Optimal. Leaf size=836 \[ \frac {68}{27} b^2 d^2 \sqrt {d-c^2 d x^2}-\frac {2}{27} b^2 c^2 d^2 x^2 \sqrt {d-c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {16 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{75 (1-c x) (1+c x)}+\frac {8 b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{225 (1-c x) (1+c x)}+\frac {2 b^2 d^2 \left (1-c^2 x^2\right )^3 \sqrt {d-c^2 d x^2}}{125 (1-c x) (1+c x)}-\frac {2 b^2 c d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{15 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {22 b c^3 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{45 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^5 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} \left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.84, antiderivative size = 836, normalized size of antiderivative = 1.00, number of steps
used = 26, number of rules used = 17, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.586, Rules used = {5930,
5926, 5947, 4265, 2611, 2320, 6724, 5879, 75, 5889, 5894, 12, 471, 200, 534, 1261, 712}
\begin {gather*} -\frac {2 b c^5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) 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} \left (a+b \cosh ^{-1}(c x)\right ) 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} \cosh ^{-1}(c x) x}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {16 b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) 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} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(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} \text {Li}_3\left (-i e^{\cosh ^{-1}(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} \text {Li}_3\left (i e^{\cosh ^{-1}(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)} \end {gather*}
Antiderivative was successfully verified.
[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*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{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 )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{5 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {2 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^2}{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 ) \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{3 \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 {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^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 (a+b \cosh ^{-1}(c x)\right ) \, 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 {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^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,\cosh ^{-1}(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 \cosh ^{-1}(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 (-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 {-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)}\\ &=-\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} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(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,\cosh ^{-1}(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,\cosh ^{-1}(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 (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 (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} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(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 \text {Li}_2\left (-i e^x\right ) \, dx,x,\cosh ^{-1}(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 \text {Li}_2\left (i e^x\right ) \, dx,x,\cosh ^{-1}(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} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(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 {\text {Li}_2(-i x)}{x} \, dx,x,e^{\cosh ^{-1}(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 {\text {Li}_2(i x)}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {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} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {16 b c d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{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} \left (a+b \cosh ^{-1}(c x)\right )}{25 \sqrt {-1+c x} \sqrt {1+c x}}+d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{3} d^2 (1-c x) (1+c x) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2+\frac {1}{5} d^2 (1-c x)^2 (1+c x)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(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} \text {Li}_3\left (-i e^{\cosh ^{-1}(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} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 5.21, size = 963, normalized size = 1.15 \begin {gather*} \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 \left (-13+\cosh \left (2 \cosh ^{-1}(c x)\right )\right )+9 \cosh ^{-1}(c x)^2 \left (-1+\cosh \left (2 \cosh ^{-1}(c x)\right )\right )+\frac {3 \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x) \left (9 c x-\cosh \left (3 \cosh ^{-1}(c x)\right )\right )}{-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 \cosh ^{-1}(c x)-\cosh \left (3 \cosh ^{-1}(c x)\right )\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}} \cosh ^{-1}(c x)+c x \sqrt {\frac {-1+c x}{1+c x}} \cosh ^{-1}(c x)+i \cosh ^{-1}(c x) \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-i \cosh ^{-1}(c x) \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+i \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-i \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(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}} \cosh ^{-1}(c x)}{1-c x}+\cosh ^{-1}(c x)^2+\frac {i \left (\cosh ^{-1}(c x)^2 \log \left (1-i e^{-\cosh ^{-1}(c x)}\right )-\cosh ^{-1}(c x)^2 \log \left (1+i e^{-\cosh ^{-1}(c x)}\right )+2 \cosh ^{-1}(c x) \text {PolyLog}\left (2,-i e^{-\cosh ^{-1}(c x)}\right )-2 \cosh ^{-1}(c x) \text {PolyLog}\left (2,i e^{-\cosh ^{-1}(c x)}\right )+2 \text {PolyLog}\left (3,-i e^{-\cosh ^{-1}(c x)}\right )-2 \text {PolyLog}\left (3,i e^{-\cosh ^{-1}(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 \left (3 \cosh ^{-1}(c x)\right )+9 \left (-50 c x+\cosh \left (5 \cosh ^{-1}(c x)\right )\right )+15 \cosh ^{-1}(c x) \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh \left (3 \cosh ^{-1}(c x)\right )-3 \sinh \left (5 \cosh ^{-1}(c x)\right )\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 \cosh ^{-1}(c x) \left (25 \cosh \left (3 \cosh ^{-1}(c x)\right )+9 \left (-50 c x+\cosh \left (5 \cosh ^{-1}(c x)\right )\right )\right )-250 \sinh \left (3 \cosh ^{-1}(c x)\right )+225 \cosh ^{-1}(c x)^2 \left (30 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)-5 \sinh \left (3 \cosh ^{-1}(c x)\right )-3 \sinh \left (5 \cosh ^{-1}(c x)\right )\right )-54 \sinh \left (5 \cosh ^{-1}(c x)\right )\right )}{54000 \sqrt {\frac {-1+c x}{1+c x}} (1+c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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