Optimal. Leaf size=63 \[ \frac{b \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a^2-b^2}}\right )}{a d \sqrt{a^2-b^2}}+\frac{x^2}{2 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.118444, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {4205, 3783, 2660, 618, 206} \[ \frac{b \tanh ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{a^2-b^2}}\right )}{a d \sqrt{a^2-b^2}}+\frac{x^2}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4205
Rule 3783
Rule 2660
Rule 618
Rule 206
Rubi steps
\begin{align*} \int \frac{x}{a+b \csc \left (c+d x^2\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{a+b \csc (c+d x)} \, dx,x,x^2\right )\\ &=\frac{x^2}{2 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\frac{a \sin (c+d x)}{b}} \, dx,x,x^2\right )}{2 a}\\ &=\frac{x^2}{2 a}-\frac{\operatorname{Subst}\left (\int \frac{1}{1+\frac{2 a x}{b}+x^2} \, dx,x,\tan \left (\frac{1}{2} \left (c+d x^2\right )\right )\right )}{a d}\\ &=\frac{x^2}{2 a}+\frac{2 \operatorname{Subst}\left (\int \frac{1}{-4 \left (1-\frac{a^2}{b^2}\right )-x^2} \, dx,x,\frac{2 a}{b}+2 \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )\right )}{a d}\\ &=\frac{x^2}{2 a}+\frac{b \tanh ^{-1}\left (\frac{b \left (\frac{a}{b}+\tan \left (\frac{1}{2} \left (c+d x^2\right )\right )\right )}{\sqrt{a^2-b^2}}\right )}{a \sqrt{a^2-b^2} d}\\ \end{align*}
Mathematica [A] time = 0.174997, size = 66, normalized size = 1.05 \[ \frac{-\frac{2 b \tan ^{-1}\left (\frac{a+b \tan \left (\frac{1}{2} \left (c+d x^2\right )\right )}{\sqrt{b^2-a^2}}\right )}{d \sqrt{b^2-a^2}}+\frac{c}{d}+x^2}{2 a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.063, size = 73, normalized size = 1.2 \begin{align*}{\frac{1}{ad}\arctan \left ( \tan \left ({\frac{d{x}^{2}}{2}}+{\frac{c}{2}} \right ) \right ) }-{\frac{b}{ad}\arctan \left ({\frac{1}{2} \left ( 2\,b\tan \left ( 1/2\,d{x}^{2}+c/2 \right ) +2\,a \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \right ){\frac{1}{\sqrt{-{a}^{2}+{b}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.52425, size = 560, normalized size = 8.89 \begin{align*} \left [\frac{2 \,{\left (a^{2} - b^{2}\right )} d x^{2} + \sqrt{a^{2} - b^{2}} b \log \left (\frac{{\left (a^{2} - 2 \, b^{2}\right )} \cos \left (d x^{2} + c\right )^{2} + 2 \, a b \sin \left (d x^{2} + c\right ) + a^{2} + b^{2} + 2 \,{\left (b \cos \left (d x^{2} + c\right ) \sin \left (d x^{2} + c\right ) + a \cos \left (d x^{2} + c\right )\right )} \sqrt{a^{2} - b^{2}}}{a^{2} \cos \left (d x^{2} + c\right )^{2} - 2 \, a b \sin \left (d x^{2} + c\right ) - a^{2} - b^{2}}\right )}{4 \,{\left (a^{3} - a b^{2}\right )} d}, \frac{{\left (a^{2} - b^{2}\right )} d x^{2} + \sqrt{-a^{2} + b^{2}} b \arctan \left (-\frac{\sqrt{-a^{2} + b^{2}}{\left (b \sin \left (d x^{2} + c\right ) + a\right )}}{{\left (a^{2} - b^{2}\right )} \cos \left (d x^{2} + c\right )}\right )}{2 \,{\left (a^{3} - a b^{2}\right )} d}\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 \frac{x}{a + b \csc{\left (c + d x^{2} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.14166, size = 113, normalized size = 1.79 \begin{align*} -\frac{{\left (\pi \left \lfloor \frac{d x^{2} + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (b\right ) + \arctan \left (\frac{b \tan \left (\frac{1}{2} \, d x^{2} + \frac{1}{2} \, c\right ) + a}{\sqrt{-a^{2} + b^{2}}}\right )\right )} b}{\sqrt{-a^{2} + b^{2}} a d} + \frac{d x^{2} + c}{2 \, a d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]