Optimal. Leaf size=492 \[ \frac{2 \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a \sin (d+e x)+b+c \cos (d+e x))^2 \text{EllipticF}\left (\frac{1}{2} \left (-\tan ^{-1}(c,a)+d+e x\right ),\frac{2 \sqrt{a^2+c^2}}{\sqrt{a^2+c^2}+b}\right )}{3 e \left (a^2-b^2+c^2\right ) \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 b (a \sin (d+e x)+b+c \cos (d+e x))^3 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{3 e \left (a^2-b^2+c^2\right )^2 \sin ^{\frac{5}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 (a b \cos (d+e x)-b c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))^2}{3 e \left (a^2-b^2+c^2\right )^2 \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{3 e \left (a^2-b^2+c^2\right ) \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}} \]
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Rubi [A] time = 0.491416, antiderivative size = 492, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3164, 3129, 3156, 3149, 3119, 2653, 3127, 2661} \[ \frac{2 \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a \sin (d+e x)+b+c \cos (d+e x))^2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{3 e \left (a^2-b^2+c^2\right ) \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 b (a \sin (d+e x)+b+c \cos (d+e x))^3 E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right )}{3 e \left (a^2-b^2+c^2\right )^2 \sin ^{\frac{5}{2}}(d+e x) \sqrt{\frac{a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt{a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}+\frac{8 (a b \cos (d+e x)-b c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))^2}{3 e \left (a^2-b^2+c^2\right )^2 \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}}-\frac{2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{3 e \left (a^2-b^2+c^2\right ) \sin ^{\frac{5}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3164
Rule 3129
Rule 3156
Rule 3149
Rule 3119
Rule 2653
Rule 3127
Rule 2661
Rubi steps
\begin{align*} \int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)} \, dx &=\frac{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2} \int \frac{1}{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2}} \, dx}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}\\ &=-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{\left (2 (b+c \cos (d+e x)+a \sin (d+e x))^{5/2}\right ) \int \frac{-\frac{3 b}{2}+\frac{1}{2} c \cos (d+e x)+\frac{1}{2} a \sin (d+e x)}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}} \, dx}{3 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}\\ &=-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{\left (4 (b+c \cos (d+e x)+a \sin (d+e x))^{5/2}\right ) \int \frac{\frac{1}{4} \left (a^2+3 b^2+c^2\right )+b c \cos (d+e x)+a b \sin (d+e x)}{\sqrt{b+c \cos (d+e x)+a \sin (d+e x)}} \, dx}{3 \left (a^2-b^2+c^2\right )^2 (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}\\ &=-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{\left (4 b (b+c \cos (d+e x)+a \sin (d+e x))^{5/2}\right ) \int \sqrt{b+c \cos (d+e x)+a \sin (d+e x)} \, dx}{3 \left (a^2-b^2+c^2\right )^2 (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{(b+c \cos (d+e x)+a \sin (d+e x))^{5/2} \int \frac{1}{\sqrt{b+c \cos (d+e x)+a \sin (d+e x)}} \, dx}{3 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}\\ &=-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{\left (4 b (b+c \cos (d+e x)+a \sin (d+e x))^3\right ) \int \sqrt{\frac{b}{b+\sqrt{a^2+c^2}}+\frac{\sqrt{a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt{a^2+c^2}}} \, dx}{3 \left (a^2-b^2+c^2\right )^2 (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x) \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}+\frac{\left ((b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}\right ) \int \frac{1}{\sqrt{\frac{b}{b+\sqrt{a^2+c^2}}+\frac{\sqrt{a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt{a^2+c^2}}}} \, dx}{3 \left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}\\ &=\frac{8 b E\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^3}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x) \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}+\frac{2 F\left (\frac{1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac{2 \sqrt{a^2+c^2}}{b+\sqrt{a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2 \sqrt{\frac{b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt{a^2+c^2}}}}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}-\frac{2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}+\frac{8 (b+c \cos (d+e x)+a \sin (d+e x))^2 (a b \cos (d+e x)-b c \sin (d+e x))}{3 \left (a^2-b^2+c^2\right )^2 e (a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)}\\ \end{align*}
Mathematica [F] time = 25.9738, size = 0, normalized size = 0. \[ \int \frac{1}{(a+c \cot (d+e x)+b \csc (d+e x))^{5/2} \sin ^{\frac{5}{2}}(d+e x)} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 1.202, size = 64189, normalized size = 130.5 \begin{align*} \text{output 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 \frac{1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac{5}{2}} \sin \left (e x + d\right )^{\frac{5}{2}}}\,{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 (-\frac{\sqrt{c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a}}{{\left (a^{3} \cos \left (e x + d\right )^{2} +{\left (c^{3} \cos \left (e x + d\right )^{2} - c^{3}\right )} \cot \left (e x + d\right )^{3} +{\left (b^{3} \cos \left (e x + d\right )^{2} - b^{3}\right )} \csc \left (e x + d\right )^{3} - a^{3} + 3 \,{\left (a c^{2} \cos \left (e x + d\right )^{2} - a c^{2}\right )} \cot \left (e x + d\right )^{2} + 3 \,{\left (a b^{2} \cos \left (e x + d\right )^{2} - a b^{2} +{\left (b^{2} c \cos \left (e x + d\right )^{2} - b^{2} c\right )} \cot \left (e x + d\right )\right )} \csc \left (e x + d\right )^{2} + 3 \,{\left (a^{2} c \cos \left (e x + d\right )^{2} - a^{2} c\right )} \cot \left (e x + d\right ) + 3 \,{\left (a^{2} b \cos \left (e x + d\right )^{2} - a^{2} b +{\left (b c^{2} \cos \left (e x + d\right )^{2} - b c^{2}\right )} \cot \left (e x + d\right )^{2} + 2 \,{\left (a b c \cos \left (e x + d\right )^{2} - a b c\right )} \cot \left (e x + d\right )\right )} \csc \left (e x + d\right )\right )} \sqrt{\sin \left (e x + d\right )}}, 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{1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac{5}{2}} \sin \left (e x + d\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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