Optimal. Leaf size=234 \[ -\frac{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
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Rubi [A] time = 0.259876, antiderivative size = 234, 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, 475, 583, 524, 426, 424, 421, 419} \[ -\frac{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{2 (a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(2 a+b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}} \]
Antiderivative was successfully verified.
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Rule 3188
Rule 475
Rule 583
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \csc ^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{\sqrt{a+b x^2}}{x^4 \sqrt{1-x^2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{2 a+b+b x^2}{x^2 \sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 f}\\ &=-\frac{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{-a b+b (2 a+b) x^2}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a f}\\ &=-\frac{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}+\frac{\left (2 (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{\left ((2 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{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{\left ((2 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 (2 (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{(2 a+b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a f}-\frac{\cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 f}-\frac{(2 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{2 (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 = 3.23071, size = 188, normalized size = 0.8 \[ \frac{\frac{\cot (e+f x) \csc ^2(e+f x) \left (4 \left (2 a^2+4 a b+b^2\right ) \cos (2 (e+f x))-(2 a+b) (8 a+b \cos (4 (e+f x))+3 b)\right )}{2 \sqrt{2}}+4 a (a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} F\left (e+f x\left |-\frac{b}{a}\right .\right )-2 a (2 a+b) \sqrt{\frac{2 a-b \cos (2 (e+f x))+b}{a}} E\left (e+f x\left |-\frac{b}{a}\right .\right )}{6 a f \sqrt{2 a-b \cos (2 (e+f x))+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.098, size = 342, 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 ( 2\,{\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}+2\,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}-2\,{\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}+2\,ab \left ( \sin \left ( fx+e \right ) \right ) ^{6}+{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{6}+2\,{a}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}-{b}^{2} \left ( \sin \left ( fx+e \right ) \right ) ^{4}-{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} \csc \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} \csc \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(-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} \csc \left (f x + e\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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