Optimal. Leaf size=193 \[ -\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 d \sqrt{a^2-b^2}}+\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))} \]
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Rubi [A] time = 1.03841, antiderivative size = 193, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.276, Rules used = {2889, 3056, 3055, 3001, 3770, 2660, 618, 204} \[ -\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^5 d \sqrt{a^2-b^2}}+\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))} \]
Antiderivative was successfully verified.
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Rule 2889
Rule 3056
Rule 3055
Rule 3001
Rule 3770
Rule 2660
Rule 618
Rule 204
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x) \csc ^2(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\int \frac{\csc ^4(c+d x) \left (1-\sin ^2(c+d x)\right )}{(a+b \sin (c+d x))^2} \, dx\\ &=\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^4(c+d x) \left (4 \left (a^2-b^2\right )-3 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^3(c+d x) \left (-12 b \left (a^2-b^2\right )-a \left (a^2-b^2\right ) \sin (c+d x)+8 b \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{3 a^2 \left (a^2-b^2\right )}\\ &=\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc ^2(c+d x) \left (-2 \left (a^4-13 a^2 b^2+12 b^4\right )+4 a b \left (a^2-b^2\right ) \sin (c+d x)-12 b^2 \left (a^2-b^2\right ) \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^3 \left (a^2-b^2\right )}\\ &=\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac{\int \frac{\csc (c+d x) \left (6 b \left (a^4-5 a^2 b^2+4 b^4\right )-12 a b^2 \left (a^2-b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{6 a^4 \left (a^2-b^2\right )}\\ &=\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac{\left (b \left (a^2-4 b^2\right )\right ) \int \csc (c+d x) \, dx}{a^5}-\frac{\left (b^2 \left (3 a^2-4 b^2\right )\right ) \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^5}\\ &=-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}-\frac{\left (2 b^2 \left (3 a^2-4 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^5 d}\\ &=-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}+\frac{\left (4 b^2 \left (3 a^2-4 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^5 d}\\ &=-\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^5 \sqrt{a^2-b^2} d}-\frac{b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cos (c+d x))}{a^5 d}+\frac{\left (a^2-12 b^2\right ) \cot (c+d x)}{3 a^4 d}+\frac{2 b \cot (c+d x) \csc (c+d x)}{a^3 d}-\frac{4 \cot (c+d x) \csc ^2(c+d x)}{3 a^2 d}+\frac{\cot (c+d x) \csc ^2(c+d x)}{a d (a+b \sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 6.34173, size = 385, normalized size = 1.99 \[ \frac{\left (a^2 b-4 b^3\right ) \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}+\frac{\left (4 b^3-a^2 b\right ) \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{a^5 d}-\frac{b^3 \cos (c+d x)}{a^4 d (a+b \sin (c+d x))}+\frac{\csc \left (\frac{1}{2} (c+d x)\right ) \left (a^2 \cos \left (\frac{1}{2} (c+d x)\right )-9 b^2 \cos \left (\frac{1}{2} (c+d x)\right )\right )}{6 a^4 d}+\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (9 b^2 \sin \left (\frac{1}{2} (c+d x)\right )-a^2 \sin \left (\frac{1}{2} (c+d x)\right )\right )}{6 a^4 d}-\frac{2 b^2 \left (3 a^2-4 b^2\right ) \tan ^{-1}\left (\frac{\sec \left (\frac{1}{2} (c+d x)\right ) \left (a \sin \left (\frac{1}{2} (c+d x)\right )+b \cos \left (\frac{1}{2} (c+d x)\right )\right )}{\sqrt{a^2-b^2}}\right )}{a^5 d \sqrt{a^2-b^2}}+\frac{b \csc ^2\left (\frac{1}{2} (c+d x)\right )}{4 a^3 d}-\frac{b \sec ^2\left (\frac{1}{2} (c+d x)\right )}{4 a^3 d}-\frac{\cot \left (\frac{1}{2} (c+d x)\right ) \csc ^2\left (\frac{1}{2} (c+d x)\right )}{24 a^2 d}+\frac{\tan \left (\frac{1}{2} (c+d x)\right ) \sec ^2\left (\frac{1}{2} (c+d x)\right )}{24 a^2 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.176, size = 390, normalized size = 2. \begin{align*}{\frac{1}{24\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{3}}-{\frac{b}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}}-{\frac{1}{8\,d{a}^{2}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }+{\frac{3\,{b}^{2}}{2\,d{a}^{4}}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{{b}^{4}\tan \left ( 1/2\,dx+c/2 \right ) }{d{a}^{5} \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}}{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 ) }}-6\,{\frac{{b}^{2}}{d{a}^{3}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tan \left ( 1/2\,dx+c/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }+8\,{\frac{{b}^{4}}{d{a}^{5}\sqrt{{a}^{2}-{b}^{2}}}\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}{24\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-3}}+{\frac{1}{8\,d{a}^{2}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{3\,{b}^{2}}{2\,d{a}^{4}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}+{\frac{b}{4\,d{a}^{3}} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-2}}+{\frac{b}{d{a}^{3}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) }-4\,{\frac{{b}^{3}\ln \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{a}^{5}}} \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 = 2.83606, size = 3228, normalized size = 16.73 \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 = 1.38767, size = 444, normalized size = 2.3 \begin{align*} \frac{\frac{24 \,{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{5}} - \frac{48 \,{\left (3 \, a^{2} b^{2} - 4 \, b^{4}\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^{5}} + \frac{a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 6 \, a^{3} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 3 \, a^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 36 \, a^{2} b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a^{6}} - \frac{48 \,{\left (b^{4} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a b^{3}\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^{5}} - \frac{44 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 176 \, b^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, a^{3} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 36 \, a b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 6 \, a^{2} b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + a^{3}}{a^{5} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3}}}{24 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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