Optimal. Leaf size=196 \[ \frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{2 f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f} \]
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Rubi [A] time = 0.170512, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {3663, 462, 453, 277, 195, 217, 206} \[ \frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}+\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{2 f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 462
Rule 453
Rule 277
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \csc ^6(e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^2 \left (a+b x^2\right )^{3/2}}{x^6} \, dx,x,\tan (e+f x)\right )}{f}\\ &=-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}+\frac{\operatorname{Subst}\left (\int \frac{\left (10 a+5 a x^2\right ) \left (a+b x^2\right )^{3/2}}{x^4} \, dx,x,\tan (e+f x)\right )}{5 a f}\\ &=-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}+\frac{(3 a+4 b) \operatorname{Subst}\left (\int \frac{\left (a+b x^2\right )^{3/2}}{x^2} \, dx,x,\tan (e+f x)\right )}{3 a f}\\ &=-\frac{(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f}-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}+\frac{(b (3 a+4 b)) \operatorname{Subst}\left (\int \sqrt{a+b x^2} \, dx,x,\tan (e+f x)\right )}{a f}\\ &=\frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}-\frac{(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f}-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}+\frac{(b (3 a+4 b)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2}} \, dx,x,\tan (e+f x)\right )}{2 f}\\ &=\frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}-\frac{(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f}-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}+\frac{(b (3 a+4 b)) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{\tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{2 f}\\ &=\frac{\sqrt{b} (3 a+4 b) \tanh ^{-1}\left (\frac{\sqrt{b} \tan (e+f x)}{\sqrt{a+b \tan ^2(e+f x)}}\right )}{2 f}+\frac{b (3 a+4 b) \tan (e+f x) \sqrt{a+b \tan ^2(e+f x)}}{2 a f}-\frac{(3 a+4 b) \cot (e+f x) \left (a+b \tan ^2(e+f x)\right )^{3/2}}{3 a f}-\frac{2 \cot ^3(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{3 a f}-\frac{\cot ^5(e+f x) \left (a+b \tan ^2(e+f x)\right )^{5/2}}{5 a f}\\ \end{align*}
Mathematica [C] time = 2.04711, size = 213, normalized size = 1.09 \[ \frac{\sqrt{\sec ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)} \left (\frac{15 \sqrt{2} (3 a+4 b) \cot (e+f x) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}{\sqrt{2}}\right ),1\right )}{\sqrt{\frac{\csc ^2(e+f x) ((a-b) \cos (2 (e+f x))+a+b)}{b}}}-\frac{2 \left (8 a^2+34 a b+3 b^2\right ) \cot (e+f x)}{a}-4 (2 a+3 b) \cot (e+f x) \csc ^2(e+f x)-6 a \cot (e+f x) \csc ^4(e+f x)+15 b \tan (e+f x)\right )}{30 \sqrt{2} f} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.868, size = 6988, normalized size = 35.7 \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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 26.6989, size = 1624, normalized size = 8.29 \begin{align*} \text{result too large to display} \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 \tan \left (f x + e\right )^{2} + a\right )}^{\frac{3}{2}} \csc \left (f x + e\right )^{6}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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