Optimal. Leaf size=321 \[ -\frac {\left (a^2-9 a b-2 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {a (a+b) \left (2 a^2+9 a b-b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) \left (a^2+6 a b+b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{35 b^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {b \sin (e+f x) \cos ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}+\frac {2 (4 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f} \]
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Rubi [A] time = 0.39, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3192, 416, 528, 524, 426, 424, 421, 419} \[ -\frac {\left (a^2-9 a b-2 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {a (a+b) \left (2 a^2+9 a b-b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) \left (a^2+6 a b+b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right )}{35 b^2 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {b \sin (e+f x) \cos ^5(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}+\frac {2 (4 a+b) \sin (e+f x) \cos ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f} \]
Antiderivative was successfully verified.
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Rule 416
Rule 419
Rule 421
Rule 424
Rule 426
Rule 524
Rule 528
Rule 3192
Rubi steps
\begin {align*} \int \cos ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \left (1-x^2\right )^{3/2} \left (a+b x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^{3/2} \left (-a (7 a+b)-2 b (4 a+b) x^2\right )}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{7 f}\\ &=\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1-x^2} \left (-3 a b (9 a+b)+3 b \left (a^2-9 a b-2 b^2\right ) x^2\right )}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {-3 a b \left (a^2+18 a b+b^2\right )+6 (a-b) b \left (a^2+6 a b+b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{105 b^2 f}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}+\frac {\left (a (a+b) \left (2 a^2+9 a b-b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{35 b^2 f}-\frac {\left (2 (a-b) \left (a^2+6 a b+b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{35 b^2 f}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {\left (2 (a-b) \left (a^2+6 a b+b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \operatorname {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (a (a+b) \left (2 a^2+9 a b-b^2\right ) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ &=-\frac {\left (a^2-9 a b-2 b^2\right ) \cos (e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b f}+\frac {2 (4 a+b) \cos ^3(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 f}-\frac {b \cos ^5(e+f x) \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{7 f}-\frac {2 (a-b) \left (a^2+6 a b+b^2\right ) \sqrt {\cos ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{35 b^2 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (a+b) \left (2 a^2+9 a b-b^2\right ) \sqrt {\cos ^2(e+f x)} F\left (\sin ^{-1}(\sin (e+f x))|-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{35 b^2 f \sqrt {a+b \sin ^2(e+f x)}}\\ \end {align*}
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Mathematica [A] time = 2.57, size = 247, normalized size = 0.77 \[ \frac {\sqrt {2} b \sin (2 (e+f x)) \left (-32 a^3+b \left (144 a^2-192 a b-37 b^2\right ) \cos (2 (e+f x))+400 a^2 b+2 b^2 (b-26 a) \cos (4 (e+f x))+212 a b^2+5 b^3 \cos (6 (e+f x))+30 b^3\right )+64 a \left (2 a^3+11 a^2 b+8 a b^2-b^3\right ) \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac {b}{a}\right .\right )-128 a \left (a^3+5 a^2 b-5 a b^2-b^3\right ) \sqrt {\frac {2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )}{2240 b^2 f \sqrt {2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (b \cos \left (f x + e\right )^{6} - {\left (a + b\right )} \cos \left (f x + e\right )^{4}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.69, size = 590, normalized size = 1.84 \[ \frac {5 b^{4} \sin \left (f x +e \right ) \left (\cos ^{8}\left (f x +e \right )\right )+\left (-13 a \,b^{3}-7 b^{4}\right ) \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (9 a^{2} b^{2}+a \,b^{3}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\left (-a^{3} b +8 a^{2} b^{2}+11 a \,b^{3}+2 b^{4}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{4}+11 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3} b +8 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b^{2}-\sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticF \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{3}-2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{4}-10 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3} b +10 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b^{2}+2 \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \EllipticE \left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{3}}{35 b^{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 [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \cos \left (f x + e\right )^{4}\,{d x} \]
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 \[ \int {\cos \left (e+f\,x\right )}^4\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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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