Optimal. Leaf size=259 \[ -\frac{\left (3 a^2-7 a b-2 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{2 (3 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}+\frac{a (3 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b f \sqrt{a+b \sin ^2(e+f x)}} \]
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Rubi [A] time = 0.284961, antiderivative size = 259, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3192, 416, 528, 524, 426, 424, 421, 419} \[ -\frac{\left (3 a^2-7 a b-2 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin (e+f x) \cos ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{2 (3 a+b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}+\frac{a (3 a-b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b f \sqrt{a+b \sin ^2(e+f x)}} \]
Antiderivative was successfully verified.
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Rule 3192
Rule 416
Rule 528
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \cos ^2(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 \sqrt{1-x^2} \left (a+b x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{b \cos ^3(e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2} \left (-a (5 a+b)-2 b (3 a+b) x^2\right )}{\sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{5 f}\\ &=\frac{2 (3 a+b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}-\frac{b \cos ^3(e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{-a b (9 a+b)+b \left (3 a^2-7 a b-2 b^2\right ) x^2}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{15 b f}\\ &=\frac{2 (3 a+b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}-\frac{b \cos ^3(e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{\left (a (3 a-b) (a+b) \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 )}{15 b f}-\frac{\left (\left (3 a^2-7 a b-2 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 )}{15 b f}\\ &=\frac{2 (3 a+b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}-\frac{b \cos ^3(e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{\left (\left (3 a^2-7 a b-2 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 )}{15 b f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}+\frac{\left (a (3 a-b) (a+b) \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 )}{15 b f \sqrt{a+b \sin ^2(e+f x)}}\\ &=\frac{2 (3 a+b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 f}-\frac{b \cos ^3(e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{\left (3 a^2-7 a b-2 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)}}{15 b f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}+\frac{a (3 a-b) (a+b) \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}}}{15 b f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.32096, size = 200, normalized size = 0.77 \[ \frac{\sqrt{2} b \sin (2 (e+f x)) \left (48 a^2-4 b (9 a+2 b) \cos (2 (e+f x))+28 a b+3 b^2 \cos (4 (e+f x))+5 b^2\right )+16 a \left (3 a^2+2 a b-b^2\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )-16 a \left (3 a^2-7 a b-2 b^2\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{240 b f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.421, size = 429, normalized size = 1.7 \begin{align*}{\frac{1}{15\,b\cos \left ( fx+e \right ) f} \left ( -3\,{b}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{7}-9\,a{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}+4\,{b}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{5}+3\,\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){a}^{3}+2\,{a}^{2}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) b-a\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){b}^{2}-3\,\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){a}^{3}+7\,\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ){a}^{2}b+2\,\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) a{b}^{2}-6\,{a}^{2}b \left ( \sin \left ( fx+e \right ) \right ) ^{3}+10\,a{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}- \left ( \sin \left ( fx+e \right ) \right ) ^{3}{b}^{3}+6\,\sin \left ( fx+e \right ){a}^{2}b-a{b}^{2}\sin \left ( fx+e \right ) \right ){\frac{1}{\sqrt{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (b \cos \left (f x + e\right )^{4} -{\left (a + b\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \cos \left (f x + e\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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