Optimal. Leaf size=164 \[ \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}-\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right )}{4 c^4}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{12 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {6299, 517, 446, 88, 63, 208} \[ \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}-\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \tanh ^{-1}\left (\sqrt {1-c^2 x^2}\right )}{4 e}-\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (2 c^2 d+e\right )}{4 c^4}+\frac {b e \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \left (1-c^2 x^2\right )^{3/2}}{12 c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 88
Rule 208
Rule 446
Rule 517
Rule 6299
Rubi steps
\begin {align*} \int x \left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {\left (d+e x^2\right )^2 \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 {\left (d+e x^2\right )^2}{x \sqrt {1-c x} \sqrt {1+c x}} \, dx}{4 e}\\ &=\frac {\left (d+e x^2\right )^2 \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 {\left (d+e x^2\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{4 e}\\ &=\frac {\left (d+e x^2\right )^2 \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 ) \operatorname {Subst}\left (\int \frac {(d+e x)^2}{x \sqrt {1-c^2 x}} \, dx,x,x^2\right )}{8 e}\\ &=\frac {\left (d+e x^2\right )^2 \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 ) \operatorname {Subst}\left (\int \left (\frac {e \left (2 c^2 d+e\right )}{c^2 \sqrt {1-c^2 x}}+\frac {d^2}{x \sqrt {1-c^2 x}}-\frac {e^2 \sqrt {1-c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{8 e}\\ &=-\frac {b \left (2 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{4 c^4}+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{12 c^4}+\frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}+\frac {\left (b d^2 \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 \left (2 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{4 c^4}+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{12 c^4}+\frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}-\frac {\left (b d^2 \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 \left (2 c^2 d+e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{4 c^4}+\frac {b e \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \left (1-c^2 x^2\right )^{3/2}}{12 c^4}+\frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{4 e}-\frac {b d^2 \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 [A] time = 0.14, size = 85, normalized size = 0.52 \[ \frac {1}{12} \left (3 a x^2 \left (2 d+e x^2\right )-\frac {b \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (c^2 \left (6 d+e x^2\right )+2 e\right )}{c^4}+3 b x^2 \text {sech}^{-1}(c x) \left (2 d+e x^2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.69, size = 125, normalized size = 0.76 \[ \frac {3 \, a c^{3} e x^{4} + 6 \, a c^{3} d x^{2} + 3 \, {\left (b c^{3} e x^{4} + 2 \, b c^{3} d x^{2}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (b c^{2} e x^{3} + 2 \, {\left (3 \, b c^{2} d + b e\right )} x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{12 \, c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} x\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.07, size = 113, normalized size = 0.69 \[ \frac {\frac {a \left (\frac {1}{4} c^{4} e \,x^{4}+\frac {1}{2} c^{4} d \,x^{2}\right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arcsech}\left (c x \right ) c^{4} x^{4} e}{4}+\frac {\mathrm {arcsech}\left (c x \right ) c^{4} x^{2} d}{2}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c^{2} x^{2} e +6 c^{2} d +2 e \right )}{12}\right )}{c^{2}}}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.32, size = 96, normalized size = 0.59 \[ \frac {1}{4} \, a e x^{4} + \frac {1}{2} \, a d x^{2} + \frac {1}{2} \, {\left (x^{2} \operatorname {arsech}\left (c x\right ) - \frac {x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c}\right )} b d + \frac {1}{12} \, {\left (3 \, x^{4} \operatorname {arsech}\left (c x\right ) + \frac {c^{2} x^{3} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, x \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{3}}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 2.19, size = 126, normalized size = 0.77 \[ \begin {cases} \frac {a d x^{2}}{2} + \frac {a e x^{4}}{4} + \frac {b d x^{2} \operatorname {asech}{\left (c x \right )}}{2} + \frac {b e x^{4} \operatorname {asech}{\left (c x \right )}}{4} - \frac {b d \sqrt {- c^{2} x^{2} + 1}}{2 c^{2}} - \frac {b e x^{2} \sqrt {- c^{2} x^{2} + 1}}{12 c^{2}} - \frac {b e \sqrt {- c^{2} x^{2} + 1}}{6 c^{4}} & \text {for}\: c \neq 0 \\\left (a + \infty b\right ) \left (\frac {d x^{2}}{2} + \frac {e x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________