Integrand size = 29, antiderivative size = 630 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=-2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {3 c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2 \arctan \left (e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {b^2 c^2 d \sqrt {d-c^2 d x^2} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x)) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \]
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Time = 0.62 (sec) , antiderivative size = 630, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.517, Rules used = {5928, 5926, 5947, 4265, 2611, 2320, 6724, 5879, 75, 5912, 14, 5921, 471, 94, 211} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\frac {3 c^2 d \sqrt {d-c^2 d x^2} \arctan \left (e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))^2}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(c x)}\right ) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {c x-1} \sqrt {c x+1}}-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}-\frac {3 i b^2 c^2 d \sqrt {d-c^2 d x^2} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(c x)}\right )}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {3 b^2 c^3 d x \text {arccosh}(c x) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {b^2 c^2 d \arctan \left (\sqrt {c x-1} \sqrt {c x+1}\right ) \sqrt {d-c^2 d x^2}}{\sqrt {c x-1} \sqrt {c x+1}}-2 b^2 c^2 d \sqrt {d-c^2 d x^2} \]
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Rule 14
Rule 75
Rule 94
Rule 211
Rule 471
Rule 2320
Rule 2611
Rule 4265
Rule 5879
Rule 5912
Rule 5921
Rule 5926
Rule 5928
Rule 5947
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}-\frac {1}{2} \left (3 c^2 d\right ) \int \frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2}{x} \, dx-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x) (1+c x) (a+b \text {arccosh}(c x))}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}-\frac {\left (b c d \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (-1+c^2 x^2\right ) (a+b \text {arccosh}(c x))}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 d \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}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b c^3 d \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 {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {\left (3 c^2 d \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 )}{2 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1+c^2 x^2}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \int \text {arccosh}(c x) \, dx}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {3 c^2 d \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 (3 i b c^2 d \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 (3 i b c^2 d \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 (b^2 c^2 d \sqrt {d-c^2 d x^2}\right ) \int \frac {1}{x \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 b^2 c^4 d \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}} \\ & = -2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {3 c^2 d \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 {3 i b c^2 d \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 {3 i b c^2 d \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 (3 i b^2 c^2 d \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 (3 i b^2 c^2 d \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 (b^2 c^3 d \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1}{c+c x^2} \, dx,x,\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \\ & = -2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {3 c^2 d \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 {b^2 c^2 d \sqrt {d-c^2 d x^2} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \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 {3 i b c^2 d \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 (3 i b^2 c^2 d \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 (3 i b^2 c^2 d \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}} \\ & = -2 b^2 c^2 d \sqrt {d-c^2 d x^2}+\frac {3 a b c^3 d x \sqrt {d-c^2 d x^2}}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b^2 c^3 d x \sqrt {d-c^2 d x^2} \text {arccosh}(c x)}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b c^3 d x \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3}{2} c^2 d \sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))^2-\frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{2 x^2}+\frac {3 c^2 d \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 {b^2 c^2 d \sqrt {d-c^2 d x^2} \arctan \left (\sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 i b c^2 d \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 {3 i b c^2 d \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 {3 i b^2 c^2 d \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 {3 i b^2 c^2 d \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*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(5444\) vs. \(2(630)=1260\).
Time = 63.65 (sec) , antiderivative size = 5444, normalized size of antiderivative = 8.64 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Result too large to show} \]
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\[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )^{2}}{x^{3}}d x\]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \]
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\[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} (a+b \text {arccosh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x^3} \,d x \]
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