Optimal. Leaf size=562 \[ \frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d x \left (d-c^2 d x^2\right )^{3/2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d x^3 \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {16 c^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.82, antiderivative size = 607, normalized size of antiderivative = 1.08, number of steps used = 34, number of rules used = 18, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.621, Rules used = {5798, 5748, 5691, 5688, 5715, 3716, 2190, 2279, 2391, 5716, 39, 5754, 5721, 5461, 4182, 5746, 103, 12} \[ -\frac {8 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {c x-1} \sqrt {c x+1} \text {PolyLog}\left (2,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (c x+1) \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {c x-1} \sqrt {c x+1} \log \left (1-e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {c x-1} \sqrt {c x+1} \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right ) \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 39
Rule 103
Rule 2190
Rule 2279
Rule 2391
Rule 3716
Rule 4182
Rule 5461
Rule 5688
Rule 5691
Rule 5715
Rule 5716
Rule 5721
Rule 5746
Rule 5748
Rule 5754
Rule 5798
Rubi steps
\begin {align*} \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (\sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^4 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x^3 \left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{x^2 (-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{x^2 (-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{5/2} (1+c x)^{5/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {8 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {2 c^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {a+b \cosh ^{-1}(c x)}{x \left (-1+c^2 x^2\right )} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (16 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b c^5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{\left (-1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}+\frac {8 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b^2 c^4 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {1}{(-1+c x)^{3/2} (1+c x)^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (32 b c^5 \sqrt {-1+c x} \sqrt {1+c x}\right ) \int \frac {x \left (a+b \cosh ^{-1}(c x)\right )}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (8 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (32 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int (a+b x) \coth (x) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (64 b c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1-e^{2 x}} \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (32 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (16 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2 c^2}{3 d^2 x \sqrt {d-c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{3 d^2 x^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}+\frac {16 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 x^3 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}-\frac {2 c^2 \left (a+b \cosh ^{-1}(c x)\right )^2}{d^2 x (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {8 c^4 x \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2}}+\frac {16 c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {32 b c^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \log \left (1-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {-1+c x} \sqrt {1+c x} \text {Li}_2\left (e^{2 \cosh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 3.68, size = 534, normalized size = 0.95 \[ \frac {\frac {a^2 \left (16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right )}{x^3 \left (c^2 x^2-1\right )}+a b c^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (\frac {1}{1-c^2 x^2}+\frac {1}{c^2 x^2}+\frac {2 \left (16 c^6 x^6-24 c^4 x^4+6 c^2 x^2+1\right ) \left (\frac {c x-1}{c x+1}\right )^{3/2} \cosh ^{-1}(c x)}{c^3 x^3 (c x-1)^3}-16 \log (c x)-16 \log \left (\sqrt {\frac {c x-1}{c x+1}} (c x+1)\right )\right )+b^2 c^3 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (\frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c^3 x^3}+\frac {\cosh ^{-1}(c x)}{1-c^2 x^2}+\frac {\cosh ^{-1}(c x)}{c^2 x^2}+8 \text {Li}_2\left (-e^{-2 \cosh ^{-1}(c x)}\right )+8 \text {Li}_2\left (e^{-2 \cosh ^{-1}(c x)}\right )-\frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c x}+\frac {c x \sqrt {\frac {c x-1}{c x+1}}}{1-c x}+\frac {8 \sqrt {\frac {c x-1}{c x+1}} (c x+1) \cosh ^{-1}(c x)^2}{c x}+\frac {8 c x \cosh ^{-1}(c x)^2}{\sqrt {\frac {c x-1}{c x+1}} (c x+1)}-\frac {c x \cosh ^{-1}(c x)^2}{\left (\frac {c x-1}{c x+1}\right )^{3/2} (c x+1)^3}-16 \cosh ^{-1}(c x)^2-16 \cosh ^{-1}(c x) \log \left (1-e^{-2 \cosh ^{-1}(c x)}\right )-16 \cosh ^{-1}(c x) \log \left (e^{-2 \cosh ^{-1}(c x)}+1\right )\right )}{3 d^2 \sqrt {d-c^2 d x^2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{10} - 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} - d^{3} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.88, size = 5251, normalized size = 9.34 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, a b c {\left (\frac {8 \, c^{2} \sqrt {-d} \log \left (c x + 1\right )}{d^{3}} + \frac {8 \, c^{2} \sqrt {-d} \log \left (c x - 1\right )}{d^{3}} + \frac {16 \, c^{2} \sqrt {-d} \log \relax (x)}{d^{3}} + \frac {\sqrt {-d}}{c^{2} d^{3} x^{4} - d^{3} x^{2}}\right )} + \frac {2}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a b \operatorname {arcosh}\left (c x\right ) + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {-c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} - \frac {6 \, c^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a^{2} + b^{2} \int \frac {\log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{4} \left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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