Optimal. Leaf size=890 \[ -\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^5}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{27} b^2 d^2 x^2 \sqrt {d-c^2 d x^2} c^4+\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 a b d^2 x \sqrt {d-c^2 d x^2} c^3}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{6} d \left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2-\frac {5}{2} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 d^2 \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right ) c^2}{(1-c x) (c x+1)}-\frac {5 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {Li}_2\left (i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {Li}_3\left (i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {170}{27} b^2 d^2 \sqrt {d-c^2 d x^2} c^2+\frac {b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} c^2}{9 (1-c x) (c x+1)}+\frac {5 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} c^2}{3 (1-c x) (c x+1)}-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.99, antiderivative size = 921, normalized size of antiderivative = 1.03, number of steps used = 27, number of rules used = 21, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.724, Rules used = {5798, 5740, 5745, 5743, 5761, 4180, 2531, 2282, 6589, 5654, 74, 5680, 12, 460, 270, 5731, 520, 1251, 897, 1153, 205} \[ -\frac {2 b d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^5}{9 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5}{27} b^2 d^2 x^2 \sqrt {d-c^2 d x^2} c^4+\frac {5 b^2 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x) c^3}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c^3}{3 \sqrt {c x-1} \sqrt {c x+1}}+\frac {5 a b d^2 x \sqrt {d-c^2 d x^2} c^3}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5}{2} d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2-\frac {5}{6} d^2 (1-c x) (c x+1) \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 c^2+\frac {5 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {b^2 d^2 \sqrt {c^2 x^2-1} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {c^2 x^2-1}\right ) c^2}{(1-c x) (c x+1)}-\frac {5 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {5 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,-i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {5 i b^2 d^2 \sqrt {d-c^2 d x^2} \text {PolyLog}\left (3,i e^{\cosh ^{-1}(c x)}\right ) c^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {170}{27} b^2 d^2 \sqrt {d-c^2 d x^2} c^2+\frac {b^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2} c^2}{9 (1-c x) (c x+1)}+\frac {5 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2} c^2}{3 (1-c x) (c x+1)}-\frac {b d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right ) c}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {d^2 (1-c x)^2 (c x+1)^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 74
Rule 205
Rule 270
Rule 460
Rule 520
Rule 897
Rule 1153
Rule 1251
Rule 2282
Rule 2531
Rule 4180
Rule 5654
Rule 5680
Rule 5731
Rule 5740
Rule 5743
Rule 5745
Rule 5761
Rule 5798
Rule 6589
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^3} \, 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^3} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {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}{2 x^2}+\frac {\left (b c d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 c^2 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}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{6} c^2 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 {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}{2 x^2}-\frac {\left (5 c^2 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}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{3 x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (5 b c^3 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 {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {\left (5 c^2 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}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{3 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b c^3 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 (5 b^2 c^4 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 {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (5 b^2 c^3 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 (5 b^2 c^4 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 (b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \int \frac {-3-6 c^2 x^2+c^4 x^4}{x \sqrt {-1+c^2 x^2}} \, dx}{3 (-1+c x) (1+c x)}\\ &=\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {5 c^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 (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 (5 i b c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 (35 b^2 c^4 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 (5 b^2 c^4 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 ) \operatorname {Subst}\left (\int \frac {-3-6 c^2 x+c^4 x^2}{x \sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{6 (-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {5 c^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 {5 i b c^2 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 {5 i b c^2 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 (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {-8-4 x^2+x^4}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 (-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {5 c^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 {5 i b c^2 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 {5 i b c^2 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 (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 (5 i b^2 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \operatorname {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 (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (-5 c^2+c^2 x^2-\frac {3}{\frac {1}{c^2}+\frac {x^2}{c^2}}\right ) \, dx,x,\sqrt {-1+c^2 x^2}\right )}{3 (-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {5 c^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 {5 i b c^2 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 {5 i b c^2 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 {5 i b^2 c^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 {5 i b^2 c^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 {\left (b^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {-1+c^2 x^2}\right )}{(-1+c x) (1+c x)}\\ &=-\frac {170}{27} b^2 c^2 d^2 \sqrt {d-c^2 d x^2}+\frac {5}{27} b^2 c^4 d^2 x^2 \sqrt {d-c^2 d x^2}+\frac {5 a b c^3 d^2 x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 b^2 c^2 d^2 \left (1-c^2 x^2\right ) \sqrt {d-c^2 d x^2}}{3 (1-c x) (1+c x)}+\frac {b^2 c^2 d^2 \left (1-c^2 x^2\right )^2 \sqrt {d-c^2 d x^2}}{9 (1-c x) (1+c x)}+\frac {5 b^2 c^3 d^2 x \sqrt {d-c^2 d x^2} \cosh ^{-1}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d^2 x \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{3 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 b c^5 d^2 x^3 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{9 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5}{2} c^2 d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2-\frac {5}{6} c^2 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 {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}{2 x^2}+\frac {5 c^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 {b^2 c^2 d^2 \sqrt {-1+c^2 x^2} \sqrt {d-c^2 d x^2} \tan ^{-1}\left (\sqrt {-1+c^2 x^2}\right )}{(1-c x) (1+c x)}-\frac {5 i b c^2 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 {5 i b c^2 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 {5 i b^2 c^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 {5 i b^2 c^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 = 90.31, size = 1384, normalized size = 1.56 \[ \text {result too large to display} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{4} d^{2} x^{4} - 2 \, a^{2} c^{2} d^{2} x^{2} + a^{2} d^{2} + {\left (b^{2} c^{4} d^{2} x^{4} - 2 \, b^{2} c^{2} d^{2} x^{2} + b^{2} d^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b c^{4} d^{2} x^{4} - 2 \, a b c^{2} d^{2} x^{2} + a b d^{2}\right )} \operatorname {arcosh}\left (c x\right )\right )} \sqrt {-c^{2} d x^{2} + d}}{x^{3}}, x\right ) \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.01, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}{x^{3}}\, dx \]
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 \[ \frac {1}{6} \, {\left (15 \, c^{2} d^{\frac {5}{2}} \log \left (\frac {2 \, \sqrt {-c^{2} d x^{2} + d} \sqrt {d}}{{\left | x \right |}} + \frac {2 \, d}{{\left | x \right |}}\right ) - 3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} c^{2} - 5 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} c^{2} d - 15 \, \sqrt {-c^{2} d x^{2} + d} c^{2} d^{2} - \frac {3 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {7}{2}}}{d x^{2}}\right )} a^{2} + \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} b^{2} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )^{2}}{x^{3}} + \frac {2 \, {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} a b \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}{x^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^3} \,d x \]
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 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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