Optimal. Leaf size=109 \[ -\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{a^{7/2} d \sqrt{a+b}}-\frac{\left (a^2-a b+b^2\right ) \cot (c+d x)}{a^3 d}-\frac{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d} \]
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Rubi [A] time = 0.124797, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3187, 461, 205} \[ -\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{a^{7/2} d \sqrt{a+b}}-\frac{\left (a^2-a b+b^2\right ) \cot (c+d x)}{a^3 d}-\frac{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\csc ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^3}{x^6 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a x^6}+\frac{2 a-b}{a^2 x^4}+\frac{a^2-a b+b^2}{a^3 x^2}+\frac{b^3}{a^3 \left (-a-(a+b) x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\left (a^2-a b+b^2\right ) \cot (c+d x)}{a^3 d}-\frac{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}+\frac{b^3 \operatorname{Subst}\left (\int \frac{1}{-a-(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{a^3 d}\\ &=-\frac{b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{a^{7/2} \sqrt{a+b} d}-\frac{\left (a^2-a b+b^2\right ) \cot (c+d x)}{a^3 d}-\frac{(2 a-b) \cot ^3(c+d x)}{3 a^2 d}-\frac{\cot ^5(c+d x)}{5 a d}\\ \end{align*}
Mathematica [A] time = 1.54195, size = 147, normalized size = 1.35 \[ -\frac{\csc ^2(c+d x) (2 a-b \cos (2 (c+d x))+b) \left (\sqrt{a} \sqrt{a+b} \cot (c+d x) \left (3 a^2 \csc ^4(c+d x)+8 a^2+a (4 a-5 b) \csc ^2(c+d x)-10 a b+15 b^2\right )+15 b^3 \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )\right )}{30 a^{7/2} d \sqrt{a+b} \left (a \csc ^2(c+d x)+b\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.123, size = 138, normalized size = 1.3 \begin{align*} -{\frac{{b}^{3}}{d{a}^{3}}\arctan \left ({ \left ( a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}} \right ){\frac{1}{\sqrt{a \left ( a+b \right ) }}}}-{\frac{1}{5\,da \left ( \tan \left ( dx+c \right ) \right ) ^{5}}}-{\frac{2}{3\,da \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}+{\frac{b}{3\,{a}^{2}d \left ( \tan \left ( dx+c \right ) \right ) ^{3}}}-{\frac{1}{da\tan \left ( dx+c \right ) }}+{\frac{b}{{a}^{2}d\tan \left ( dx+c \right ) }}-{\frac{{b}^{2}}{d{a}^{3}\tan \left ( dx+c \right ) }} \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 [B] time = 1.88241, size = 1400, normalized size = 12.84 \begin{align*} \left [-\frac{4 \,{\left (8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} - 20 \,{\left (4 \, a^{4} - a^{3} b + a^{2} b^{2} + 6 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} + 15 \,{\left (b^{3} \cos \left (d x + c\right )^{4} - 2 \, b^{3} \cos \left (d x + c\right )^{2} + b^{3}\right )} \sqrt{-a^{2} - a b} \log \left (\frac{{\left (8 \, a^{2} + 8 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{4} - 2 \,{\left (4 \, a^{2} + 5 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} - 4 \,{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{3} -{\left (a + b\right )} \cos \left (d x + c\right )\right )} \sqrt{-a^{2} - a b} \sin \left (d x + c\right ) + a^{2} + 2 \, a b + b^{2}}{b^{2} \cos \left (d x + c\right )^{4} - 2 \,{\left (a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + a^{2} + 2 \, a b + b^{2}}\right ) \sin \left (d x + c\right ) + 60 \,{\left (a^{4} + a b^{3}\right )} \cos \left (d x + c\right )}{60 \,{\left ({\left (a^{5} + a^{4} b\right )} d \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{5} + a^{4} b\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{5} + a^{4} b\right )} d\right )} \sin \left (d x + c\right )}, -\frac{2 \,{\left (8 \, a^{4} - 2 \, a^{3} b + 5 \, a^{2} b^{2} + 15 \, a b^{3}\right )} \cos \left (d x + c\right )^{5} - 10 \,{\left (4 \, a^{4} - a^{3} b + a^{2} b^{2} + 6 \, a b^{3}\right )} \cos \left (d x + c\right )^{3} - 15 \,{\left (b^{3} \cos \left (d x + c\right )^{4} - 2 \, b^{3} \cos \left (d x + c\right )^{2} + b^{3}\right )} \sqrt{a^{2} + a b} \arctan \left (\frac{{\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b}{2 \, \sqrt{a^{2} + a b} \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) \sin \left (d x + c\right ) + 30 \,{\left (a^{4} + a b^{3}\right )} \cos \left (d x + c\right )}{30 \,{\left ({\left (a^{5} + a^{4} b\right )} d \cos \left (d x + c\right )^{4} - 2 \,{\left (a^{5} + a^{4} b\right )} d \cos \left (d x + c\right )^{2} +{\left (a^{5} + a^{4} b\right )} d\right )} \sin \left (d x + c\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 [A] time = 1.16892, size = 209, normalized size = 1.92 \begin{align*} -\frac{\frac{15 \,{\left (\pi \left \lfloor \frac{d x + c}{\pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac{a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt{a^{2} + a b}}\right )\right )} b^{3}}{\sqrt{a^{2} + a b} a^{3}} + \frac{15 \, a^{2} \tan \left (d x + c\right )^{4} - 15 \, a b \tan \left (d x + c\right )^{4} + 15 \, b^{2} \tan \left (d x + c\right )^{4} + 10 \, a^{2} \tan \left (d x + c\right )^{2} - 5 \, a b \tan \left (d x + c\right )^{2} + 3 \, a^{2}}{a^{3} \tan \left (d x + c\right )^{5}}}{15 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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