Optimal. Leaf size=167 \[ \frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b e x^4 \sqrt {-c^2 x^2-1}}{20 c \sqrt {-c^2 x^2}}+\frac {b x \left (20 c^2 d-9 e\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{120 c^4 \sqrt {-c^2 x^2}}+\frac {b x^2 \sqrt {-c^2 x^2-1} \left (20 c^2 d-9 e\right )}{120 c^3 \sqrt {-c^2 x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {14, 6302, 12, 459, 321, 217, 203} \[ \frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b x^2 \sqrt {-c^2 x^2-1} \left (20 c^2 d-9 e\right )}{120 c^3 \sqrt {-c^2 x^2}}+\frac {b x \left (20 c^2 d-9 e\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{120 c^4 \sqrt {-c^2 x^2}}+\frac {b e x^4 \sqrt {-c^2 x^2-1}}{20 c \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 14
Rule 203
Rule 217
Rule 321
Rule 459
Rule 6302
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (5 d+3 e x^2\right )}{15 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {x^2 \left (5 d+3 e x^2\right )}{\sqrt {-1-c^2 x^2}} \, dx}{15 \sqrt {-c^2 x^2}}\\ &=\frac {b e x^4 \sqrt {-1-c^2 x^2}}{20 c \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c \left (20 d-\frac {9 e}{c^2}\right ) x\right ) \int \frac {x^2}{\sqrt {-1-c^2 x^2}} \, dx}{60 \sqrt {-c^2 x^2}}\\ &=\frac {b \left (20 c^2 d-9 e\right ) x^2 \sqrt {-1-c^2 x^2}}{120 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^4 \sqrt {-1-c^2 x^2}}{20 c \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {\left (b \left (20 d-\frac {9 e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{120 c \sqrt {-c^2 x^2}}\\ &=\frac {b \left (20 c^2 d-9 e\right ) x^2 \sqrt {-1-c^2 x^2}}{120 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^4 \sqrt {-1-c^2 x^2}}{20 c \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {\left (b \left (20 d-\frac {9 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 )}{120 c \sqrt {-c^2 x^2}}\\ &=\frac {b \left (20 c^2 d-9 e\right ) x^2 \sqrt {-1-c^2 x^2}}{120 c^3 \sqrt {-c^2 x^2}}+\frac {b e x^4 \sqrt {-1-c^2 x^2}}{20 c \sqrt {-c^2 x^2}}+\frac {1}{3} d x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{5} e x^5 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b \left (20 c^2 d-9 e\right ) x \tan ^{-1}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{120 c^4 \sqrt {-c^2 x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 119, normalized size = 0.71 \[ \frac {c^2 x^2 \left (8 a c^3 x \left (5 d+3 e x^2\right )+b \sqrt {\frac {1}{c^2 x^2}+1} \left (c^2 \left (20 d+6 e x^2\right )-9 e\right )\right )+8 b c^5 x^3 \text {csch}^{-1}(c x) \left (5 d+3 e x^2\right )+b \left (9 e-20 c^2 d\right ) \log \left (x \left (\sqrt {\frac {1}{c^2 x^2}+1}+1\right )\right )}{120 c^5} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 3.02, size = 273, normalized size = 1.63 \[ \frac {24 \, a c^{5} e x^{5} + 40 \, a c^{5} d x^{3} + 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + {\left (20 \, b c^{2} d - 9 \, b e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 8 \, {\left (5 \, b c^{5} d + 3 \, b c^{5} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 8 \, {\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3} - 5 \, b c^{5} d - 3 \, b c^{5} e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} e x^{4} + {\left (20 \, b c^{4} d - 9 \, b c^{2} e\right )} x^{2}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{120 \, c^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.05, size = 171, normalized size = 1.02 \[ \frac {\frac {a \left (\frac {1}{5} c^{5} x^{5} e +\frac {1}{3} c^{5} x^{3} d \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) c^{5} x^{5} e}{5}+\frac {\mathrm {arccsch}\left (c x \right ) c^{5} x^{3} d}{3}-\frac {\sqrt {c^{2} x^{2}+1}\, \left (-6 e \,c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-20 c^{3} d x \sqrt {c^{2} x^{2}+1}+20 c^{2} d \arcsinh \left (c x \right )+9 e c x \sqrt {c^{2} x^{2}+1}-9 e \arcsinh \left (c x \right )\right )}{120 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.38, size = 227, normalized size = 1.36 \[ \frac {1}{5} \, a e x^{5} + \frac {1}{3} \, a d x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b d + \frac {1}{80} \, {\left (16 \, x^{5} \operatorname {arcsch}\left (c x\right ) - \frac {\frac {2 \, {\left (3 \, {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 5 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}\right )}}{c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{2} - 2 \, c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} + c^{4}} - \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{4}} + \frac {3 \, \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{4}}}{c}\right )} b e \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________