Optimal. Leaf size=158 \[ \frac{6 b \sqrt{a^2-b^2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^4 d}-\frac{\left (a^2-b^2\right ) \cos (c+d x)}{a^3 d (a+b \sin (c+d x))}+\frac{3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{2 b \cot (c+d x)}{a^3 d}-\frac{\cos (c+d x)}{2 a^2 d \left (1-\cos ^2(c+d x)\right )} \]
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Rubi [A] time = 0.449638, antiderivative size = 180, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2890, 3055, 3001, 3770, 2660, 618, 204} \[ \frac{6 b \sqrt{a^2-b^2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^4 d}-\frac{\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac{3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2890
Rule 3055
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cos (c+d x) \cot ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (2 \left (a^2-3 b^2\right )-a b \sin (c+d x)+3 b^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^2 b}\\ &=-\frac{\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (-3 b \left (a^2-2 b^2\right )+3 a b^2 \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{2 a^3 b}\\ &=-\frac{\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}-\frac{\left (3 \left (a^2-2 b^2\right )\right ) \int \csc (c+d x) \, dx}{2 a^4}+\frac{\left (3 b \left (a^2-b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^4}\\ &=\frac{3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}-\frac{\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}+\frac{\left (6 b \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 d}\\ &=\frac{3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}-\frac{\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}-\frac{\left (12 b \left (a^2-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac{1}{2} (c+d x)\right )\right )}{a^4 d}\\ &=\frac{6 b \sqrt{a^2-b^2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^4 d}+\frac{3 \left (a^2-2 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{2 a^4 d}-\frac{\left (a^2-3 b^2\right ) \cot (c+d x)}{a^3 b d}+\frac{\left (2 a^2-3 b^2\right ) \cot (c+d x)}{2 a^2 b d (a+b \sin (c+d x))}-\frac{\cot (c+d x) \csc (c+d x)}{2 a d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 4.16124, size = 191, normalized size = 1.21 \[ \frac{48 b \sqrt{a^2-b^2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )-12 \left (a^2-2 b^2\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )+12 \left (a^2-2 b^2\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )+\frac{8 a \left (b^2-a^2\right ) \cos (c+d x)}{a+b \sin (c+d x)}-a^2 \csc ^2\left (\frac{1}{2} (c+d x)\right )+a^2 \sec ^2\left (\frac{1}{2} (c+d x)\right )-8 a b \tan \left (\frac{1}{2} (c+d x)\right )+8 a b \cot \left (\frac{1}{2} (c+d x)\right )}{8 a^4 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.171, size = 339, normalized size = 2.2 \begin{align*}{\frac{1}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{b}{d{a}^{3}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) b}{d{a}^{2} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+2\,{\frac{{b}^{3}\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{4} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}-2\,{\frac{1}{da \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+2\,{\frac{{b}^{2}}{d{a}^{3} \left ( \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a+2\,\tan \left ( 1/2\,dx+c/2 \right ) b+a \right ) }}+6\,{\frac{b\sqrt{{a}^{2}-{b}^{2}}}{d{a}^{4}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{1}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}-{\frac{3}{2\,d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }+3\,{\frac{\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ){b}^{2}}{d{a}^{4}}}+{\frac{b}{d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}} \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 = 2.61294, size = 1891, normalized size = 11.97 \begin{align*} \text{result too large to display} \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 = 1.28195, size = 371, normalized size = 2.35 \begin{align*} -\frac{\frac{12 \,{\left (a^{2} - 2 \, b^{2}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{4}} - \frac{48 \,{\left (a^{2} b - b^{3}\right )}{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (a\right ) + \arctan \left (\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + b}{\sqrt{a^{2} - b^{2}}}\right )\right )}}{\sqrt{a^{2} - b^{2}} a^{4}} - \frac{a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 8 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{4}} + \frac{16 \,{\left (a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3} - a b^{2}\right )}}{{\left (a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 2 \, b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a\right )} a^{4}} - \frac{18 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 36 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 8 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a^{2}}{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2}}}{8 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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