Optimal. Leaf size=312 \[ \frac {e n (a x-1)^{3/2} (a x+1)^{3/2}}{27 a^3}+\frac {2 e n \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}-\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right ) \log \left (c x^n\right )}{9 a^3}+\frac {n \sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right )}{9 a^3}-\frac {n \left (9 a^2 d+2 e\right ) \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )}{9 a^3}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )-\frac {e x^2 \sqrt {a x-1} \sqrt {a x+1} \log \left (c x^n\right )}{9 a}+\frac {d n \sqrt {a x-1} \sqrt {a x+1}}{a}-d n x \cosh ^{-1}(a x)-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)+\frac {e n x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 312, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 11, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.611, Rules used = {5705, 460, 74, 2387, 101, 92, 205, 5654, 5662, 100, 12} \[ -\frac {\sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right ) \log \left (c x^n\right )}{9 a^3}+\frac {n \sqrt {a x-1} \sqrt {a x+1} \left (9 a^2 d+2 e\right )}{9 a^3}-\frac {n \left (9 a^2 d+2 e\right ) \tan ^{-1}\left (\sqrt {a x-1} \sqrt {a x+1}\right )}{9 a^3}+\frac {e n (a x-1)^{3/2} (a x+1)^{3/2}}{27 a^3}+\frac {2 e n \sqrt {a x-1} \sqrt {a x+1}}{27 a^3}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )-\frac {e x^2 \sqrt {a x-1} \sqrt {a x+1} \log \left (c x^n\right )}{9 a}+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {d n \sqrt {a x-1} \sqrt {a x+1}}{a}-d n x \cosh ^{-1}(a x)+\frac {e n x^2 \sqrt {a x-1} \sqrt {a x+1}}{27 a}-\frac {1}{9} e n x^3 \cosh ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 74
Rule 92
Rule 100
Rule 101
Rule 205
Rule 460
Rule 2387
Rule 5654
Rule 5662
Rule 5705
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \cosh ^{-1}(a x) \log \left (c x^n\right ) \, dx &=-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )-n \int \left (-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3 x}-\frac {e x \sqrt {-1+a x} \sqrt {1+a x}}{9 a}+d \cosh ^{-1}(a x)+\frac {1}{3} e x^2 \cosh ^{-1}(a x)\right ) \, dx\\ &=-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )-(d n) \int \cosh ^{-1}(a x) \, dx-\frac {1}{3} (e n) \int x^2 \cosh ^{-1}(a x) \, dx+\frac {(e n) \int x \sqrt {-1+a x} \sqrt {1+a x} \, dx}{9 a}+\frac {\left (\left (9 a^2 d+2 e\right ) n\right ) \int \frac {\sqrt {-1+a x} \sqrt {1+a x}}{x} \, dx}{9 a^3}\\ &=\frac {\left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}+\frac {e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+(a d n) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx+\frac {1}{9} (a e n) \int \frac {x^3}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx-\frac {\left (\left (9 a^2 d+2 e\right ) n\right ) \int \frac {1}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{9 a^3}\\ &=\frac {d n \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {\left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}+\frac {e n x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {(e n) \int \frac {2 x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a}-\frac {\left (\left (9 a^2 d+2 e\right ) n\right ) \operatorname {Subst}\left (\int \frac {1}{a+a x^2} \, dx,x,\sqrt {-1+a x} \sqrt {1+a x}\right )}{9 a^2}\\ &=\frac {d n \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {\left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}+\frac {e n x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)-\frac {\left (9 a^2 d+2 e\right ) n \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{9 a^3}-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {(2 e n) \int \frac {x}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{27 a}\\ &=\frac {d n \sqrt {-1+a x} \sqrt {1+a x}}{a}+\frac {2 e n \sqrt {-1+a x} \sqrt {1+a x}}{27 a^3}+\frac {\left (9 a^2 d+2 e\right ) n \sqrt {-1+a x} \sqrt {1+a x}}{9 a^3}+\frac {e n x^2 \sqrt {-1+a x} \sqrt {1+a x}}{27 a}+\frac {e n (-1+a x)^{3/2} (1+a x)^{3/2}}{27 a^3}-d n x \cosh ^{-1}(a x)-\frac {1}{9} e n x^3 \cosh ^{-1}(a x)-\frac {\left (9 a^2 d+2 e\right ) n \tan ^{-1}\left (\sqrt {-1+a x} \sqrt {1+a