Optimal. Leaf size=79 \[ \frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 b^{3/2} f}-\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 b f} \]
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Rubi [A] time = 0.09, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3190, 388, 217, 206} \[ \frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 b^{3/2} f}-\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 b f} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 388
Rule 3190
Rubi steps
\begin {align*} \int \frac {\cos ^3(e+f x)}{\sqrt {a+b \sin ^2(e+f x)}} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 b f}+\frac {(a+2 b) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 b f}\\ &=-\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 b f}+\frac {(a+2 b) \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 )}{2 b f}\\ &=\frac {(a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 b^{3/2} f}-\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 b f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 79, normalized size = 1.00 \[ \frac {-\frac {(-a-2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 b^{3/2}}-\frac {\sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 b}}{f} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 461, normalized size = 5.84 \[ \left [\frac {{\left (a + 2 \, b\right )} \sqrt {b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left (a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{6} - 24 \, {\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left (5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b} \sin \left (f x + e\right )\right ) - 8 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b \sin \left (f x + e\right )}{16 \, b^{2} f}, -\frac {{\left (a + 2 \, b\right )} \sqrt {-b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} b \sin \left (f x + e\right )}{8 \, b^{2} f}\right ] \]
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 [A] time = 1.49, size = 98, normalized size = 1.24 \[ -\frac {\sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{2 b f}+\frac {a \ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{2 f \,b^{\frac {3}{2}}}+\frac {\ln \left (\sin \left (f x +e \right ) \sqrt {b}+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{f \sqrt {b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 69, normalized size = 0.87 \[ \frac {\frac {a \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{b^{\frac {3}{2}}} + \frac {2 \, \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {b}} - \frac {\sqrt {b \sin \left (f x + e\right )^{2} + a} \sin \left (f x + e\right )}{b}}{2 \, f} \]
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 \[ \int \frac {{\cos \left (e+f\,x\right )}^3}{\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}} \,d x \]
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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