Optimal. Leaf size=117 \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{b^3 d \sqrt{a+b}}+\frac{x \left (8 a^2-4 a b+3 b^2\right )}{8 b^3}+\frac{(4 a-3 b) \sin (c+d x) \cos (c+d x)}{8 b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d} \]
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Rubi [A] time = 0.223681, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {3187, 470, 578, 522, 203, 205} \[ -\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{b^3 d \sqrt{a+b}}+\frac{x \left (8 a^2-4 a b+3 b^2\right )}{8 b^3}+\frac{(4 a-3 b) \sin (c+d x) \cos (c+d x)}{8 b^2 d}-\frac{\sin ^3(c+d x) \cos (c+d x)}{4 b d} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 470
Rule 578
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^6(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (1+x^2\right )^3 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}+\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (3 a+(-a+3 b) x^2\right )}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{4 b d}\\ &=\frac{(4 a-3 b) \cos (c+d x) \sin (c+d x)}{8 b^2 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}-\frac{\operatorname{Subst}\left (\int \frac{a (4 a-3 b)+\left (-4 a^2+a b-3 b^2\right ) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{8 b^2 d}\\ &=\frac{(4 a-3 b) \cos (c+d x) \sin (c+d x)}{8 b^2 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}-\frac{a^3 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{b^3 d}+\frac{\left (8 a^2-4 a b+3 b^2\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{8 b^3 d}\\ &=\frac{\left (8 a^2-4 a b+3 b^2\right ) x}{8 b^3}-\frac{a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{b^3 \sqrt{a+b} d}+\frac{(4 a-3 b) \cos (c+d x) \sin (c+d x)}{8 b^2 d}-\frac{\cos (c+d x) \sin ^3(c+d x)}{4 b d}\\ \end{align*}
Mathematica [A] time = 0.445376, size = 95, normalized size = 0.81 \[ \frac{4 \left (8 a^2-4 a b+3 b^2\right ) (c+d x)-\frac{32 a^{5/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a+b}}+8 b (a-b) \sin (2 (c+d x))+b^2 \sin (4 (c+d x))}{32 b^3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.092, size = 196, normalized size = 1.7 \begin{align*}{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}a}{2\,{b}^{2}d \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{5\, \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{8\,bd \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{a\tan \left ( dx+c \right ) }{2\,{b}^{2}d \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) ^{2}}}-{\frac{3\,\tan \left ( dx+c \right ) }{8\,bd \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) ^{2}}}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}}{d{b}^{3}}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a}{2\,{b}^{2}d}}+{\frac{3\,\arctan \left ( \tan \left ( dx+c \right ) \right ) }{8\,bd}}-{\frac{{a}^{3}}{d{b}^{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 ) }}}} \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 = 1.92646, size = 884, normalized size = 7.56 \begin{align*} \left [\frac{2 \, a^{2} \sqrt{-\frac{a}{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^{2} + 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{3} -{\left (a^{2} + 2 \, a b + b^{2}\right )} \cos \left (d x + c\right )\right )} \sqrt{-\frac{a}{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 ) +{\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} d x +{\left (2 \, b^{2} \cos \left (d x + c\right )^{3} +{\left (4 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, b^{3} d}, \frac{4 \, a^{2} \sqrt{\frac{a}{a + b}} \arctan \left (\frac{{\left ({\left (2 \, a + b\right )} \cos \left (d x + c\right )^{2} - a - b\right )} \sqrt{\frac{a}{a + b}}}{2 \, a \cos \left (d x + c\right ) \sin \left (d x + c\right )}\right ) +{\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )} d x +{\left (2 \, b^{2} \cos \left (d x + c\right )^{3} +{\left (4 \, a b - 5 \, b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8 \, b^{3} d}\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 = 1.2068, size = 212, normalized size = 1.81 \begin{align*} -\frac{\frac{8 \,{\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 )} a^{3}}{\sqrt{a^{2} + a b} b^{3}} - \frac{{\left (8 \, a^{2} - 4 \, a b + 3 \, b^{2}\right )}{\left (d x + c\right )}}{b^{3}} - \frac{4 \, a \tan \left (d x + c\right )^{3} - 5 \, b \tan \left (d x + c\right )^{3} + 4 \, a \tan \left (d x + c\right ) - 3 \, b \tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )}^{2} b^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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