Optimal. Leaf size=87 \[ \frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d} e}-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.245075, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {6299, 517, 446, 93, 207} \[ \frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d} e}-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6299
Rule 517
Rule 446
Rule 93
Rule 207
Rubi steps
\begin{align*} \int \frac{x \left (a+b \text{sech}^{-1}(c x)\right )}{\left (d+e x^2\right )^{3/2}} \, dx &=-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c x} \sqrt{1+c x} \sqrt{d+e x^2}} \, dx}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c^2 x^2} \sqrt{d+e x^2}} \, dx}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x} \sqrt{d+e x}} \, dx,x,x^2\right )}{2 e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{-d+x^2} \, dx,x,\frac{\sqrt{d+e x^2}}{\sqrt{1-c^2 x^2}}\right )}{e}\\ &=-\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\frac{\sqrt{d+e x^2}}{\sqrt{d} \sqrt{1-c^2 x^2}}\right )}{\sqrt{d} e}\\ \end{align*}
Mathematica [A] time = 0.59144, size = 135, normalized size = 1.55 \[ -\frac{a+b \text{sech}^{-1}(c x)}{e \sqrt{d+e x^2}}-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sqrt{-d-e x^2} \tan ^{-1}\left (\frac{\sqrt{d} \sqrt{1-c^2 x^2}}{\sqrt{-d-e x^2}}\right )}{\sqrt{d} e (c x-1) \sqrt{d+e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.918, size = 0, normalized size = 0. \begin{align*} \int{x \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \left ( e{x}^{2}+d \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} b \int \frac{x \log \left (\sqrt{\frac{1}{c x} + 1} \sqrt{\frac{1}{c x} - 1} + \frac{1}{c x}\right )}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} - \frac{a}{\sqrt{e x^{2} + d} e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.39633, size = 829, normalized size = 9.53 \begin{align*} \left [-\frac{4 \, \sqrt{e x^{2} + d} b d \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 4 \, \sqrt{e x^{2} + d} a d -{\left (b e x^{2} + b d\right )} \sqrt{d} \log \left (\frac{{\left (c^{4} d^{2} - 6 \, c^{2} d e + e^{2}\right )} x^{4} - 8 \,{\left (c^{2} d^{2} - d e\right )} x^{2} - 4 \,{\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt{e x^{2} + d} \sqrt{d} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 8 \, d^{2}}{x^{4}}\right )}{4 \,{\left (d e^{2} x^{2} + d^{2} e\right )}}, -\frac{2 \, \sqrt{e x^{2} + d} b d \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 2 \, \sqrt{e x^{2} + d} a d -{\left (b e x^{2} + b d\right )} \sqrt{-d} \arctan \left (-\frac{{\left ({\left (c^{3} d - c e\right )} x^{3} - 2 \, c d x\right )} \sqrt{e x^{2} + d} \sqrt{-d} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \,{\left (c^{2} d e x^{4} +{\left (c^{2} d^{2} - d e\right )} x^{2} - d^{2}\right )}}\right )}{2 \,{\left (d e^{2} x^{2} + d^{2} 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 \frac{x \left (a + b \operatorname{asech}{\left (c x \right )}\right )}{\left (d + e x^{2}\right )^{\frac{3}{2}}}\, 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 \frac{{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x}{{\left (e x^{2} + d\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]