Integrand size = 29, antiderivative size = 836 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (1+c x)}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
Time = 0.92 (sec) , antiderivative size = 836, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {5930, 5926, 5947, 4265, 2611, 2320, 6724, 5879, 75, 5889, 5894, 12, 471, 200, 534, 1261, 712} \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=-\frac {2 b c^5 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x^5}{25 \sqrt {c x-1} \sqrt {c x+1}}+\frac {22 b c^3 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x^3}{45 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2} x^2-\frac {2 b^2 c d^2 \sqrt {d-c^2 d x^2} \text {arccosh}(c x) x}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {16 b c d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) x}{15 \sqrt {c x-1} \sqrt {c x+1}}-\frac {2 a b c d^2 \sqrt {d-c^2 d x^2} x}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (c x+1)}+\frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (c x+1)}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (c x+1)} \]
[In]
[Out]
Rule 12
Rule 75
Rule 200
Rule 471
Rule 534
Rule 712
Rule 1261
Rule 2320
Rule 2611
Rule 4265
Rule 5879
Rule 5889
Rule 5894
Rule 5926
Rule 5930
Rule 5947
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+d \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x} \, dx-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x)^2 (1+c x)^2 (a+b \text {arccosh}(c x)) \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = \frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2+d^2 \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x} \, dx-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right )^2 (a+b \text {arccosh}(c x)) \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (-1+c x) (1+c x) (a+b \text {arccosh}(c x)) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{5 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(a+b \text {arccosh}(c x))^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x)) \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b c d^2 \sqrt {d-c^2 d x^2}\right ) \int (a+b \text {arccosh}(c x)) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c d^2 \sqrt {d-c^2 d x^2}\right ) \int \text {arccosh}(c x) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{75 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 i b d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (-3+c^2 x^2\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{9 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {x \left (15-10 c^2 x^2+3 c^4 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{75 (-1+c x) (1+c x)} \\ & = 2 b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arccosh}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (14 b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{27 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {15-10 c^2 x+3 c^4 x^2}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)} \\ & = \frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 i b^2 d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {8}{\sqrt {-1+c^2 x}}-4 \sqrt {-1+c^2 x}+3 \left (-1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 (-1+c x) (1+c x)} \\ & = \frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (1+c x)}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ \end{align*}
Time = 5.84 (sec) , antiderivative size = 963, normalized size of antiderivative = 1.15 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\frac {1}{15} a^2 d^2 \sqrt {d-c^2 d x^2} \left (23-11 c^2 x^2+3 c^4 x^4\right )-\frac {1}{27} b^2 d^2 \sqrt {d-c^2 d x^2} \left (2 (-13+\cosh (2 \text {arccosh}(c x)))+9 \text {arccosh}(c x)^2 (-1+\cosh (2 \text {arccosh}(c x)))+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x) (9 c x-\cosh (3 \text {arccosh}(c x)))}{-1+c x}\right )-\frac {a b d^2 \sqrt {d-c^2 d x^2} \left (9 c x+12 \left (\frac {-1+c x}{1+c x}\right )^{3/2} (1+c x)^3 \text {arccosh}(c x)-\cosh (3 \text {arccosh}(c x))\right )}{9 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+a^2 d^{5/2} \log (c x)-a^2 d^{5/2} \log \left (d+\sqrt {d} \sqrt {d-c^2 d x^2}\right )+\frac {2 a b d^2 \sqrt {d-c^2 d x^2} \left (-c x+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)+i \text {arccosh}(c x) \log \left (1-i e^{-\text {arccosh}(c x)}\right )-i \text {arccosh}(c x) \log \left (1+i e^{-\text {arccosh}(c x)}\right )+i \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-i \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}+b^2 d^2 \sqrt {d-c^2 d x^2} \left (2+\frac {2 c x \sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)}{1-c x}+\text {arccosh}(c x)^2+\frac {i \left (\text {arccosh}(c x)^2 \log \left (1-i e^{-\text {arccosh}(c x)}\right )-\text {arccosh}(c x)^2 \log \left (1+i e^{-\text {arccosh}(c x)}\right )+2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(c x)}\right )-2 \text {arccosh}(c x) \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(c x)}\right )+2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(c x)}\right )-2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(c x)}\right )\right )}{\sqrt {\frac {-1+c x}{1+c x}} (1+c x)}\right )-\frac {a b d^2 \sqrt {d-c^2 d x^2} \left (25 \cosh (3 \text {arccosh}(c x))+9 (-50 c x+\cosh (5 \text {arccosh}(c x)))+15 \text {arccosh}(c x) \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh (3 \text {arccosh}(c x))-3 \sinh (5 \text {arccosh}(c x))\right )\right )}{1800 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)}-\frac {b^2 d^2 \sqrt {d-c^2 d x^2} \left (13500 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)+30 \text {arccosh}(c x) (25 \cosh (3 \text {arccosh}(c x))+9 (-50 c x+\cosh (5 \text {arccosh}(c x))))-250 \sinh (3 \text {arccosh}(c x))+225 \text {arccosh}(c x)^2 \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh (3 \text {arccosh}(c x))-3 \sinh (5 \text {arccosh}(c x))\right )-54 \sinh (5 \text {arccosh}(c x))\right )}{54000 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \]
[In]
[Out]
\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x}d x\]
[In]
[Out]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
[In]
[Out]
Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
[In]
[Out]
Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x} \,d x \]
[In]
[Out]