Optimal. Leaf size=264 \[ \frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (9 c^2 d^2+e^2\right )}{6 c^4}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (2 c^2 d^2+e^2\right ) \sin ^{-1}(c x)}{2 c^3}-\frac{b d^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{4 e}-\frac{b d e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{12 c^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.362204, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438, Rules used = {6288, 1809, 844, 216, 266, 63, 208} \[ \frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}-\frac{b e \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2} \left (9 c^2 d^2+e^2\right )}{6 c^4}+\frac{b d \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \left (2 c^2 d^2+e^2\right ) \sin ^{-1}(c x)}{2 c^3}-\frac{b d^4 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{4 e}-\frac{b d e^2 x \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sqrt{1-c^2 x^2}}{12 c^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 6288
Rule 1809
Rule 844
Rule 216
Rule 266
Rule 63
Rule 208
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{(d+e x)^4}{x \sqrt{1-c^2 x^2}} \, dx}{4 e}\\ &=-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-3 c^2 d^4-12 c^2 d^3 e x-2 e^2 \left (9 c^2 d^2+e^2\right ) x^2-12 c^2 d e^3 x^3}{x \sqrt{1-c^2 x^2}} \, dx}{12 c^2 e}\\ &=-\frac{b d e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}+\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{6 c^4 d^4+12 c^2 d e \left (2 c^2 d^2+e^2\right ) x+4 c^2 e^2 \left (9 c^2 d^2+e^2\right ) x^2}{x \sqrt{1-c^2 x^2}} \, dx}{24 c^4 e}\\ &=-\frac{b e \left (9 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^4}-\frac{b d e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}-\frac{\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{-6 c^6 d^4-12 c^4 d e \left (2 c^2 d^2+e^2\right ) x}{x \sqrt{1-c^2 x^2}} \, dx}{24 c^6 e}\\ &=-\frac{b e \left (9 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^4}-\frac{b d e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}+\frac{\left (b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{x \sqrt{1-c^2 x^2}} \, dx}{4 e}+\frac{\left (b d \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{2 c^2}\\ &=-\frac{b e \left (9 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^4}-\frac{b d e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}+\frac{b d \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{2 c^3}+\frac{\left (b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{1-c^2 x}} \, dx,x,x^2\right )}{8 e}\\ &=-\frac{b e \left (9 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^4}-\frac{b d e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}+\frac{b d \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{2 c^3}-\frac{\left (b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{1}{c^2}-\frac{x^2}{c^2}} \, dx,x,\sqrt{1-c^2 x^2}\right )}{4 c^2 e}\\ &=-\frac{b e \left (9 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{6 c^4}-\frac{b d e^2 x \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{2 c^2}-\frac{b e^3 x^2 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sqrt{1-c^2 x^2}}{12 c^2}+\frac{(d+e x)^4 \left (a+b \text{sech}^{-1}(c x)\right )}{4 e}+\frac{b d \left (2 c^2 d^2+e^2\right ) \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{2 c^3}-\frac{b d^4 \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \tanh ^{-1}\left (\sqrt{1-c^2 x^2}\right )}{4 e}\\ \end{align*}
Mathematica [C] time = 0.396662, size = 190, normalized size = 0.72 \[ \frac{1}{4} \left (6 a d^2 e x^2+4 a d^3 x+4 a d e^2 x^3+a e^3 x^4-\frac{b e \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (c^2 \left (18 d^2+6 d e x+e^2 x^2\right )+2 e^2\right )}{3 c^4}+\frac{2 i b d \left (2 c^2 d^2+e^2\right ) \log \left (2 \sqrt{\frac{1-c x}{c x+1}} (c x+1)-2 i c x\right )}{c^3}+b x \text{sech}^{-1}(c x) \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.196, size = 283, normalized size = 1.1 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( cxe+cd \right ) ^{4}a}{4\,{c}^{3}e}}+{\frac{b}{{c}^{3}} \left ({\frac{{e}^{3}{\rm arcsech} \left (cx\right ){c}^{4}{x}^{4}}{4}}+{e}^{2}{\rm arcsech} \left (cx\right ){c}^{4}{x}^{3}d+{\frac{3\,e{\rm arcsech} \left (cx\right ){c}^{4}{x}^{2}{d}^{2}}{2}}+{\rm arcsech} \left (cx\right ){c}^{4}x{d}^{3}+{\frac{{\rm arcsech} \left (cx\right ){c}^{4}{d}^{4}}{4\,e}}+{\frac{cx}{12\,e}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}} \left ( -3\,{c}^{4}{d}^{4}{\it Artanh} \left ({\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}} \right ) +12\,{c}^{3}{d}^{3}e\arcsin \left ( cx \right ) -{c}^{2}{x}^{2}{e}^{4}\sqrt{-{c}^{2}{x}^{2}+1}-6\,{c}^{2}d{e}^{3}x\sqrt{-{c}^{2}{x}^{2}+1}-18\,{c}^{2}{d}^{2}{e}^{2}\sqrt{-{c}^{2}{x}^{2}+1}+6\,cd{e}^{3}\arcsin \left ( cx \right ) -2\,{e}^{4}\sqrt{-{c}^{2}{x}^{2}+1} \right ){\frac{1}{\sqrt{-{c}^{2}{x}^{2}+1}}}} \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.51568, size = 298, normalized size = 1.13 \begin{align*} \frac{1}{4} \, a e^{3} x^{4} + a d e^{2} x^{3} + \frac{3}{2} \, a d^{2} e x^{2} + \frac{3}{2} \,{\left (x^{2} \operatorname{arsech}\left (c x\right ) - \frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c}\right )} b d^{2} e + \frac{1}{2} \,{\left (2 \, x^{3} \operatorname{arsech}\left (c x\right ) - \frac{\frac{\sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{2}{\left (\frac{1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac{\arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} b d e^{2} + \frac{1}{12} \,{\left (3 \, x^{4} \operatorname{arsech}\left (c x\right ) + \frac{c^{2} x^{3}{\left (\frac{1}{c^{2} x^{2}} - 1\right )}^{\frac{3}{2}} - 3 \, x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b e^{3} + a d^{3} x + \frac{{\left (c x \operatorname{arsech}\left (c x\right ) - \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )\right )} b d^{3}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.41672, size = 764, normalized size = 2.89 \begin{align*} \frac{3 \, a c^{3} e^{3} x^{4} + 12 \, a c^{3} d e^{2} x^{3} + 18 \, a c^{3} d^{2} e x^{2} + 12 \, a c^{3} d^{3} x - 12 \,{\left (2 \, b c^{2} d^{3} + b d e^{2}\right )} \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) - 3 \,{\left (4 \, b c^{3} d^{3} + 6 \, b c^{3} d^{2} e + 4 \, b c^{3} d e^{2} + b c^{3} e^{3}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) + 3 \,{\left (b c^{3} e^{3} x^{4} + 4 \, b c^{3} d e^{2} x^{3} + 6 \, b c^{3} d^{2} e x^{2} + 4 \, b c^{3} d^{3} x - 4 \, b c^{3} d^{3} - 6 \, b c^{3} d^{2} e - 4 \, b c^{3} d e^{2} - b c^{3} e^{3}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) -{\left (b c^{2} e^{3} x^{3} + 6 \, b c^{2} d e^{2} x^{2} + 2 \,{\left (9 \, b c^{2} d^{2} e + b e^{3}\right )} x\right )} \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, 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{asech}{\left (c x \right )}\right ) \left (d + e x\right )^{3}\, 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 + d\right )}^{3}{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]