Optimal. Leaf size=164 \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {-c^2 x^2-1}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {2 b c d \sqrt {-c^2 x^2-1} \left (c^2 d-9 e\right )}{9 \sqrt {-c^2 x^2}}-\frac {b e^2 x \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{\sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.14, antiderivative size = 164, 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, 451, 217, 203} \[ -\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {b c d^2 \sqrt {-c^2 x^2-1}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {2 b c d \sqrt {-c^2 x^2-1} \left (c^2 d-9 e\right )}{9 \sqrt {-c^2 x^2}}-\frac {b e^2 x \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{\sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 217
Rule 270
Rule 451
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^4} \, dx &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{3 x^4 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {-d^2-6 d e x^2+3 e^2 x^4}{x^4 \sqrt {-1-c^2 x^2}} \, dx}{3 \sqrt {-c^2 x^2}}\\ &=\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {2 d \left (c^2 d-9 e\right )+9 e^2 x^2}{x^2 \sqrt {-1-c^2 x^2}} \, dx}{9 \sqrt {-c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c e^2 x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c e^2 x\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1-c^2 x^2}}\right )}{\sqrt {-c^2 x^2}}\\ &=-\frac {2 b c d \left (c^2 d-9 e\right ) \sqrt {-1-c^2 x^2}}{9 \sqrt {-c^2 x^2}}+\frac {b c d^2 \sqrt {-1-c^2 x^2}}{9 x^2 \sqrt {-c^2 x^2}}-\frac {d^2 \left (a+b \text {csch}^{-1}(c x)\right )}{3 x^3}-\frac {2 d e \left (a+b \text {csch}^{-1}(c x)\right )}{x}+e^2 x \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b e^2 x \tan ^{-1}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{\sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.27, size = 123, normalized size = 0.75 \[ \frac {b c d x \sqrt {\frac {1}{c^2 x^2}+1} \left (-2 c^2 d x^2+d+18 e x^2\right )-3 a \left (d^2+6 d e x^2-3 e^2 x^4\right )}{9 x^3}+\frac {b e^2 \log \left (x \left (\sqrt {\frac {1}{c^2 x^2}+1}+1\right )\right )}{c}-\frac {b \text {csch}^{-1}(c x) \left (d^2+6 d e x^2-3 e^2 x^4\right )}{3 x^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 334, normalized size = 2.04 \[ \frac {9 \, a c e^{2} x^{4} - 9 \, b e^{2} x^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 18 \, a c d e x^{2} - 3 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) + 3 \, {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) - 3 \, a c d^{2} - 2 \, {\left (b c^{4} d^{2} - 9 \, b c^{2} d e\right )} x^{3} + 3 \, {\left (3 \, b c e^{2} x^{4} - 6 \, b c d e x^{2} - b c d^{2} + {\left (b c d^{2} + 6 \, b c d e - 3 \, b c e^{2}\right )} x^{3}\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right ) + {\left (b c^{2} d^{2} x - 2 \, {\left (b c^{4} d^{2} - 9 \, b c^{2} d e\right )} x^{3}\right )} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}}}{9 \, c x^{3}} \]
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 \[ \int \frac {{\left (e x^{2} + d\right )}^{2} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 190, normalized size = 1.16 \[ c^{3} \left (\frac {a \left (c x \,e^{2}-\frac {2 c d e}{x}-\frac {d^{2} c}{3 x^{3}}\right )}{c^{4}}+\frac {b \left (\mathrm {arccsch}\left (c x \right ) c x \,e^{2}-\frac {2 \,\mathrm {arccsch}\left (c x \right ) c d e}{x}-\frac {\mathrm {arccsch}\left (c x \right ) d^{2} c}{3 x^{3}}+\frac {\sqrt {c^{2} x^{2}+1}\, \left (-2 \sqrt {c^{2} x^{2}+1}\, c^{6} x^{2} d^{2}+18 c^{4} d e \sqrt {c^{2} x^{2}+1}\, x^{2}+d^{2} c^{4} \sqrt {c^{2} x^{2}+1}+9 e^{2} \arcsinh \left (c x \right ) c^{3} x^{3}\right )}{9 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c^{4} x^{4}}\right )}{c^{4}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 152, normalized size = 0.93 \[ 2 \, {\left (c \sqrt {\frac {1}{c^{2} x^{2}} + 1} - \frac {\operatorname {arcsch}\left (c x\right )}{x}\right )} b d e + a e^{2} x + \frac {1}{9} \, b d^{2} {\left (\frac {c^{4} {\left (\frac {1}{c^{2} x^{2}} + 1\right )}^{\frac {3}{2}} - 3 \, c^{4} \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c} - \frac {3 \, \operatorname {arcsch}\left (c x\right )}{x^{3}}\right )} + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b e^{2}}{2 \, c} - \frac {2 \, a d e}{x} - \frac {a d^{2}}{3 \, x^{3}} \]
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 \[ \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{4}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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