Optimal. Leaf size=267 \[ -\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{32} b c^3 d^3 x \sqrt {c x-1} \sqrt {c x+1}+\frac {3}{2} b c^2 d^3 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)+\frac {b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.32, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5729, 97, 12, 38, 52, 5727, 5660, 3718, 2190, 2279, 2391} \[ -\frac {3}{2} b c^2 d^3 \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \log \left (e^{2 \cosh ^{-1}(c x)}+1\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {7}{16} b c^3 d^3 x (c x-1)^{3/2} (c x+1)^{3/2}-\frac {3}{32} b c^3 d^3 x \sqrt {c x-1} \sqrt {c x+1}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)+\frac {b c d^3 (c x-1)^{5/2} (c x+1)^{5/2}}{2 x} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
Rule 12
Rule 38
Rule 52
Rule 97
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 5660
Rule 5727
Rule 5729
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{x^3} \, dx &=-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d\right ) \int \frac {\left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x} \, dx-\frac {1}{2} \left (b c d^3\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2} \, dx\\ &=\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^2\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^3\right ) \int 5 c^2 (-1+c x)^{3/2} (1+c x)^{3/2} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=\frac {3}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x} \, dx-\frac {1}{16} \left (9 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx-\frac {1}{2} \left (3 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx-\frac {1}{2} \left (5 b c^3 d^3\right ) \int (-1+c x)^{3/2} (1+c x)^{3/2} \, dx\\ &=-\frac {33}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}-\left (3 c^2 d^3\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\cosh ^{-1}(c x)\right )+\frac {1}{32} \left (9 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{4} \left (3 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx+\frac {1}{8} \left (15 b c^3 d^3\right ) \int \sqrt {-1+c x} \sqrt {1+c x} \, dx\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {33}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-\left (6 c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )-\frac {1}{16} \left (15 b c^3 d^3\right ) \int \frac {1}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\left (3 b c^2 d^3\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )+\frac {1}{2} \left (3 b c^2 d^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )\\ &=-\frac {3}{32} b c^3 d^3 x \sqrt {-1+c x} \sqrt {1+c x}-\frac {7}{16} b c^3 d^3 x (-1+c x)^{3/2} (1+c x)^{3/2}+\frac {b c d^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{2 x}+\frac {3}{32} b c^2 d^3 \cosh ^{-1}(c x)-\frac {3}{2} c^2 d^3 \left (1-c^2 x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )-\frac {3}{4} c^2 d^3 \left (1-c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )-\frac {d^3 \left (1-c^2 x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right )}{2 x^2}+\frac {3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b}-3 c^2 d^3 \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1+e^{2 \cosh ^{-1}(c x)}\right )-\frac {3}{2} b c^2 d^3 \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.37, size = 226, normalized size = 0.85 \[ -\frac {d^3 \left (8 a c^6 x^6-48 a c^4 x^4+96 a c^2 x^2 \log (x)+16 a-2 b c^5 x^5 \sqrt {c x-1} \sqrt {c x+1}+21 b c^3 x^3 \sqrt {c x-1} \sqrt {c x+1}-48 b c^2 x^2 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+48 b c^2 x^2 \cosh ^{-1}(c x)^2+42 b c^2 x^2 \tanh ^{-1}\left (\sqrt {\frac {c x-1}{c x+1}}\right )+8 b \cosh ^{-1}(c x) \left (c^6 x^6-6 c^4 x^4+12 c^2 x^2 \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )+2\right )-16 b c x \sqrt {c x-1} \sqrt {c x+1}\right )}{32 x^2} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a c^{6} d^{3} x^{6} - 3 \, a c^{4} d^{3} x^{4} + 3 \, a c^{2} d^{3} x^{2} - a d^{3} + {\left (b c^{6} d^{3} x^{6} - 3 \, b c^{4} d^{3} x^{4} + 3 \, b c^{2} d^{3} x^{2} - b d^{3}\right )} \operatorname {arcosh}\left (c x\right )}{x^{3}}, x\right ) \]
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 = 0.78, size = 275, normalized size = 1.03 \[ -\frac {c^{6} d^{3} a \,x^{4}}{4}+\frac {3 c^{4} d^{3} a \,x^{2}}{2}-3 c^{2} d^{3} a \ln \left (c x \right )-\frac {d^{3} a}{2 x^{2}}-\frac {d^{3} b \,c^{2}}{2}-\frac {21 b \,c^{2} d^{3} \mathrm {arccosh}\left (c x \right )}{32}+\frac {c \,d^{3} b \sqrt {c x +1}\, \sqrt {c x -1}}{2 x}+\frac {c^{5} d^{3} b \sqrt {c x -1}\, \sqrt {c x +1}\, x^{3}}{16}-\frac {21 b \,c^{3} d^{3} x \sqrt {c x -1}\, \sqrt {c x +1}}{32}-\frac {c^{6} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{4}}{4}+\frac {3 c^{4} d^{3} b \,\mathrm {arccosh}\left (c x \right ) x^{2}}{2}-\frac {d^{3} b \,\mathrm {arccosh}\left (c x \right )}{2 x^{2}}-\frac {3 c^{2} d^{3} b \polylog \left (2, -\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right )}{2}+\frac {3 c^{2} d^{3} b \mathrm {arccosh}\left (c x \right )^{2}}{2}-3 c^{2} d^{3} b \,\mathrm {arccosh}\left (c x \right ) \ln \left (1+\left (c x +\sqrt {c x -1}\, \sqrt {c x +1}\right )^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {1}{4} \, a c^{6} d^{3} x^{4} + \frac {3}{2} \, a c^{4} d^{3} x^{2} - 3 \, a c^{2} d^{3} \log \relax (x) + \frac {1}{2} \, b d^{3} {\left (\frac {\sqrt {c^{2} x^{2} - 1} c}{x} - \frac {\operatorname {arcosh}\left (c x\right )}{x^{2}}\right )} - \frac {a d^{3}}{2 \, x^{2}} - \int b c^{6} d^{3} x^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) - 3 \, b c^{4} d^{3} x \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right ) + \frac {3 \, b c^{2} d^{3} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^3}{x^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - d^{3} \left (\int \left (- \frac {a}{x^{3}}\right )\, dx + \int \frac {3 a c^{2}}{x}\, dx + \int \left (- 3 a c^{4} x\right )\, dx + \int a c^{6} x^{3}\, dx + \int \left (- \frac {b \operatorname {acosh}{\left (c x \right )}}{x^{3}}\right )\, dx + \int \frac {3 b c^{2} \operatorname {acosh}{\left (c x \right )}}{x}\, dx + \int \left (- 3 b c^{4} x \operatorname {acosh}{\left (c x \right )}\right )\, dx + \int b c^{6} x^{3} \operatorname {acosh}{\left (c x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________