Optimal. Leaf size=348 \[ \frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(5 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^4 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rubi [A] time = 0.522608, antiderivative size = 348, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.36, Rules used = {3196, 468, 579, 583, 524, 426, 424, 421, 419} \[ \frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(5 a+8 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{\frac{b \sin ^2(e+f x)}{a}+1} F\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^3 f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \sqrt{\cos ^2(e+f x)} \sec (e+f x) \sqrt{a+b \sin ^2(e+f x)} E\left (\sin ^{-1}(\sin (e+f x))|-\frac{b}{a}\right )}{3 a^4 f \sqrt{\frac{b \sin ^2(e+f x)}{a}+1}}+\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3196
Rule 468
Rule 579
Rule 583
Rule 524
Rule 426
Rule 424
Rule 421
Rule 419
Rubi steps
\begin{align*} \int \frac{\cot ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, 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}}{x^4 \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{-3 (a+2 b)+(2 a+5 b) x^2}{x^4 \sqrt{1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{3 a b f}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{3 (a+b) (3 a+8 b)-6 (a+b) (a+3 b) x^2}{x^4 \sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 a^2 b (a+b) f}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}-\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{24 b (a+b) (a+2 b)-3 b (a+b) (3 a+8 b) x^2}{x^2 \sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^3 b (a+b) f}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{\left (\sqrt{\cos ^2(e+f x)} \sec (e+f x)\right ) \operatorname{Subst}\left (\int \frac{3 a b (a+b) (3 a+8 b)+24 b^2 (a+b) (a+2 b) x^2}{\sqrt{1-x^2} \sqrt{a+b x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^4 b (a+b) f}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{\left (8 (a+2 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^4 f}-\frac{\left ((5 a+8 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 a^3 f}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{\left (8 (a+2 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^4 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{\left ((5 a+8 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 a^3 f \sqrt{a+b \sin ^2(e+f x)}}\\ &=\frac{(a+b) \cot (e+f x) \csc ^2(e+f x)}{3 a b f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac{2 (a+3 b) \cot (e+f x) \csc ^2(e+f x)}{3 a^2 b f \sqrt{a+b \sin ^2(e+f x)}}+\frac{8 (a+2 b) \cot (e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^4 f}-\frac{(3 a+8 b) \cot (e+f x) \csc ^2(e+f x) \sqrt{a+b \sin ^2(e+f x)}}{3 a^3 b f}+\frac{8 (a+2 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^4 f \sqrt{1+\frac{b \sin ^2(e+f x)}{a}}}-\frac{(5 a+8 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 a^3 f \sqrt{a+b \sin ^2(e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.11643, size = 226, normalized size = 0.65 \[ \frac{2 a^2 b \left (\frac{2 a-b \cos (2 (e+f x))+b}{a}\right )^{3/2} \left (8 (a+2 b) E\left (e+f x\left |-\frac{b}{a}\right .\right )-(5 a+8 b) F\left (e+f x\left |-\frac{b}{a}\right .\right )\right )+\sqrt{2} b \left (2 a b (a+b) \sin (2 (e+f x))+4 b (a+2 b) \sin (2 (e+f x)) (2 a-b \cos (2 (e+f x))+b)+4 (a+2 b) \cot (e+f x) (2 a-b \cos (2 (e+f x))+b)^2-a \cot (e+f x) \csc ^2(e+f x) (2 a-b \cos (2 (e+f x))+b)^2\right )}{6 a^4 b f (2 a-b \cos (2 (e+f x))+b)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 1.55, size = 633, normalized size = 1.8 \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(-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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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \cot \left (f x + e\right )^{4}}{b^{3} \cos \left (f x + e\right )^{6} - 3 \,{\left (a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 3 \, a^{2} b - 3 \, a b^{2} - b^{3} + 3 \,{\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \cos \left (f x + e\right )^{2}}, 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 \frac{\cot \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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