x}\right )}{9 a^3}-\frac {\left (9 a^2 d+2 e\right ) \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a^3}-\frac {e x^2 \sqrt {-1+a x} \sqrt {1+a x} \log \left (c x^n\right )}{9 a}+d x \cosh ^{-1}(a x) \log \left (c x^n\right )+\frac {1}{3} e x^3 \cosh ^{-1}(a x) \log \left (c x^n\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.24, size = 145, normalized size = 0.46 \[ \frac {-3 a^3 x \cosh ^{-1}(a x) \left (n \left (9 d+e x^2\right )-3 \left (3 d+e x^2\right ) \log \left (c x^n\right )\right )+\sqrt {a x-1} \sqrt {a x+1} \left (n \left (2 a^2 \left (27 d+e x^2\right )+7 e\right )-3 \left (a^2 \left (9 d+e x^2\right )+2 e\right ) \log \left (c x^n\right )\right )+3 n \left (9 a^2 d+2 e\right ) \tan ^{-1}\left (\frac {1}{\sqrt {a x-1} \sqrt {a x+1}}\right )}{27 a^3} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 1.05, size = 275, normalized size = 0.88 \[ -\frac {6 \, {\left (9 \, a^{2} d + 2 \, e\right )} n \arctan \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) + 3 \, {\left (a^{3} e n x^{3} + 9 \, a^{3} d n x - {\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x - 3 \, a^{3} d - a^{3} e\right )} \log \relax (c) - 3 \, {\left (a^{3} e n x^{3} + 3 \, a^{3} d n x\right )} \log \relax (x)\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - 3 \, {\left ({\left (9 \, a^{3} d + a^{3} e\right )} n - 3 \, {\left (3 \, a^{3} d + a^{3} e\right )} \log \relax (c)\right )} \log \left (-a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (2 \, a^{2} e n x^{2} + {\left (54 \, a^{2} d + 7 \, e\right )} n - 3 \, {\left (a^{2} e x^{2} + 9 \, a^{2} d + 2 \, e\right )} \log \relax (c) - 3 \, {\left (a^{2} e n x^{2} + {\left (9 \, a^{2} d + 2 \, e\right )} n\right )} \log \relax (x)\right )} \sqrt {a^{2} x^{2} - 1}}{27 \, a^{3}} \]
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 [F(-2)] time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (e \,x^{2}+d \right ) \mathrm {arccosh}\left (a x \right ) \ln \left (c \,x^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (3 \, a^{2} d n + e n\right )} {\left (\log \left (a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (-a x\right )\right )}}{6 \, a^{3}} - \frac {{\left (3 \, a^{2} d n + e n\right )} {\left (\log \left (-a x + 1\right ) \log \relax (x) + {\rm Li}_2\left (a x\right )\right )}}{6 \, a^{3}} - \frac {{\left (9 \, {\left (d n - d \log \relax (c)\right )} a^{2} + e n - 3 \, e \log \relax (c)\right )} \log \left (a x + 1\right )}{18 \, a^{3}} + \frac {{\left (9 \, {\left (d n - d \log \relax (c)\right )} a^{2} + e n - 3 \, e \log \relax (c)\right )} \log \left (a x - 1\right )}{18 \, a^{3}} + \frac {2 \, {\left (2 \, e n - 3 \, e \log \relax (c)\right )} a^{3} x^{3} - 9 \, {\left (3 \, a^{2} d n + e n\right )} \log \left (a x + 1\right ) \log \relax (x) + 9 \, {\left (3 \, a^{2} d n + e n\right )} \log \left (a x - 1\right ) \log \relax (x) + 6 \, {\left (9 \, {\left (2 \, d n - d \log \relax (c)\right )} a^{3} + {\left (4 \, e n - 3 \, e \log \relax (c)\right )} a\right )} x - 6 \, {\left ({\left (e n - 3 \, e \log \relax (c)\right )} a^{3} x^{3} + 9 \, {\left (d n - d \log \relax (c)\right )} a^{3} x - 3 \, {\left (a^{3} e x^{3} + 3 \, a^{3} d x\right )} \log \left (x^{n}\right )\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right ) - 3 \, {\left (2 \, a^{3} e x^{3} + 6 \, {\left (3 \, a^{3} d + a e\right )} x - 3 \, {\left (3 \, a^{2} d + e\right )} \log \left (a x + 1\right ) + 3 \, {\left (3 \, a^{2} d + e\right )} \log \left (a x - 1\right )\right )} \log \left (x^{n}\right )}{54 \, a^{3}} + \int -\frac {{\left (e n - 3 \, e \log \relax (c)\right )} a x^{3} + 9 \, {\left (d n - d \log \relax (c)\right )} a x - 3 \, {\left (a e x^{3} + 3 \, a d x\right )} \log \left (x^{n}\right )}{9 \, {\left (a^{3} x^{3} + {\left (a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \ln \left (c\,x^n\right )\,\mathrm {acosh}\left (a\,x\right )\,\left (e\,x^2+d\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d + e x^{2}\right ) \log {\left (c x^{n} \right )} \operatorname {acosh}{\left (a x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________