Optimal. Leaf size=185 \[ \frac {3 a^2 b x \sqrt {a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{2 \left (a^2 \sin (d+e x)+a b\right )}-\frac {a^2 b \sin (d+e x) \cos (d+e x) \sqrt {a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{2 e \left (a^2 \sin (d+e x)+a b\right )}-\frac {\left (a^2+b^2\right ) \cos (d+e x) \sqrt {a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{e (a \sin (d+e x)+b)} \]
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Rubi [A] time = 0.11, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.049, Rules used = {3290, 2734} \[ \frac {3 a^2 b x \sqrt {a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{2 \left (a^2 \sin (d+e x)+a b\right )}-\frac {a^2 b \sin (d+e x) \cos (d+e x) \sqrt {a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{2 e \left (a^2 \sin (d+e x)+a b\right )}-\frac {\left (a^2+b^2\right ) \cos (d+e x) \sqrt {a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2}}{e (a \sin (d+e x)+b)} \]
Antiderivative was successfully verified.
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Rule 2734
Rule 3290
Rubi steps
\begin {align*} \int (a+b \sin (d+e x)) \sqrt {b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \, dx &=\frac {\sqrt {b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)} \int \left (2 a b+2 a^2 \sin (d+e x)\right ) (a+b \sin (d+e x)) \, dx}{2 a b+2 a^2 \sin (d+e x)}\\ &=-\frac {\left (a^2+b^2\right ) \cos (d+e x) \sqrt {b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)}}{e (b+a \sin (d+e x))}+\frac {3 a^2 b x \sqrt {b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)}}{2 \left (a b+a^2 \sin (d+e x)\right )}-\frac {a^2 b \cos (d+e x) \sin (d+e x) \sqrt {b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)}}{2 e \left (a b+a^2 \sin (d+e x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 70, normalized size = 0.38 \[ -\frac {\sqrt {(a \sin (d+e x)+b)^2} \left (4 \left (a^2+b^2\right ) \cos (d+e x)+a b (\sin (2 (d+e x))-6 (d+e x))\right )}{4 e (a \sin (d+e x)+b)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 43, normalized size = 0.23 \[ \frac {3 \, a b e x - a b \cos \left (e x + d\right ) \sin \left (e x + d\right ) - 2 \, {\left (a^{2} + b^{2}\right )} \cos \left (e x + d\right )}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 98, normalized size = 0.53 \[ -a^{2} \cos \left (x e + d\right ) e^{\left (-1\right )} \mathrm {sgn}\left (a \sin \left (x e + d\right ) + b\right ) - b^{2} \cos \left (x e + d\right ) e^{\left (-1\right )} \mathrm {sgn}\left (a \sin \left (x e + d\right ) + b\right ) - \frac {1}{4} \, a b e^{\left (-1\right )} \mathrm {sgn}\left (a \sin \left (x e + d\right ) + b\right ) \sin \left (2 \, x e + 2 \, d\right ) + \frac {3}{2} \, a b x \mathrm {sgn}\left (a \sin \left (x e + d\right ) + b\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.59, size = 107, normalized size = 0.58 \[ -\frac {\sqrt {-a^{2} \left (\cos ^{2}\left (e x +d \right )\right )+2 a b \sin \left (e x +d \right )+a^{2}+b^{2}}\, \left (\sin \left (e x +d \right ) \cos \left (e x +d \right ) a b +2 a^{2} \cos \left (e x +d \right )+2 \cos \left (e x +d \right ) b^{2}-3 \left (e x +d \right ) a b +2 a^{2}+2 b^{2}\right )}{2 e \left (b +a \sin \left (e x +d \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 187, normalized size = 1.01 \[ \frac {2 \, {\left (b \arctan \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right ) - \frac {a}{\frac {\sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + 1}\right )} a + {\left (a \arctan \left (\frac {\sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1}\right ) - \frac {2 \, b + \frac {a \sin \left (e x + d\right )}{\cos \left (e x + d\right ) + 1} + \frac {2 \, b \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} - \frac {a \sin \left (e x + d\right )^{3}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{3}}}{\frac {2 \, \sin \left (e x + d\right )^{2}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{2}} + \frac {\sin \left (e x + d\right )^{4}}{{\left (\cos \left (e x + d\right ) + 1\right )}^{4}} + 1}\right )} b}{e} \]
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 \[ \int \left (a+b\,\sin \left (d+e\,x\right )\right )\,\sqrt {a^2\,{\sin \left (d+e\,x\right )}^2+2\,a\,b\,\sin \left (d+e\,x\right )+b^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \sin {\left (d + e x \right )}\right ) \sqrt {\left (a \sin {\left (d + e x \right )} + b\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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