Optimal. Leaf size=184 \[ \frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \cosh ^{-1}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} b d^2 \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.17, antiderivative size = 184, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5919, 5882,
3799, 2221, 2317, 2438, 38, 54} \begin {gather*} \frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{2} b d^2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac {11}{32} b c d^2 x \sqrt {c x-1} \sqrt {c x+1}-\frac {11}{32} b d^2 \cosh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 38
Rule 54
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5882
Rule 5919
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx &=\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+d \int \frac {\left (d-c^2 d x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{4} \left (b c d^2\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx+\frac {1}{16} \left (3 b c d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx+\frac {1}{2} \left (b c d^2\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=\frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+d^2 \text {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{32} \left (3 b c d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx-\frac {1}{4} \left (b c d^2\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \cosh ^{-1}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+\left (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 {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \cosh ^{-1}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\left (b d^2\right ) \text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=\frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \cosh ^{-1}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac {1}{2} \left (b d^2\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=\frac {11}{32} b c d^2 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {1}{16} b c d^2 x (-1+c x)^{3/2} (1+c x)^{3/2}-\frac {11}{32} b d^2 \cosh ^{-1}(c x)+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} b d^2 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.39, size = 208, normalized size = 1.13 \begin {gather*} \frac {1}{4} d^2 \left (-4 a c^2 x^2+a c^4 x^4-4 b c^2 x^2 \cosh ^{-1}(c x)+b c^4 x^4 \cosh ^{-1}(c x)+2 b \left (c x \sqrt {-1+c x} \sqrt {1+c x}+2 \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )-\frac {1}{8} b \left (c x \sqrt {\frac {-1+c x}{1+c x}} \left (3+3 c x+2 c^2 x^2+2 c^3 x^3\right )+6 \tanh ^{-1}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )\right )+2 b \cosh ^{-1}(c x) \left (\cosh ^{-1}(c x)+2 \log \left (1+e^{-2 \cosh ^{-1}(c x)}\right )\right )+4 a \log (x)-2 b \text {PolyLog}\left (2,-e^{-2 \cosh ^{-1}(c x)}\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 5.30, size = 201, normalized size = 1.09
method | result | size |
derivativedivides | \(\frac {a \,d^{2} c^{4} x^{4}}{4}-a \,d^{2} c^{2} x^{2}+a \,d^{2} \ln \left (c x \right )+d^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \mathrm {arccosh}\left (c x \right )^{2}}{2}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{16}+\frac {13 b \,d^{2} \mathrm {arccosh}\left (c x \right )}{32}+\frac {d^{2} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}\) | \(201\) |
default | \(\frac {a \,d^{2} c^{4} x^{4}}{4}-a \,d^{2} c^{2} x^{2}+a \,d^{2} \ln \left (c x \right )+d^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \mathrm {arccosh}\left (c x \right )^{2}}{2}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{16}+\frac {13 b \,d^{2} \mathrm {arccosh}\left (c x \right )}{32}+\frac {d^{2} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}\) | \(201\) |
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}\, dx + \int \left (- 2 a c^{2} x\right )\, dx + \int a c^{4} x^{3}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x}\, dx + \int \left (- 2 b c^{2} x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{3} \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 = -0.01 \begin {gather*} \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________