Optimal. Leaf size=259 \[ -\frac{\left (2 a^2-3 a b-8 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (a-2 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{(a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f} \]
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Rubi [A] time = 0.315668, 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 = {3188, 478, 582, 524, 426, 424, 421, 419} \[ -\frac{\left (2 a^2-3 a b-8 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{2 a (a-2 b) (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{15 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{\sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{(a+4 b) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 478
Rule 582
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \sin ^4(e+f x) \sqrt{a+b \sin ^2(e+f x)} \, dx &=\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{x^4 \sqrt{a+b x^2}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\cos (e+f x) \sin ^3(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{x^2 \left (3 a+(a+4 b) x^2\right )}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{5 f}\\ &=-\frac{(a+4 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}-\frac{\cos (e+f x) \sin ^3(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 (a+4 b)+\left (-2 a^2+3 a b+8 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{(a+4 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}-\frac{\cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{\left (2 a (a-2 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^2 f}+\frac{\left (\left (-2 a^2+3 a b+8 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^2 f}\\ &=-\frac{(a+4 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}-\frac{\cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}+\frac{\left (\left (-2 a^2+3 a b+8 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^2 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}+\frac{\left (2 a (a-2 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^2 f \sqrt{a+b \sin ^2(e+f x)}}\\ &=-\frac{(a+4 b) \cos (e+f x) \sin (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{15 b f}-\frac{\cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{5 f}-\frac{\left (2 a^2-3 a b-8 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^2 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}+\frac{2 a (a-2 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^2 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.4529, size = 199, normalized size = 0.77 \[ \frac{-\sqrt{2} b \sin (2 (e+f x)) \left (8 a^2-4 b (4 a+7 b) \cos (2 (e+f x))+48 a b+3 b^2 \cos (4 (e+f x))+25 b^2\right )+32 a \left (a^2-a b-2 b^2\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )-16 a \left (2 a^2-3 a b-8 b^2\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{240 b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.037, size = 413, normalized size = 1.6 \begin{align*}{\frac{1}{15\,{b}^{2}\cos \left ( fx+e \right ) f} \left ( 3\,{b}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{7}+4\,a{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{5}+{b}^{3} \left ( \sin \left ( fx+e \right ) \right ) ^{5}+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 ){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-4\,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}-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}^{3}+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}^{2}b+8\,\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}+{a}^{2}b \left ( \sin \left ( fx+e \right ) \right ) ^{3}-4\, \left ( \sin \left ( fx+e \right ) \right ) ^{3}{b}^{3}-\sin \left ( fx+e \right ){a}^{2}b-4\,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 \sqrt{b \sin \left (f x + e\right )^{2} + a} \sin \left (f x + e\right )^{4}\,{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 (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1\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 \sqrt{b \sin \left (f x + e\right )^{2} + a} \sin \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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