Optimal. Leaf size=125 \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{b} d}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{b} d} \]
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Rubi [A] time = 0.109314, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {3224, 1093, 205} \[ \frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{b} d}-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 3224
Rule 1093
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{a-b \sin ^4(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{a-\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt{a} \sqrt{b} d}-\frac{(a-b) \operatorname{Subst}\left (\int \frac{1}{a+\sqrt{a} \sqrt{b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 \sqrt{a} \sqrt{b} d}\\ &=-\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}-\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{b} d}+\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan ^{-1}\left (\frac{\sqrt{\sqrt{a}+\sqrt{b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 a^{3/4} \sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.281721, size = 158, normalized size = 1.26 \[ \frac{\frac{\left (\sqrt{a} \sqrt{b}+b\right ) \tan ^{-1}\left (\frac{\left (\sqrt{a}+\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}+a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}+a}}+\frac{\left (\sqrt{a} \sqrt{b}-b\right ) \tanh ^{-1}\left (\frac{\left (\sqrt{a}-\sqrt{b}\right ) \tan (c+d x)}{\sqrt{\sqrt{a} \sqrt{b}-a}}\right )}{\sqrt{\sqrt{a} \sqrt{b}-a}}}{2 \sqrt{a} b d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.112, size = 226, normalized size = 1.8 \begin{align*} -{\frac{a}{2\,d}\arctan \left ({ \left ( a-b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) \left ( a-b \right ) }}}}+{\frac{a}{2\,d}{\it Artanh} \left ({ \left ( -a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) \left ( a-b \right ) }}}}+{\frac{b}{2\,d}\arctan \left ({ \left ( a-b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}+a \right ) \left ( a-b \right ) }}}}-{\frac{b}{2\,d}{\it Artanh} \left ({ \left ( -a+b \right ) \tan \left ( dx+c \right ){\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) \left ( a-b \right ) }}}} \right ){\frac{1}{\sqrt{ab}}}{\frac{1}{\sqrt{ \left ( \sqrt{ab}-a \right ) \left ( a-b \right ) }}}} \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 \frac{\cos \left (d x + c\right )^{2}}{b \sin \left (d x + c\right )^{4} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.76512, size = 1277, normalized size = 10.22 \begin{align*} -\frac{1}{8} \, \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \log \left (\frac{1}{2} \, a d \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4} \,{\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt{\frac{1}{a^{3} b d^{4}}} - \frac{1}{4}\right ) + \frac{1}{8} \, \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \log \left (-\frac{1}{2} \, a d \sqrt{-\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} + 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + \frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4} \,{\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt{\frac{1}{a^{3} b d^{4}}} - \frac{1}{4}\right ) + \frac{1}{8} \, \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \log \left (\frac{1}{2} \, a d \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4} \,{\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt{\frac{1}{a^{3} b d^{4}}} + \frac{1}{4}\right ) - \frac{1}{8} \, \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \log \left (-\frac{1}{2} \, a d \sqrt{\frac{a b d^{2} \sqrt{\frac{1}{a^{3} b d^{4}}} - 1}{a b d^{2}}} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - \frac{1}{4} \, \cos \left (d x + c\right )^{2} + \frac{1}{4} \,{\left (2 \, a^{2} d^{2} \cos \left (d x + c\right )^{2} - a^{2} d^{2}\right )} \sqrt{\frac{1}{a^{3} b d^{4}}} + \frac{1}{4}\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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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