \left (a^2+6 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.24, antiderivative size = 160, 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 = {2691, 2862, 2669, 2640, 2639} \[ \frac {2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac {2 a \left (a^2+6 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac {2 (a \sin (c+d x)+b) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rule 2691
Rule 2862
Rubi steps
\begin {align*} \int \frac {(a+b \sin (c+d x))^3}{(e \cos (c+d x))^{3/2}} \, dx &=\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}}-\frac {2 \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \left (\frac {a^2}{2}+2 b^2+\frac {5}{2} a b \sin (c+d x)\right ) \, dx}{e^2}\\ &=\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}}-\frac {4 \int \sqrt {e \cos (c+d x)} \left (\frac {5}{4} a \left (a^2+6 b^2\right )+\frac {5}{4} b \left (3 a^2+4 b^2\right ) \sin (c+d x)\right ) \, dx}{5 e^2}\\ &=\frac {2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (a \left (a^2+6 b^2\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{e^2}\\ &=\frac {2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}}-\frac {\left (a \left (a^2+6 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{e^2 \sqrt {\cos (c+d x)}}\\ &=\frac {2 b \left (3 a^2+4 b^2\right ) (e \cos (c+d x))^{3/2}}{3 d e^3}-\frac {2 a \left (a^2+6 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d e^2 \sqrt {\cos (c+d x)}}+\frac {2 a b (e \cos (c+d x))^{3/2} (a+b \sin (c+d x))}{d e^3}+\frac {2 (b+a \sin (c+d x)) (a+b \sin (c+d x))^2}{d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 98, normalized size = 0.61 \[ \frac {-6 a \left (a^2+6 b^2\right ) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+6 \left (a \left (a^2+3 b^2\right ) \sin (c+d x)+3 a^2 b+b^3\right )+2 b^3 \cos ^2(c+d x)}{3 d e \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.09, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (3 \, a b^{2} \cos \left (d x + c\right )^{2} - a^{3} - 3 \, a b^{2} + {\left (b^{3} \cos \left (d x + c\right )^{2} - 3 \, a^{2} b - b^{3}\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{e^{2} \cos \left (d x + c\right )^{2}}, x\right ) \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 2.69, size = 248, normalized size = 1.55 \[ -\frac {2 \left (-4 b^{3} \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}+18 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}-6 a^{3} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-18 a \,b^{2} \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b^{3} \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-9 a^{2} b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )-4 b^{3} \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 e \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) d} \]
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 \[ \int \frac {{\left (b \sin \left (d x + c\right ) + a\right )}^{3}}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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