Optimal. Leaf size=232 \[ -\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{4 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.267956, antiderivative size = 232, 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 = {3196, 473, 580, 524, 426, 424, 421, 419} \[ -\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{4 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{(7 a-b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3196
Rule 473
Rule 580
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \cot ^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{\left (1-x^2\right )^{3/2} \sqrt{a+b x^2}}{x^4} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left (2 \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-x^2} \left (\frac{1}{2} (-3 a+b)-2 b x^2\right )}{x^2 \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 f}\\ &=\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left (2 \sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{\frac{1}{2} a (3 a-5 b)+\frac{1}{2} (7 a-b) b x^2}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a f}\\ &=\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left ((7 a-b) \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 )}{3 a f}-\frac{\left (4 (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 )}{3 f}\\ &=\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left ((7 a-b) \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 )}{3 a f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{\left (4 (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 )}{3 f \sqrt{a+b \sin ^2(e+f x)}}\\ &=\frac{(3 a-b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot ^3(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{(7 a-b) \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)}}{3 a f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{4 (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}}}{3 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.29504, size = 197, normalized size = 0.85 \[ \frac{-\frac{\cot (e+f x) \csc ^2(e+f x) \left (4 \left (4 a^2+2 a b-b^2\right ) \cos (2 (e+f x))-8 a^2+b (b-4 a) \cos (4 (e+f x))-4 a b+3 b^2\right )}{2 \sqrt{2}}-8 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )+2 a (7 a-b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{6 a f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.322, size = 351, normalized size = 1.5 \begin{align*} -{\frac{1}{3\,a \left ( \sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f} \left ( 4\,{\it EllipticF} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+4\,b\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 \left ( \sin \left ( fx+e \right ) \right ) ^{3}-7\,{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\it EllipticE} \left ( \sin \left ( fx+e \right ) ,\sqrt{-{\frac{b}{a}}} \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2}}\sqrt{{\frac{a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{a}}}ab \left ( \sin \left ( fx+e \right ) \right ) ^{3}+4\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{6}-{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{6}+4\,{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}-6\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}-5\,{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+2\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{2}+{a}^{2} \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} \cot \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 (\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \cot \left (f x + e\right )^{4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \sin ^{2}{\left (e + f x \right )}} \cot ^{4}{\left (e + f x \right )}\, dx \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} \cot \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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