Optimal. Leaf size=213 \[ \frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \left (6 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}+\frac {b \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x} \]
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Rubi [A]
time = 0.11, antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 6436, 12,
1279, 464, 270} \begin {gather*} -\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {b d^2 \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (6 c^2 d+25 e\right )}{225 x^3}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (24 c^4 d^2+100 c^2 d e+225 e^2\right )}{225 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 270
Rule 276
Rule 464
Rule 1279
Rule 6436
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right )^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x^6} \, dx &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{15 x^6 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {1}{15} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-3 d^2-10 d e x^2-15 e^2 x^4}{x^6 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {1}{75} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {2 d \left (6 c^2 d+25 e\right )+75 e^2 x^2}{x^4 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \left (6 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}-\frac {1}{225} \left (b \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{x^2 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d^2 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{25 x^5}+\frac {2 b d \left (6 c^2 d+25 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x^3}+\frac {b \left (24 c^4 d^2+100 c^2 d e+225 e^2\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{225 x}-\frac {d^2 \left (a+b \text {sech}^{-1}(c x)\right )}{5 x^5}-\frac {2 d e \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e^2 \left (a+b \text {sech}^{-1}(c x)\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 134, normalized size = 0.63 \begin {gather*} \frac {-15 a \left (3 d^2+10 d e x^2+15 e^2 x^4\right )+b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (225 e^2 x^4+50 d e x^2 \left (1+2 c^2 x^2\right )+3 d^2 \left (3+4 c^2 x^2+8 c^4 x^4\right )\right )-15 b \left (3 d^2+10 d e x^2+15 e^2 x^4\right ) \text {sech}^{-1}(c x)}{225 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 193, normalized size = 0.91
method | result | size |
derivativedivides | \(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {2 d e}{3 c \,x^{3}}-\frac {d^{2}}{5 c \,x^{5}}\right )}{c^{4}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) e^{2}}{c x}-\frac {2 \,\mathrm {arcsech}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) d^{2}}{5 c \,x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{4} x^{4}}\right )}{c^{4}}\right )\) | \(193\) |
default | \(c^{5} \left (\frac {a \left (-\frac {e^{2}}{c x}-\frac {2 d e}{3 c \,x^{3}}-\frac {d^{2}}{5 c \,x^{5}}\right )}{c^{4}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) e^{2}}{c x}-\frac {2 \,\mathrm {arcsech}\left (c x \right ) d e}{3 c \,x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) d^{2}}{5 c \,x^{5}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (24 c^{8} d^{2} x^{4}+100 c^{6} d e \,x^{4}+12 c^{6} d^{2} x^{2}+225 c^{4} e^{2} x^{4}+50 c^{4} d e \,x^{2}+9 c^{4} d^{2}\right )}{225 c^{4} x^{4}}\right )}{c^{4}}\right )\) | \(193\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 175, normalized size = 0.82 \begin {gather*} \frac {1}{75} \, b d^{2} {\left (\frac {3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 10 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 15 \, c^{6} \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c} - \frac {15 \, \operatorname {arsech}\left (c x\right )}{x^{5}}\right )} + \frac {2}{9} \, b d {\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 {arsech}\left (c x\right )}{x^{3}}\right )} e + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b e^{2} - \frac {a e^{2}}{x} - \frac {2 \, a d e}{3 \, x^{3}} - \frac {a d^{2}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 278 vs.
\(2 (138) = 276\).
time = 0.45, size = 278, normalized size = 1.31 \begin {gather*} -\frac {225 \, a x^{4} \cosh \left (1\right )^{2} + 225 \, a x^{4} \sinh \left (1\right )^{2} + 150 \, a d x^{2} \cosh \left (1\right ) + 45 \, a d^{2} + 15 \, {\left (15 \, b x^{4} \cosh \left (1\right )^{2} + 15 \, b x^{4} \sinh \left (1\right )^{2} + 10 \, b d x^{2} \cosh \left (1\right ) + 3 \, b d^{2} + 10 \, {\left (3 \, b x^{4} \cosh \left (1\right ) + b d x^{2}\right )} \sinh \left (1\right )\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + 150 \, {\left (3 \, a x^{4} \cosh \left (1\right ) + a d x^{2}\right )} \sinh \left (1\right ) - {\left (24 \, b c^{5} d^{2} x^{5} + 12 \, b c^{3} d^{2} x^{3} + 225 \, b c x^{5} \cosh \left (1\right )^{2} + 225 \, b c x^{5} \sinh \left (1\right )^{2} + 9 \, b c d^{2} x + 50 \, {\left (2 \, b c^{3} d x^{5} + b c d x^{3}\right )} \cosh \left (1\right ) + 50 \, {\left (2 \, b c^{3} d x^{5} + 9 \, b c x^{5} \cosh \left (1\right ) + b c d x^{3}\right )} \sinh \left (1\right )\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{225 \, x^{5}} \end {gather*}
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 \begin {gather*} \int \frac {\left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x^{2}\right )^{2}}{x^{6}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (e\,x^2+d\right )}^2\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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