Optimal. Leaf size=169 \[ \frac {(a-5 b) (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{16 b^{3/2} f}-\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 b f}+\frac {(a-5 b) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{24 b f}+\frac {(a-5 b) (a+b) \cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{16 b f} \]
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Rubi [A] time = 0.15, antiderivative size = 169, 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 = {3186, 388, 195, 217, 203} \[ \frac {(a-5 b) (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a-b \cos ^2(e+f x)+b}}\right )}{16 b^{3/2} f}-\frac {\cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{5/2}}{6 b f}+\frac {(a-5 b) \cos (e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{24 b f}+\frac {(a-5 b) (a+b) \cos (e+f x) \sqrt {a-b \cos ^2(e+f x)+b}}{16 b f} \]
Antiderivative was successfully verified.
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Rule 195
Rule 203
Rule 217
Rule 388
Rule 3186
Rubi steps
\begin {align*} \int \sin ^3(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=-\frac {\operatorname {Subst}\left (\int \left (1-x^2\right ) \left (a+b-b x^2\right )^{3/2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {(a-5 b) \operatorname {Subst}\left (\int \left (a+b-b x^2\right )^{3/2} \, dx,x,\cos (e+f x)\right )}{6 b f}\\ &=\frac {(a-5 b) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {((a-5 b) (a+b)) \operatorname {Subst}\left (\int \sqrt {a+b-b x^2} \, dx,x,\cos (e+f x)\right )}{8 b f}\\ &=\frac {(a-5 b) (a+b) \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{16 b f}+\frac {(a-5 b) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {\left ((a-5 b) (a+b)^2\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{16 b f}\\ &=\frac {(a-5 b) (a+b) \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{16 b f}+\frac {(a-5 b) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {\left ((a-5 b) (a+b)^2\right ) \operatorname {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{16 b f}\\ &=\frac {(a-5 b) (a+b)^2 \tan ^{-1}\left (\frac {\sqrt {b} \cos (e+f x)}{\sqrt {a+b-b \cos ^2(e+f x)}}\right )}{16 b^{3/2} f}+\frac {(a-5 b) (a+b) \cos (e+f x) \sqrt {a+b-b \cos ^2(e+f x)}}{16 b f}+\frac {(a-5 b) \cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\cos (e+f x) \left (a+b-b \cos ^2(e+f x)\right )^{5/2}}{6 b f}\\ \end {align*}
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Mathematica [A] time = 0.78, size = 152, normalized size = 0.90 \[ \frac {\frac {(a+b)^2 (5 b-a) \log \left (\sqrt {2 a-b \cos (2 (e+f x))+b}+\sqrt {2} \sqrt {-b} \cos (e+f x)\right )}{(-b)^{3/2}}-\frac {\cos (e+f x) \sqrt {2 a-b \cos (2 (e+f x))+b} \left (3 a^2-b (7 a+9 b) \cos (2 (e+f x))+29 a b+b^2 \cos (4 (e+f x))+23 b^2\right )}{3 \sqrt {2} b}}{16 f} \]
Warning: Unable to verify antiderivative.
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fricas [A] time = 1.95, size = 579, normalized size = 3.43 \[ \left [\frac {3 \, {\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} \sqrt {-b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{6} + 160 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4} - 32 \, {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} \cos \left (f x + e\right )^{2} + 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{7} - 24 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{5} + 10 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} - {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}\right ) - 8 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (7 \, a b^{2} + 13 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{2} b + 12 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{384 \, b^{2} f}, -\frac {3 \, {\left (a^{3} - 3 \, a^{2} b - 9 \, a b^{2} - 5 \, b^{3}\right )} \sqrt {b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{5} - 3 \, {\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{3} + {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )\right )}}\right ) + 4 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{5} - 2 \, {\left (7 \, a b^{2} + 13 \, b^{3}\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (a^{2} b + 12 \, a b^{2} + 11 \, b^{3}\right )} \cos \left (f x + e\right )\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{192 \, b^{2} f}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.82, size = 446, normalized size = 2.64 \[ -\frac {\sqrt {\left (\cos ^{2}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}\, \left (16 \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b^{\frac {7}{2}} \left (\cos ^{4}\left (f x +e \right )\right )-4 b^{\frac {5}{2}} \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (13 b +7 a \right ) \left (\cos ^{2}\left (f x +e \right )\right )+66 b^{\frac {7}{2}} \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}+72 a \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b^{\frac {5}{2}}+6 a^{2} \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b^{\frac {3}{2}}+3 \arctan \left (\frac {-2 b \left (\cos ^{2}\left (f x +e \right )\right )+a +b}{2 \sqrt {b}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right ) a^{3} b -9 \arctan \left (\frac {-2 b \left (\cos ^{2}\left (f x +e \right )\right )+a +b}{2 \sqrt {b}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right ) a^{2} b^{2}-27 b^{3} a \arctan \left (\frac {-2 b \left (\cos ^{2}\left (f x +e \right )\right )+a +b}{2 \sqrt {b}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )-15 b^{4} \arctan \left (\frac {-2 b \left (\cos ^{2}\left (f x +e \right )\right )+a +b}{2 \sqrt {b}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}}\right )\right )}{96 b^{\frac {5}{2}} \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 248, normalized size = 1.47 \[ \frac {\frac {3 \, {\left (a + b\right )}^{2} a \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{b^{\frac {3}{2}}} + \frac {3 \, {\left (a + b\right )}^{2} \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {b}} - \frac {18 \, {\left (a + b\right )} a \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right )}{\sqrt {b}} - 18 \, {\left (a + b\right )} \sqrt {b} \arcsin \left (\frac {b \cos \left (f x + e\right )}{\sqrt {{\left (a + b\right )} b}}\right ) - 12 \, {\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} \cos \left (f x + e\right ) - 18 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \cos \left (f x + e\right ) - \frac {8 \, {\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {5}{2}} \cos \left (f x + e\right )}{b} + \frac {2 \, {\left (-b \cos \left (f x + e\right )^{2} + a + b\right )}^{\frac {3}{2}} {\left (a + b\right )} \cos \left (f x + e\right )}{b} + \frac {3 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )}^{2} \cos \left (f x + e\right )}{b}}{48 \, f} \]
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 \[ \int {\sin \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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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