Optimal. Leaf size=325 \[ -\frac{\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 b f}+\frac{a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{35 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{35 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin ^5(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}-\frac{2 (4 a+3 b) \sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f} \]
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Rubi [A] time = 0.482311, antiderivative size = 325, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {3188, 477, 582, 524, 426, 424, 421, 419} \[ -\frac{\left (a^2+11 a b+8 b^2\right ) \sin (e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 b f}+\frac{a (a+b) \left (2 a^2-5 a b-8 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{35 b^2 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{2 (a+2 b) \left (a^2-4 a b-4 b^2\right ) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{35 b^2 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}-\frac{b \sin ^5(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}-\frac{2 (4 a+3 b) \sin ^3(e+f x) \cos (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 477
Rule 582
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \sin ^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 \frac{x^4 \left (a+b x^2\right )^{3/2}}{\sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{b \cos (e+f x) \sin ^5(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{x^4 \left (-a (7 a+5 b)-2 b (4 a+3 b) x^2\right )}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{7 f}\\ &=-\frac{2 (4 a+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}-\frac{b \cos (e+f x) \sin ^5(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{x^2 \left (-6 a b (4 a+3 b)-3 b \left (a^2+11 a b+8 b^2\right ) x^2\right )}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{35 b f}\\ &=-\frac{\left (a^2+11 a b+8 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+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}-\frac{b \cos (e+f x) \sin ^5(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+11 a b+8 b^2\right )+6 b (a+2 b) \left (a^2-4 a b-4 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+11 a b+8 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+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}-\frac{b \cos (e+f x) \sin ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}+\frac{\left (a (a+b) \left (2 a^2-5 a b-8 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+2 b) \left (a^2-4 a b-4 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+11 a b+8 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+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}-\frac{b \cos (e+f x) \sin ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}-\frac{\left (2 (a+2 b) \left (a^2-4 a b-4 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^2-5 a b-8 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+11 a b+8 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+3 b) \cos (e+f x) \sin ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{35 f}-\frac{b \cos (e+f x) \sin ^5(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{7 f}-\frac{2 (a+2 b) \left (a^2-4 a b-4 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^2-5 a b-8 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*}
Mathematica [A] time = 2.74001, size = 249, normalized size = 0.77 \[ \frac{\sqrt{2} b \sin (2 (e+f x)) \left (b \left (144 a^2+480 a b+299 b^2\right ) \cos (2 (e+f x))-496 a^2 b-32 a^3-2 b^2 (26 a+27 b) \cos (4 (e+f x))-684 a b^2+5 b^3 \cos (6 (e+f x))-250 b^3\right )+64 a \left (-3 a^2 b+2 a^3-13 a b^2-8 b^3\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )-128 a \left (-2 a^2 b+a^3-12 a b^2-8 b^3\right ) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{2240 b^2 f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.324, size = 602, normalized size = 1.9 \begin{align*} \text{result too large to display} \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}} \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 (b \cos \left (f x + e\right )^{6} -{\left (a + 3 \, b\right )} \cos \left (f x + e\right )^{4} +{\left (2 \, a + 3 \, b\right )} \cos \left (f x + e\right )^{2} - a - b\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}} \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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