Optimal. Leaf size=190 \[ \frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 190, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6420, 5556,
3378, 3384, 3379, 3382} \begin {gather*} -\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 6420
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b \text {sech}^{-1}(c x)\right )^2} \, dx &=-\left (c^3 \text {Subst}\left (\int \frac {\cosh ^2(x) \sinh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (c^3 \text {Subst}\left (\int \left (\frac {\sinh (x)}{4 (a+b x)^2}+\frac {\sinh (3 x)}{4 (a+b x)^2}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\right )\\ &=-\left (\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sinh (x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\right )-\frac {1}{4} c^3 \text {Subst}\left (\int \frac {\sinh (3 x)}{(a+b x)^2} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}-\frac {\left (3 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {\left (c^3 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}-\frac {\left (3 c^3 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}+\frac {\left (c^3 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}+\frac {\left (3 c^3 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\text {sech}^{-1}(c x)\right )}{4 b}\\ &=\frac {c^2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)}{4 b x \left (a+b \text {sech}^{-1}(c x)\right )}-\frac {c^3 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}-\frac {3 c^3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {c^3 \sinh \left (3 \text {sech}^{-1}(c x)\right )}{4 b \left (a+b \text {sech}^{-1}(c x)\right )}+\frac {c^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )}{4 b^2}+\frac {3 c^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \text {sech}^{-1}(c x)\right )}{4 b^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.38, size = 250, normalized size = 1.32 \begin {gather*} \frac {4 b \sqrt {\frac {1-c x}{1+c x}}+4 b c x \sqrt {\frac {1-c x}{1+c x}}-c^3 x^3 \left (a+b \text {sech}^{-1}(c x)\right ) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )-3 c^3 x^3 \left (a+b \text {sech}^{-1}(c x)\right ) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+a c^3 x^3 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+b c^3 x^3 \text {sech}^{-1}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )+3 a c^3 x^3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )+3 b c^3 x^3 \text {sech}^{-1}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {sech}^{-1}(c x)\right )\right )}{4 b^2 x^3 \left (a+b \text {sech}^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(419\) vs.
\(2(176)=352\).
time = 0.55, size = 420, normalized size = 2.21
method | result | size |
derivativedivides | \(c^{3} \left (-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -3 c^{2} x^{2}+4}{8 c^{3} x^{3} b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1}{8 c x b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{8 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4}{8 b \,c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}\right )\) | \(420\) |
default | \(c^{3} \left (-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -3 c^{2} x^{2}+4}{8 c^{3} x^{3} b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, \frac {3 a}{b}+3 \,\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x -1}{8 c x b \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \frac {a}{b}+\mathrm {arcsech}\left (c x \right )\right )}{8 b^{2}}+\frac {\sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +1}{8 b c x \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\mathrm {arcsech}\left (c x \right )-\frac {a}{b}\right )}{8 b^{2}}-\frac {\sqrt {\frac {c x +1}{c x}}\, \sqrt {-\frac {c x -1}{c x}}\, c^{3} x^{3}-4 \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c x +3 c^{2} x^{2}-4}{8 b \,c^{3} x^{3} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )}+\frac {3 \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \,\mathrm {arcsech}\left (c x \right )-\frac {3 a}{b}\right )}{8 b^{2}}\right )\) | \(420\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{x^{4} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{x^4\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________