Optimal. Leaf size=170 \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {-c^2 x^2-1}}{\sqrt {-c^2 x^2}}-\frac {b e x \left (12 c^2 d-e\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{6 c^2 \sqrt {-c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-c^2 x^2-1}}{6 c \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {270, 6302, 12, 1265, 388, 217, 203} \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {-c^2 x^2-1}}{\sqrt {-c^2 x^2}}-\frac {b e x \left (12 c^2 d-e\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{6 c^2 \sqrt {-c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-c^2 x^2-1}}{6 c \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 217
Rule 270
Rule 388
Rule 1265
Rule 6302
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{3 x^2 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {-3 d^2+6 d e x^2+e^2 x^4}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{3 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{\sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {6 d e+e^2 x^2}{\sqrt {-1-c^2 x^2}} \, dx}{3 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{\sqrt {-c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b \left (12 c^2 d e-e^2\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{6 c \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{\sqrt {-c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b \left (12 c^2 d e-e^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 c \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{\sqrt {-c^2 x^2}}+\frac {b e^2 x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{x}+2 d e x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e^2 x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (12 c^2 d-e\right ) e x \tan ^{-1}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{6 c^2 \sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 134, normalized size = 0.79 \[ \frac {c^2 \left (2 a c \left (-3 d^2+6 d e x^2+e^2 x^4\right )+b x \sqrt {\frac {1}{c^2 x^2}+1} \left (6 c^2 d^2+e^2 x^2\right )\right )+2 b c^3 \text {csch}^{-1}(c x) \left (-3 d^2+6 d e x^2+e^2 x^4\right )+b e x \left (12 c^2 d-e\right ) \log \left (x \left (\sqrt {\frac {1}{c^2 x^2}+1}+1\right )\right )}{6 c^3 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.22, size = 347, normalized size = 2.04 \[ \frac {2 \, a c^{3} e^{2} x^{4} + 6 \, b c^{4} d^{2} x + 12 \, a c^{3} d e x^{2} - 6 \, a c^{3} d^{2} - 2 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (12 \, b c^{2} d e - b e^{2}\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) + 2 \, {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{3} e^{2} x^{4} + 6 \, b c^{3} d e x^{2} - 3 \, b c^{3} d^{2} + {\left (3 \, b c^{3} d^{2} - 6 \, b c^{3} d e - b c^{3} e^{2}\right )} x\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (6 \, b c^{4} d^{2} x + b c^{2} e^{2} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{6 \, c^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 189, normalized size = 1.11 \[ c \left (\frac {a \left (\frac {c^{3} x^{3} e^{2}}{3}+2 c^{3} d e x -\frac {d^{2} c^{3}}{x}\right )}{c^{4}}+\frac {b \left (\frac {e^{2} \mathrm {arccsch}\left (c x \right ) c^{3} x^{3}}{3}+2 \,\mathrm {arccsch}\left (c x \right ) c^{3} d e x -\frac {\mathrm {arccsch}\left (c x \right ) d^{2} c^{3}}{x}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 d^{2} c^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} d e \arcsinh \left (c x \right ) x +\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2} e^{2}-\arcsinh \left (c x \right ) c x \,e^{2}\right )}{6 c^{2} x^{2} \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}}\right )}{c^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.41, size = 191, normalized size = 1.12 \[ \frac {1}{3} \, a e^{2} x^{3} + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b d^{2} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e^{2} + 2 \, a d e x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d e}{c} - \frac {a d^{2}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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