Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2},x\right ) \]
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Rubi [A] time = 0.0425309, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx &=\int \frac{\cos ^2(c+d x)}{\left (a+b \sin ^3(c+d x)\right )^2} \, dx\\ \end{align*}
Mathematica [A] time = 0.245492, size = 273, normalized size = 10.92 \[ \frac{\frac{12 \sin (2 (c+d x))}{4 a+3 b \sin (c+d x)-b \sin (3 (c+d x))}-i \text{RootSum}\left [8 \text{$\#$1}^3 a+i \text{$\#$1}^6 b-3 i \text{$\#$1}^4 b+3 i \text{$\#$1}^2 b-i b\& ,\frac{-i \text{$\#$1}^4 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-6 i \text{$\#$1}^2 \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )-i \log \left (\text{$\#$1}^2-2 \text{$\#$1} \cos (c+d x)+1\right )+2 \text{$\#$1}^4 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+12 \text{$\#$1}^2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )+2 \tan ^{-1}\left (\frac{\sin (c+d x)}{\cos (c+d x)-\text{$\#$1}}\right )}{-4 i \text{$\#$1}^2 a+\text{$\#$1}^5 b-2 \text{$\#$1}^3 b+\text{$\#$1} b}\& \right ]}{18 a d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.247, size = 236, normalized size = 9.4 \begin{align*} -{\frac{2}{3\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{5} \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+8\,b \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+a \right ) ^{-1}}+{\frac{2}{3\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{6}a+3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{4}a+8\,b \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{3}+3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+a \right ) ^{-1}}+{\frac{2}{9\,da}\sum _{{\it \_R}={\it RootOf} \left ( a{{\it \_Z}}^{6}+3\,a{{\it \_Z}}^{4}+8\,b{{\it \_Z}}^{3}+3\,a{{\it \_Z}}^{2}+a \right ) }{\frac{{{\it \_R}}^{4}+1}{{{\it \_R}}^{5}a+2\,{{\it \_R}}^{3}a+4\,{{\it \_R}}^{2}b+{\it \_R}\,a}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) -{\it \_R} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [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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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 = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{2}}{{\left (b \sin \left (d x + c\right )^{3} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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