Optimal. Leaf size=239 \[ -\frac {\cos (d+e x) (a \sin (d+e x)+b)}{2 e \left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^{3/2}}-\frac {\left (a^2 \sin (d+e x)+a b\right )^3 \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 e \left (a^2-b^2\right )^{3/2} \left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^{3/2}}+\frac {b \cos (d+e x) \left (a^2 \sin (d+e x)+a b\right )^3}{2 e \left (a^2-b^2\right ) \left (a^4 \sin (d+e x)+a^3 b\right ) \left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.27, antiderivative size = 239, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.146, Rules used = {3290, 2754, 12, 2660, 618, 206} \[ \frac {b \cos (d+e x) \left (a^2 \sin (d+e x)+a b\right )^3}{2 e \left (a^2-b^2\right ) \left (a^3 b+a^4 \sin (d+e x)\right ) \left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^{3/2}}-\frac {\cos (d+e x) (a \sin (d+e x)+b)}{2 e \left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^{3/2}}-\frac {\left (a^2 \sin (d+e x)+a b\right )^3 \tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2}}\right )}{a^2 e \left (a^2-b^2\right )^{3/2} \left (a^2 \sin ^2(d+e x)+2 a b \sin (d+e x)+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 618
Rule 2660
Rule 2754
Rule 3290
Rubi steps
\begin {align*} \int \frac {a+b \sin (d+e x)}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}} \, dx &=\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \int \frac {a+b \sin (d+e x)}{\left (2 a b+2 a^2 \sin (d+e x)\right )^3} \, dx}{\left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \int \frac {2 a \left (a^2-b^2\right ) \sin (d+e x)}{\left (2 a b+2 a^2 \sin (d+e x)\right )^2} \, dx}{8 a^2 \left (a^2-b^2\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \int \frac {\sin (d+e x)}{\left (2 a b+2 a^2 \sin (d+e x)\right )^2} \, dx}{4 a \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {b \cos (d+e x) \left (a b+a^2 \sin (d+e x)\right )^3}{2 \left (a^2-b^2\right ) e \left (a^3 b+a^4 \sin (d+e x)\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \int \frac {2 a^2}{2 a b+2 a^2 \sin (d+e x)} \, dx}{16 a^3 \left (a^2-b^2\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {b \cos (d+e x) \left (a b+a^2 \sin (d+e x)\right )^3}{2 \left (a^2-b^2\right ) e \left (a^3 b+a^4 \sin (d+e x)\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \int \frac {1}{2 a b+2 a^2 \sin (d+e x)} \, dx}{8 a \left (a^2-b^2\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {b \cos (d+e x) \left (a b+a^2 \sin (d+e x)\right )^3}{2 \left (a^2-b^2\right ) e \left (a^3 b+a^4 \sin (d+e x)\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \operatorname {Subst}\left (\int \frac {1}{2 a b+4 a^2 x+2 a b x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{4 a \left (a^2-b^2\right ) e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {b \cos (d+e x) \left (a b+a^2 \sin (d+e x)\right )^3}{2 \left (a^2-b^2\right ) e \left (a^3 b+a^4 \sin (d+e x)\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}-\frac {\left (2 a b+2 a^2 \sin (d+e x)\right )^3 \operatorname {Subst}\left (\int \frac {1}{16 a^2 \left (a^2-b^2\right )-x^2} \, dx,x,4 a^2+4 a b \tan \left (\frac {1}{2} (d+e x)\right )\right )}{2 a \left (a^2-b^2\right ) e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ &=-\frac {\cos (d+e x) (b+a \sin (d+e x))}{2 e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}-\frac {\tanh ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2-b^2}}\right ) \left (a b+a^2 \sin (d+e x)\right )^3}{a^2 \left (a^2-b^2\right )^{3/2} e \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}+\frac {b \cos (d+e x) \left (a b+a^2 \sin (d+e x)\right )^3}{2 \left (a^2-b^2\right ) e \left (a^3 b+a^4 \sin (d+e x)\right ) \left (b^2+2 a b \sin (d+e x)+a^2 \sin ^2(d+e x)\right )^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 144, normalized size = 0.60 \[ \frac {\sqrt {b^2-a^2} \cos (d+e x) \left (a^2-a b \sin (d+e x)-2 b^2\right )-2 a (a \sin (d+e x)+b)^2 \tan ^{-1}\left (\frac {a+b \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {b^2-a^2}}\right )}{2 e (b-a) (a+b) \sqrt {b^2-a^2} (a \sin (d+e x)+b) \sqrt {(a \sin (d+e x)+b)^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.65, size = 527, normalized size = 2.21 \[ \left [-\frac {2 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (a^{3} \cos \left (e x + d\right )^{2} - 2 \, a^{2} b \sin \left (e x + d\right ) - a^{3} - a b^{2}\right )} \sqrt {a^{2} - b^{2}} \log \left (\frac {{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (e x + d\right )^{2} + 2 \, a b \sin \left (e x + d\right ) + a^{2} + b^{2} + 2 \, {\left (b \cos \left (e x + d\right ) \sin \left (e x + d\right ) + a \cos \left (e x + d\right )\right )} \sqrt {a^{2} - b^{2}}}{a^{2} \cos \left (e x + d\right )^{2} - 2 \, a b \sin \left (e x + d\right ) - a^{2} - b^{2}}\right ) - 2 \, {\left (a^{4} - 3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (e x + d\right )}{4 \, {\left ({\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} e \cos \left (e x + d\right )^{2} - 