Optimal. Leaf size=157 \[ \frac {a^2 (a+6 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{16 b^{3/2} f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}+\frac {a (a+6 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{16 b f} \]
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Rubi [A] time = 0.14, antiderivative size = 157, 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 = {3190, 388, 195, 217, 206} \[ \frac {a^2 (a+6 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{16 b^{3/2} f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}+\frac {a (a+6 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{16 b f} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 388
Rule 3190
Rubi steps
\begin {align*} \int \cos ^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 x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {(a+6 b) \operatorname {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{6 b f}\\ &=\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {(a (a+6 b)) \operatorname {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\sin (e+f x)\right )}{8 b f}\\ &=\frac {a (a+6 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{16 b f}+\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {\left (a^2 (a+6 b)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{16 b f}\\ &=\frac {a (a+6 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{16 b f}+\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}+\frac {\left (a^2 (a+6 b)\right ) \operatorname {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{16 b f}\\ &=\frac {a^2 (a+6 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{16 b^{3/2} f}+\frac {a (a+6 b) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{16 b f}+\frac {(a+6 b) \sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{24 b f}-\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{6 b f}\\ \end {align*}
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Mathematica [A] time = 0.81, size = 149, normalized size = 0.95 \[ \frac {\sqrt {a+b \sin ^2(e+f x)} \left (3 a^{3/2} (a+6 b) \sinh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a}}\right )+\sqrt {b} \sin (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \left (-2 b (7 a-6 b) \sin ^2(e+f x)-3 a (a-10 b)-8 b^2 \sin ^4(e+f x)\right )\right )}{48 b^{3/2} f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 2.13, size = 577, normalized size = 3.68 \[ \left [\frac {3 \, {\left (a^{3} + 6 \, a^{2} b\right )} \sqrt {b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left (a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{6} - 24 \, {\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left (5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b} \sin \left (f x + e\right )\right ) - 8 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b - 16 \, a b^{2} - 4 \, b^{3} - 2 \, {\left (7 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{384 \, b^{2} f}, -\frac {3 \, {\left (a^{3} + 6 \, a^{2} b\right )} \sqrt {-b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 8 \, a b + 8 \, 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 )^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (8 \, b^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b - 16 \, a b^{2} - 4 \, b^{3} - 2 \, {\left (7 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{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 = 1.97, size = 277, normalized size = 1.76 \[ -\frac {b \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )}{6 f}+\frac {7 \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) a}{24 f}+\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) b}{12 f}-\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sin \left (f x +e \right ) a^{2}}{16 b f}+\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sin \left (f x +e \right ) a}{3 f}+\frac {b \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sin \left (f x +e \right )}{12 f}+\frac {a^{3} \ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\right )}{16 b^{\frac {3}{2}} f}+\frac {3 a^{2} \ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\right )}{8 \sqrt {b}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.37, size = 174, normalized size = 1.11 \[ \frac {\frac {3 \, a^{3} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {18 \, a^{2} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {b}} + 12 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right ) + 18 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right ) - \frac {8 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}} \sin \left (f x + e\right )}{b} + \frac {2 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} a \sin \left (f x + e\right )}{b} + \frac {3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a^{2} \sin \left (f x + e\right )}{b}}{48 \, f} \]
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 \[ \int {\cos \left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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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