Optimal. Leaf size=115 \[ d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (6 c^2 d-e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{6 c^2 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1}}{6 c \sqrt{-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0556023, antiderivative size = 115, 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 = {6292, 12, 388, 217, 203} \[ d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b x \left (6 c^2 d-e\right ) \tan ^{-1}\left (\frac{c x}{\sqrt{-c^2 x^2-1}}\right )}{6 c^2 \sqrt{-c^2 x^2}}+\frac{b e x^2 \sqrt{-c^2 x^2-1}}{6 c \sqrt{-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6292
Rule 12
Rule 388
Rule 217
Rule 203
Rubi steps
\begin{align*} \int \left (d+e x^2\right ) \left (a+b \text{csch}^{-1}(c x)\right ) \, dx &=d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{3 d+e x^2}{3 \sqrt{-1-c^2 x^2}} \, dx}{\sqrt{-c^2 x^2}}\\ &=d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{(b c x) \int \frac{3 d+e x^2}{\sqrt{-1-c^2 x^2}} \, dx}{3 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^2 \sqrt{-1-c^2 x^2}}{6 c \sqrt{-c^2 x^2}}+d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c \left (6 d-\frac{e}{c^2}\right ) x\right ) \int \frac{1}{\sqrt{-1-c^2 x^2}} \, dx}{6 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^2 \sqrt{-1-c^2 x^2}}{6 c \sqrt{-c^2 x^2}}+d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{\left (b c \left (6 d-\frac{e}{c^2}\right ) x\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x^2} \, dx,x,\frac{x}{\sqrt{-1-c^2 x^2}}\right )}{6 \sqrt{-c^2 x^2}}\\ &=\frac{b e x^2 \sqrt{-1-c^2 x^2}}{6 c \sqrt{-c^2 x^2}}+d x \left (a+b \text{csch}^{-1}(c x)\right )+\frac{1}{3} e x^3 \left (a+b \text{csch}^{-1}(c x)\right )-\frac{b \left (6 d-\frac{e}{c^2}\right ) x \tan ^{-1}\left (\frac{c x}{\sqrt{-1-c^2 x^2}}\right )}{6 \sqrt{-c^2 x^2}}\\ \end{align*}
Mathematica [A] time = 0.262972, size = 135, normalized size = 1.17 \[ a d x+\frac{1}{3} a e x^3+\frac{b d x \sqrt{\frac{1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt{c^2 x^2+1}}+\frac{b e x^2 \sqrt{\frac{c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac{b e \log \left (x \left (\sqrt{\frac{c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{6 c^3}+b d x \text{csch}^{-1}(c x)+\frac{1}{3} b e x^3 \text{csch}^{-1}(c x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.209, size = 126, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{3}{x}^{3}}{3}}+x{c}^{3}d \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{{\rm arccsch} \left (cx\right )e{c}^{3}{x}^{3}}{3}}+{\rm arccsch} \left (cx\right ){c}^{3}dx+{\frac{1}{6\,cx}\sqrt{{c}^{2}{x}^{2}+1} \left ( 6\,{c}^{2}d{\it Arcsinh} \left ( cx \right ) +ecx\sqrt{{c}^{2}{x}^{2}+1}-e{\it Arcsinh} \left ( cx \right ) \right ){\frac{1}{\sqrt{{\frac{{c}^{2}{x}^{2}+1}{{c}^{2}{x}^{2}}}}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02023, size = 200, normalized size = 1.74 \begin{align*} \frac{1}{3} \, a e x^{3} + \frac{1}{12} \,{\left (4 \, x^{3} \operatorname{arcsch}\left (c x\right ) + \frac{\frac{2 \, \sqrt{\frac{1}{c^{2} x^{2}} + 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac{\log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e + a d x + \frac{{\left (2 \, c x \operatorname{arcsch}\left (c x\right ) + \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt{\frac{1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.95878, size = 544, normalized size = 4.73 \begin{align*} \frac{2 \, a c^{3} e x^{3} + b c^{2} e x^{2} \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 6 \, a c^{3} d x + 2 \,{\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) -{\left (6 \, b c^{2} d - b e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \,{\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \,{\left (b c^{3} e x^{3} + 3 \, b c^{3} d x - 3 \, b c^{3} d - b c^{3} e\right )} \log \left (\frac{c x \sqrt{\frac{c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \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 \left (a + b \operatorname{acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\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{\left (e x^{2} + d\right )}{\left (b \operatorname{arcsch}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]