Optimal. Leaf size=200 \[ \frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )+b c^2 d^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5920, 99, 12,
38, 54, 5919, 5882, 3799, 2221, 2317, 2438} \begin {gather*} -c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} b c^3 d^2 x \sqrt {c x-1} \sqrt {c x+1}+b c^2 d^2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-\frac {b c d^2 (c x-1)^{3/2} (c x+1)^{3/2}}{2 x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 38
Rule 54
Rule 99
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5919
Rule 5920
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx+\frac {1}{2} \left (b c d^2\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2}}{x^2} \, dx\\ &=-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {1}{2} \left (b c d^2\right ) \int 3 c^2 \sqrt {-1+c x} \sqrt {1+c x} \, dx-\left (2 c^2 d^2\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx-\left (b c^3 d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=-\frac {1}{2} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (2 c^2 d^2\right ) \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )+\frac {1}{2} \left (b c^3 d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{2} \left (3 b c^3 d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}+\frac {1}{2} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-\left (4 c^2 d^2\right ) \text {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{4} \left (3 b c^3 d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (2 b c^2 d^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (b c^2 d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac {1}{4} b c^3 d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {b c d^2 (-1+c x)^{3/2} (1+c x)^{3/2}}{2 x}-\frac {1}{4} b c^2 d^2 \cosh ^{-1}(c x)-c^2 d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{b}-2 c^2 d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-b c^2 d^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.16, size = 182, normalized size = 0.91 \begin {gather*} \frac {d^2 \left (-2 a+2 a c^4 x^4+2 b c x \sqrt {-1+c x} \sqrt {1+c x}-b c^3 x^3 \sqrt {-1+c x} \sqrt {1+c x}-4 b c^2 x^2 \cosh ^{-1}(c x)^2-2 b c^2 x^2 \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )+2 b \cosh ^{-1}(c x) \left (-1+c^4 x^4-4 c^2 x^2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )-8 a c^2 x^2 \log (x)+4 b c^2 x^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right )}{4 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 7.14, size = 212, normalized size = 1.06
method | result | size |
derivativedivides | \(c^{2} \left (\frac {a \,d^{2} c^{2} x^{2}}{2}-\frac {a \,d^{2}}{2 c^{2} x^{2}}-2 a \,d^{2} \ln \left (c x \right )+d^{2} b \mathrm {arccosh}\left (c x \right )^{2}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b \,d^{2} \mathrm {arccosh}\left (c x \right )}{4}-\frac {d^{2} b}{2}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-2 d^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\right )\) | \(212\) |
default | \(c^{2} \left (\frac {a \,d^{2} c^{2} x^{2}}{2}-\frac {a \,d^{2}}{2 c^{2} x^{2}}-2 a \,d^{2} \ln \left (c x \right )+d^{2} b \mathrm {arccosh}\left (c x \right )^{2}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}}{2}-\frac {b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{4}-\frac {b \,d^{2} \mathrm {arccosh}\left (c x \right )}{4}-\frac {d^{2} b}{2}+\frac {d^{2} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 c x}-\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right )}{2 c^{2} x^{2}}-2 d^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )-d^{2} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )\right )\) | \(212\) |
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*} d^{2} \left (\int \frac {a}{x^{3}}\, dx + \int \left (- \frac {2 a c^{2}}{x}\right )\, dx + \int a c^{4} x\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\, dx + \int \left (- \frac {2 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\right )\, dx + \int b c^{4} x \operatorname {acosh}{\left (c x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________