Optimal. Leaf size=126 \[ \frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x^3}+\frac {b \left (2 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{x} \]
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Rubi [A]
time = 0.05, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {14, 6436, 12,
464, 270} \begin {gather*} -\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2} \left (2 c^2 d+9 e\right )}{9 x}+\frac {b d \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \sqrt {1-c^2 x^2}}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 270
Rule 464
Rule 6436
Rubi steps
\begin {align*} \int \frac {\left (d+e x^2\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{x^4} \, dx &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e \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 {-d-3 e x^2}{3 x^4 \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {1}{3} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {-d-3 e x^2}{x^4 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b d \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x^3}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{x}+\frac {1}{9} \left (b \left (-2 c^2 d-9 e\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 \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x^3}+\frac {b \left (2 c^2 d+9 e\right ) \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sqrt {1-c^2 x^2}}{9 x}-\frac {d \left (a+b \text {sech}^{-1}(c x)\right )}{3 x^3}-\frac {e \left (a+b \text {sech}^{-1}(c x)\right )}{x}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 76, normalized size = 0.60 \begin {gather*} \frac {-3 a \left (d+3 e x^2\right )+b \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (d+2 c^2 d x^2+9 e x^2\right )-3 b \left (d+3 e x^2\right ) \text {sech}^{-1}(c x)}{9 x^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 123, normalized size = 0.98
method | result | size |
derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) e}{c x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{4} d \,x^{2}+9 e \,c^{2} x^{2}+c^{2} d \right )}{9 c^{2} x^{2}}\right )}{c^{2}}\right )\) | \(123\) |
default | \(c^{3} \left (\frac {a \left (-\frac {d}{3 c \,x^{3}}-\frac {e}{c x}\right )}{c^{2}}+\frac {b \left (-\frac {\mathrm {arcsech}\left (c x \right ) d}{3 c \,x^{3}}-\frac {\mathrm {arcsech}\left (c x \right ) e}{c x}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, \left (2 c^{4} d \,x^{2}+9 e \,c^{2} x^{2}+c^{2} d \right )}{9 c^{2} x^{2}}\right )}{c^{2}}\right )\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 93, normalized size = 0.74 \begin {gather*} \frac {1}{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 )} + {\left (c \sqrt {\frac {1}{c^{2} x^{2}} - 1} - \frac {\operatorname {arsech}\left (c x\right )}{x}\right )} b e - \frac {a e}{x} - \frac {a d}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.52, size = 135, normalized size = 1.07 \begin {gather*} -\frac {9 \, a x^{2} \cosh \left (1\right ) + 9 \, a x^{2} \sinh \left (1\right ) + 3 \, a d + 3 \, {\left (3 \, b x^{2} \cosh \left (1\right ) + 3 \, b x^{2} \sinh \left (1\right ) + b d\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (2 \, b c^{3} d x^{3} + 9 \, b c x^{3} \cosh \left (1\right ) + 9 \, b c x^{3} \sinh \left (1\right ) + b c d x\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{9 \, x^{3}} \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 )}{x^{4}}\, 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 = -0.01 \begin {gather*} \int \frac {\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}{x^4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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