Optimal. Leaf size=77 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{b^2 d \sqrt{a+b}}-\frac{x (2 a-b)}{2 b^2}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d} \]
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Rubi [A] time = 0.114091, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3187, 470, 522, 203, 205} \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{b^2 d \sqrt{a+b}}-\frac{x (2 a-b)}{2 b^2}-\frac{\sin (c+d x) \cos (c+d x)}{2 b d} \]
Antiderivative was successfully verified.
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Rule 3187
Rule 470
Rule 522
Rule 203
Rule 205
Rubi steps
\begin{align*} \int \frac{\sin ^4(c+d x)}{a+b \sin ^2(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1+x^2\right )^2 \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{\operatorname{Subst}\left (\int \frac{a+(-a+b) x^2}{\left (1+x^2\right ) \left (a+(a+b) x^2\right )} \, dx,x,\tan (c+d x)\right )}{2 b d}\\ &=-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}+\frac{a^2 \operatorname{Subst}\left (\int \frac{1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac{(2 a-b) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}\\ &=-\frac{(2 a-b) x}{2 b^2}+\frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{b^2 \sqrt{a+b} d}-\frac{\cos (c+d x) \sin (c+d x)}{2 b d}\\ \end{align*}
Mathematica [A] time = 0.310148, size = 69, normalized size = 0.9 \[ -\frac{-\frac{4 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{a+b} \tan (c+d x)}{\sqrt{a}}\right )}{\sqrt{a+b}}+2 (2 a-b) (c+d x)+b \sin (2 (c+d x))}{4 b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.079, size = 94, normalized size = 1.2 \begin{align*} -{\frac{\tan \left ( dx+c \right ) }{2\,bd \left ( \left ( \tan \left ( dx+c \right ) \right ) ^{2}+1 \right ) }}+{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) }{2\,bd}}-{\frac{\arctan \left ( \tan \left ( dx+c \right ) \right ) a}{{b}^{2}d}}+{\frac{{a}^{2}}{{b}^{2}d}\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 = 1.87633, size = 738, normalized size = 9.58 \begin{align*} \left [-\frac{2 \,{\left (2 \, a - b\right )} d x + 2 \, b \cos \left (d x + c\right ) \sin \left (d x + c\right ) - a \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 )}{4 \, b^{2} d}, -\frac{{\left (2 \, a - b\right )} d x + b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a \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 )}{2 \, b^{2} 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 = 1.19799, size = 154, normalized size = 2. \begin{align*} \frac{\frac{2 \,{\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^{2}}{\sqrt{a^{2} + a b} b^{2}} - \frac{{\left (d x + c\right )}{\left (2 \, a - b\right )}}{b^{2}} - \frac{\tan \left (d x + c\right )}{{\left (\tan \left (d x + c\right )^{2} + 1\right )} b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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