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} e \sin \left (e x + d\right ) - {\left (a^{6} - a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} e\right )}}, -\frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (e x + d\right ) \sin \left (e x + d\right ) + {\left (a^{3} \cos \left (e x + d\right )^{2} - 2 \, a^{2} b \sin \left (e x + d\right ) - a^{3} - a b^{2}\right )} \sqrt {-a^{2} + b^{2}} \arctan \left (-\frac {\sqrt {-a^{2} + b^{2}} {\left (b \sin \left (e x + d\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (e x + d\right )}\right ) - {\left (a^{4} - 3 \, a^{2} b^{2} + 2 \, b^{4}\right )} \cos \left (e x + d\right )}{2 \, {\left ({\left (a^{6} - 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} e \cos \left (e x + d\right )^{2} - 2 \, {\left (a^{5} b - 2 \, a^{3} b^{3} + a b^{5}\right )} e \sin \left (e x + d\right ) - {\left (a^{6} - a^{4} b^{2} - a^{2} b^{4} + b^{6}\right )} e\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.75, size = 479, normalized size = 2.00 \[ {\left (\frac {a \arctan \left (\frac {b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a}{\sqrt {-a^{2} + b^{2}}}\right )}{{\left (a^{2} \mathrm {sgn}\left (b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b\right ) - b^{2} \mathrm {sgn}\left (b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b\right )\right )} \sqrt {-a^{2} + b^{2}}} - \frac {2 \, a^{3} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 3 \, a b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 2 \, a^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 3 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 2 \, b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a^{3} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 5 \, a b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + a^{2} b^{2} - 2 \, b^{4}}{{\left (a^{2} b^{2} \mathrm {sgn}\left (b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b\right ) - b^{4} \mathrm {sgn}\left (b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 2 \, b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b\right )\right )} {\left (b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 2 \, a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + b\right )}^{2}}\right )} e^{\left (-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 741, normalized size = 3.10 \[ \frac {2 \sin \left (e x +d \right ) \left (\cos ^{2}\left (e x +d \right )\right ) \arctan \left (\frac {b \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b}{\sin \left (e x +d \right ) \sqrt {-a^{2}+b^{2}}}\right ) a^{4} b^{2}+\sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) \left (\cos ^{2}\left (e x +d \right )\right ) a^{5}-2 \sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) \left (\cos ^{2}\left (e x +d \right )\right ) a^{3} b^{2}-\sqrt {-a^{2}+b^{2}}\, \left (\cos ^{3}\left (e x +d \right )\right ) a^{2} b^{3}+6 \left (\cos ^{2}\left (e x +d \right )\right ) \arctan \left (\frac {b \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b}{\sin \left (e x +d \right ) \sqrt {-a^{2}+b^{2}}}\right ) a^{3} b^{3}-\sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) \cos \left (e x +d \right ) a^{3} b^{2}+3 \sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) \cos \left (e x +d \right ) a \,b^{4}+3 \sqrt {-a^{2}+b^{2}}\, \left (\cos ^{2}\left (e x +d \right )\right ) a^{4} b -6 \sqrt {-a^{2}+b^{2}}\, \left (\cos ^{2}\left (e x +d \right )\right ) a^{2} b^{3}-2 \sin \left (e x +d \right ) \arctan \left (\frac {b \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b}{\sin \left (e x +d \right ) \sqrt {-a^{2}+b^{2}}}\right ) a^{4} b^{2}-6 \sin \left (e x +d \right ) \arctan \left (\frac {b \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b}{\sin \left (e x +d \right ) \sqrt {-a^{2}+b^{2}}}\right ) a^{2} b^{4}-\sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) a^{5}-\sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) a^{3} b^{2}+6 \sqrt {-a^{2}+b^{2}}\, \sin \left (e x +d \right ) a \,b^{4}+2 \sqrt {-a^{2}+b^{2}}\, \cos \left (e x +d \right ) b^{5}-6 \arctan \left (\frac {b \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b}{\sin \left (e x +d \right ) \sqrt {-a^{2}+b^{2}}}\right ) a^{3} b^{3}-2 \arctan \left (\frac {b \cos \left (e x +d \right )-a \sin \left (e x +d \right )-b}{\sin \left (e x +d \right ) \sqrt {-a^{2}+b^{2}}}\right ) a \,b^{5}-3 \sqrt {-a^{2}+b^{2}}\, a^{4} b +5 \sqrt {-a^{2}+b^{2}}\, a^{2} b^{3}+2 \sqrt {-a^{2}+b^{2}}\, b^{5}}{2 e \left (-a^{2} \left (\cos ^{2}\left (e x +d \right )\right )+2 a b \sin \left (e x +d \right )+a^{2}+b^{2}\right )^{\frac {3}{2}} \sqrt {-a^{2}+b^{2}}\, \left (a^{2}-b^{2}\right ) b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
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 \[ \int \frac {a+b\,\sin \left (d+e\,x\right )}{{\left (a^2\,{\sin \left (d+e\,x\right )}^2+2\,a\,b\,\sin \left (d+e\,x\right )+b^2\right )}^{3/2}} \,d x \]
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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