Optimal. Leaf size=184 \[ \frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{2} b d^2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )-\frac {1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac {11}{32} b c d^2 x \sqrt {c x-1} \sqrt {c x+1}-\frac {11}{32} b d^2 \cosh ^{-1}(c x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.20, 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 = {5727, 5660, 3718, 2190, 2279, 2391, 38, 52} \[ \frac {1}{2} b d^2 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{4} d^2 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{2} d^2 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}+d^2 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{16} b c d^2 x (c x-1)^{3/2} (c x+1)^{3/2}+\frac {11}{32} b c d^2 x \sqrt {c x-1} \sqrt {c x+1}-\frac {11}{32} b d^2 \cosh ^{-1}(c x) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 38
Rule 52
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5727
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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 0.28, size = 162, normalized size = 0.88 \[ \frac {1}{32} d^2 \left (8 a c^4 x^4-32 a c^2 x^2+32 a \log (x)-2 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1}+8 b \cosh ^{-1}(c x) \left (c^4 x^4-4 c^2 x^2+4 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )-16 b \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+13 b c x \sqrt {c x-1} \sqrt {c x+1}+16 b \cosh ^{-1}(c x)^2+26 b \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a c^{4} d^{2} x^{4} - 2 \, a c^{2} d^{2} x^{2} + a d^{2} + {\left (b c^{4} d^{2} x^{4} - 2 \, b c^{2} d^{2} x^{2} + b d^{2}\right )} \operatorname {arcosh}\left (c x\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 201, normalized size = 1.09 \[ \frac {d^{2} a \,c^{4} x^{4}}{4}-d^{2} a \,c^{2} x^{2}+d^{2} a \ln \left (c x \right )+\frac {13 b \,d^{2} \mathrm {arccosh}\left (c x \right )}{32}+d^{2} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )+\frac {d^{2} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}-\frac {d^{2} b \sqrt {c x -1}\, \sqrt {c x +1}\, c^{3} x^{3}}{16}+\frac {13 b c \,d^{2} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}+\frac {d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{4} x^{4}}{4}-d^{2} b \,\mathrm {arccosh}\left (c x \right ) c^{2} x^{2}-\frac {d^{2} b \mathrm {arccosh}\left (c x \right )^{2}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{4} \, a c^{4} d^{2} x^{4} - a c^{2} d^{2} x^{2} + a d^{2} \log \relax (x) + \int b c^{4} d^{2} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 2 \, b c^{2} d^{2} x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \frac {b d^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ d^{2} \left (\int \frac {a}{x}\, dx + \int \left (- 2 a c^{2} x\right )\, dx + \int a c^{4} x^{3}\, dx + \int \frac {b \operatorname {acosh}{\left (c x \right )}}{x}\, dx + \int \left (- 2 b c^{2} x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{4} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________