Optimal. Leaf size=192 \[ \frac{\sqrt{d} \log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.137801, antiderivative size = 192, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.474, Rules used = {16, 3476, 329, 211, 1165, 628, 1162, 617, 204} \[ \frac{\sqrt{d} \log \left (\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left (\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}+\sqrt{d}\right )}{2 \sqrt{2} f}+\frac{\sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}+1\right )}{\sqrt{2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 16
Rule 3476
Rule 329
Rule 211
Rule 1165
Rule 628
Rule 1162
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \sqrt{d \cot (e+f x)} \tan (e+f x) \, dx &=d \int \frac{1}{\sqrt{d \cot (e+f x)}} \, dx\\ &=-\frac{d^2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (d^2+x^2\right )} \, dx,x,d \cot (e+f x)\right )}{f}\\ &=-\frac{\left (2 d^2\right ) \operatorname{Subst}\left (\int \frac{1}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=-\frac{d \operatorname{Subst}\left (\int \frac{d-x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}-\frac{d \operatorname{Subst}\left (\int \frac{d+x^2}{d^2+x^4} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{f}\\ &=\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}+2 x}{-d-\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} f}+\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{d}-2 x}{-d+\sqrt{2} \sqrt{d} x-x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} f}-\frac{d \operatorname{Subst}\left (\int \frac{1}{d-\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 f}-\frac{d \operatorname{Subst}\left (\int \frac{1}{d+\sqrt{2} \sqrt{d} x+x^2} \, dx,x,\sqrt{d \cot (e+f x)}\right )}{2 f}\\ &=\frac{\sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} f}-\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}+\frac{\sqrt{d} \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}\\ &=\frac{\sqrt{d} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}-\frac{\sqrt{d} \tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{d \cot (e+f x)}}{\sqrt{d}}\right )}{\sqrt{2} f}+\frac{\sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)-\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} f}-\frac{\sqrt{d} \log \left (\sqrt{d}+\sqrt{d} \cot (e+f x)+\sqrt{2} \sqrt{d \cot (e+f x)}\right )}{2 \sqrt{2} f}\\ \end{align*}
Mathematica [A] time = 0.183199, size = 132, normalized size = 0.69 \[ \frac{d \sqrt{\cot (e+f x)} \left (\log \left (\cot (e+f x)-\sqrt{2} \sqrt{\cot (e+f x)}+1\right )-\log \left (\cot (e+f x)+\sqrt{2} \sqrt{\cot (e+f x)}+1\right )+2 \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (e+f x)}\right )-2 \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (e+f x)}+1\right )\right )}{2 \sqrt{2} f \sqrt{d \cot (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.124, size = 287, normalized size = 1.5 \begin{align*} -{\frac{\sqrt{2} \left ( \cos \left ( fx+e \right ) -1 \right ) \left ( \cos \left ( fx+e \right ) +1 \right ) ^{2}}{2\,f \left ( \sin \left ( fx+e \right ) \right ) ^{2}\cos \left ( fx+e \right ) }\sqrt{{\frac{d\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) -1+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{{\frac{\cos \left ( fx+e \right ) -1}{\sin \left ( fx+e \right ) }}} \left ( i{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -i{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}-{\frac{i}{2}},{\frac{\sqrt{2}}{2}} \right ) -{\it EllipticPi} \left ( \sqrt{{\frac{1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}},{\frac{1}{2}}+{\frac{i}{2}},{\frac{\sqrt{2}}{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 [B] time = 1.88023, size = 1230, normalized size = 6.41 \begin{align*} \sqrt{2} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} f^{3} \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{3}{4}} - \sqrt{2} f^{3} \sqrt{\frac{f^{2} \sqrt{\frac{d^{2}}{f^{4}}} \sin \left (f x + e\right ) + \sqrt{2} f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{3}{4}} + d^{2}}{d^{2}}\right ) + \sqrt{2} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} f^{3} \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{3}{4}} - \sqrt{2} f^{3} \sqrt{\frac{f^{2} \sqrt{\frac{d^{2}}{f^{4}}} \sin \left (f x + e\right ) - \sqrt{2} f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{3}{4}} - d^{2}}{d^{2}}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \log \left (\frac{f^{2} \sqrt{\frac{d^{2}}{f^{4}}} \sin \left (f x + e\right ) + \sqrt{2} f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \log \left (\frac{f^{2} \sqrt{\frac{d^{2}}{f^{4}}} \sin \left (f x + e\right ) - \sqrt{2} f \sqrt{\frac{d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}} \left (\frac{d^{2}}{f^{4}}\right )^{\frac{1}{4}} \sin \left (f x + e\right ) + d \cos \left (f x + e\right )}{\sin \left (f x + e\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \cot{\left (e + f x \right )}} \tan{\left (e + f x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{d \cot \left (f x + e\right )} \tan \left (f x + e\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]