Optimal. Leaf size=128 \[ -\frac{3 (a+b)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right )}{8 \sqrt{a} f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{4 f}-\frac{3 (a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 f} \]
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Rubi [A] time = 0.125199, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3186, 378, 377, 206} \[ -\frac{3 (a+b)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a-b \cos ^2(e+f x)+b}}\right )}{8 \sqrt{a} f}-\frac{\cot (e+f x) \csc ^3(e+f x) \left (a-b \cos ^2(e+f x)+b\right )^{3/2}}{4 f}-\frac{3 (a+b) \cot (e+f x) \csc (e+f x) \sqrt{a-b \cos ^2(e+f x)+b}}{8 f} \]
Antiderivative was successfully verified.
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Rule 3186
Rule 378
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \csc ^5(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (a+b-b x^2\right )^{3/2}}{\left (1-x^2\right )^3} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{(3 (a+b)) \operatorname{Subst}\left (\int \frac{\sqrt{a+b-b x^2}}{\left (1-x^2\right )^2} \, dx,x,\cos (e+f x)\right )}{4 f}\\ &=-\frac{3 (a+b) \sqrt{a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{8 f}-\frac{\left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{\left (3 (a+b)^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{a+b-b x^2}} \, dx,x,\cos (e+f x)\right )}{8 f}\\ &=-\frac{3 (a+b) \sqrt{a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{8 f}-\frac{\left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{4 f}-\frac{\left (3 (a+b)^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-a x^2} \, dx,x,\frac{\cos (e+f x)}{\sqrt{a+b-b \cos ^2(e+f x)}}\right )}{8 f}\\ &=-\frac{3 (a+b)^2 \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+b-b \cos ^2(e+f x)}}\right )}{8 \sqrt{a} f}-\frac{3 (a+b) \sqrt{a+b-b \cos ^2(e+f x)} \cot (e+f x) \csc (e+f x)}{8 f}-\frac{\left (a+b-b \cos ^2(e+f x)\right )^{3/2} \cot (e+f x) \csc ^3(e+f x)}{4 f}\\ \end{align*}
Mathematica [A] time = 0.703615, size = 114, normalized size = 0.89 \[ -\frac{\frac{6 (a+b)^2 \tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \cos (e+f x)}{\sqrt{2 a-b \cos (2 (e+f x))+b}}\right )}{\sqrt{a}}+\sqrt{2} \cot (e+f x) \csc (e+f x) \sqrt{2 a-b \cos (2 (e+f x))+b} \left (2 a \csc ^2(e+f x)+3 a+5 b\right )}{16 f} \]
Antiderivative was successfully verified.
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Maple [B] time = 1.58, size = 376, normalized size = 2.9 \begin{align*} -{\frac{1}{16\, \left ( \sin \left ( fx+e \right ) \right ) ^{4}\cos \left ( fx+e \right ) f}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) } \left ( 3\,{a}^{2}\ln \left ({\frac{ \left ( a-b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+a+b}{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{4}+6\,ab\ln \left ({\frac{ \left ( a-b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+a+b}{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{4}+3\,{b}^{2}\ln \left ({\frac{ \left ( a-b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sqrt{a}\sqrt{-b \left ( \cos \left ( fx+e \right ) \right ) ^{4}+ \left ( a+b \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{2}}+a+b}{ \left ( \sin \left ( fx+e \right ) \right ) ^{2}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{4}+6\,{a}^{3/2}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) } \left ( \sin \left ( fx+e \right ) \right ) ^{2}+10\,b\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) }\sqrt{a} \left ( \sin \left ( fx+e \right ) \right ) ^{2}+4\,{a}^{3/2}\sqrt{ \left ( \cos \left ( fx+e \right ) \right ) ^{2} \left ( a+b \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) } \right ){\frac{1}{\sqrt{a}}}{\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{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \csc \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 4.33269, size = 1216, normalized size = 9.5 \begin{align*} \left [\frac{3 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt{a} \log \left (\frac{2 \,{\left ({\left (a^{2} - 6 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} + 2 \,{\left (3 \, a^{2} + 2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \,{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{3} +{\left (a + b\right )} \cos \left (f x + e\right )\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{a} + a^{2} + 2 \, a b + b^{2}\right )}}{\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} + 1}\right ) + 4 \,{\left ({\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{3} - 5 \,{\left (a^{2} + a b\right )} \cos \left (f x + e\right )\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{32 \,{\left (a f \cos \left (f x + e\right )^{4} - 2 \, a f \cos \left (f x + e\right )^{2} + a f\right )}}, \frac{3 \,{\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{4} - 2 \,{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )} \sqrt{-a} \arctan \left (-\frac{{\left ({\left (a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b} \sqrt{-a}}{2 \,{\left (a b \cos \left (f x + e\right )^{3} -{\left (a^{2} + a b\right )} \cos \left (f x + e\right )\right )}}\right ) + 2 \,{\left ({\left (3 \, a^{2} + 5 \, a b\right )} \cos \left (f x + e\right )^{3} - 5 \,{\left (a^{2} + a b\right )} \cos \left (f x + e\right )\right )} \sqrt{-b \cos \left (f x + e\right )^{2} + a + b}}{16 \,{\left (a f \cos \left (f x + e\right )^{4} - 2 \, a f \cos \left (f x + e\right )^{2} + a f\right )}}\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}} \csc \left (f x + e\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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