Optimal. Leaf size=104 \[ \frac{2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 b^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a x}{b^2}-\frac{\cot (c+d x)}{a d}-\frac{\cos (c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.269578, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2894, 3057, 2660, 618, 204, 3770} \[ \frac{2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )}{a^2 b^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{a x}{b^2}-\frac{\cot (c+d x)}{a d}-\frac{\cos (c+d x)}{b d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2894
Rule 3057
Rule 2660
Rule 618
Rule 204
Rule 3770
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=-\frac{\cos (c+d x)}{b d}-\frac{\cot (c+d x)}{a d}-\frac{\int \frac{\csc (c+d x) \left (b^2+2 a b \sin (c+d x)+a^2 \sin ^2(c+d x)\right )}{a+b \sin (c+d x)} \, dx}{a b}\\ &=-\frac{a x}{b^2}-\frac{\cos (c+d x)}{b d}-\frac{\cot (c+d x)}{a d}-\frac{b \int \csc (c+d x) \, dx}{a^2}+\frac{\left (a^2-b^2\right )^2 \int \frac{1}{a+b \sin (c+d x)} \, dx}{a^2 b^2}\\ &=-\frac{a x}{b^2}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cos (c+d x)}{b d}-\frac{\cot (c+d x)}{a d}+\frac{\left (2 \left (a^2-b^2\right )^2\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^2 b^2 d}\\ &=-\frac{a x}{b^2}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cos (c+d x)}{b d}-\frac{\cot (c+d x)}{a d}-\frac{\left (4 \left (a^2-b^2\right )^2\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^2 b^2 d}\\ &=-\frac{a x}{b^2}+\frac{2 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{b+a \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a^2-b^2}}\right )}{a^2 b^2 d}+\frac{b \tanh ^{-1}(\cos (c+d x))}{a^2 d}-\frac{\cos (c+d x)}{b d}-\frac{\cot (c+d x)}{a d}\\ \end{align*}
Mathematica [A] time = 0.757968, size = 146, normalized size = 1.4 \[ -\frac{-4 \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac{a \tan \left (\frac{1}{2} (c+d x)\right )+b}{\sqrt{a^2-b^2}}\right )+2 a^2 b \cos (c+d x)+2 a^3 c+2 a^3 d x-a b^2 \tan \left (\frac{1}{2} (c+d x)\right )+a b^2 \cot \left (\frac{1}{2} (c+d x)\right )+2 b^3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )\right )-2 b^3 \log \left (\cos \left (\frac{1}{2} (c+d x)\right )\right )}{2 a^2 b^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.112, size = 249, normalized size = 2.4 \begin{align*}{\frac{1}{2\,da}\tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) }-2\,{\frac{1}{bd \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{a\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{d{b}^{2}}}+2\,{\frac{{a}^{2}}{d{b}^{2}\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 ) }-4\,{\frac{1}{d\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 ) }+2\,{\frac{{b}^{2}}{d{a}^{2}\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}{2\,da} \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{-1}}-{\frac{b}{d{a}^{2}}\ln \left ( \tan \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.52279, size = 981, normalized size = 9.43 \begin{align*} \left [\frac{b^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - b^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 2 \, a b^{2} \cos \left (d x + c\right ) -{\left (a^{2} - b^{2}\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2} + 2 \,{\left (a \cos \left (d x + c\right ) \sin \left (d x + c\right ) + b \cos \left (d x + c\right )\right )} \sqrt{-a^{2} + b^{2}}}{b^{2} \cos \left (d x + c\right )^{2} - 2 \, a b \sin \left (d x + c\right ) - a^{2} - b^{2}}\right ) \sin \left (d x + c\right ) - 2 \,{\left (a^{3} d x + a^{2} b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, a^{2} b^{2} d \sin \left (d x + c\right )}, \frac{b^{3} \log \left (\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - b^{3} \log \left (-\frac{1}{2} \, \cos \left (d x + c\right ) + \frac{1}{2}\right ) \sin \left (d x + c\right ) - 2 \, a b^{2} \cos \left (d x + c\right ) - 2 \,{\left (a^{2} - b^{2}\right )}^{\frac{3}{2}} \arctan \left (-\frac{a \sin \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \cos \left (d x + c\right )}\right ) \sin \left (d x + c\right ) - 2 \,{\left (a^{3} d x + a^{2} b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, a^{2} b^{2} d \sin \left (d x + c\right )}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.19248, size = 298, normalized size = 2.87 \begin{align*} -\frac{\frac{6 \,{\left (d x + c\right )} a}{b^{2}} + \frac{6 \, b \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{a^{2}} - \frac{3 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a} - \frac{12 \,{\left (a^{4} - 2 \, a^{2} b^{2} + 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^{2} b^{2}} - \frac{2 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 3 \, a b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 12 \, a^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 2 \, b^{2} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 3 \, a b}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} a^{2} b}}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]