Integrand size = 28, antiderivative size = 635 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\frac {598}{225} b^2 d^2 \sqrt {d+c^2 d x^2}-\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \]
[Out]
Time = 0.64 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {5808, 5806, 5816, 4267, 2611, 2320, 6724, 5772, 267, 5784, 455, 45, 200, 12, 1261, 712} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=-\frac {2 d^2 \sqrt {c^2 d x^2+d} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}-\frac {2 b d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {2 b d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}-\frac {16 b c d^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{15 \sqrt {c^2 x^2+1}}+d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c^5 d^2 x^5 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{25 \sqrt {c^2 x^2+1}}-\frac {22 b c^3 d^2 x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{45 \sqrt {c^2 x^2+1}}-\frac {2 a b c d^2 x \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {2 b^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 b^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {c^2 x^2+1}}-\frac {2 b^2 c d^2 x \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{\sqrt {c^2 x^2+1}}+\frac {598}{225} b^2 d^2 \sqrt {c^2 d x^2+d}+\frac {2}{125} b^2 d^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}+\frac {74}{675} b^2 d^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} \]
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Rule 12
Rule 45
Rule 200
Rule 267
Rule 455
Rule 712
Rule 1261
Rule 2320
Rule 2611
Rule 4267
Rule 5772
Rule 5784
Rule 5806
Rule 5808
Rule 5816
Rule 6724
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2+d \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(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 {arcsinh}(c x)) \, dx}{5 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 \sqrt {1+c^2 x^2}}-\frac {4 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2+d^2 \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(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 ) (a+b \text {arcsinh}(c x)) \, dx}{3 \sqrt {1+c^2 x^2}}+\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^2 x^2}} \, dx}{5 \sqrt {1+c^2 x^2}} \\ & = -\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b c d^2 \sqrt {d+c^2 d x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\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^2 x^2}} \, dx}{75 \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x \left (1+\frac {c^2 x^2}{3}\right )}{\sqrt {1+c^2 x^2}} \, dx}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {\left (d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \text {csch}(x) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 c d^2 \sqrt {d+c^2 d x^2}\right ) \int \text {arcsinh}(c x) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^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 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {1+\frac {c^2 x}{3}}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 a b c d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (2 b d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^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 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1+c^2 x}}+\frac {1}{3} \sqrt {1+c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {598}{225} b^2 d^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^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {598}{225} b^2 d^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^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (2 b^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \\ & = \frac {598}{225} b^2 d^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^2 x^2}}+\frac {74}{675} b^2 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}+\frac {2}{125} b^2 d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {2 b^2 c d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {16 b c d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{15 \sqrt {1+c^2 x^2}}-\frac {22 b c^3 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{3} d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{5} \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {2 b^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {2 b^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 3.56 (sec) , antiderivative size = 757, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(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 {4 a b d^2 \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{9 \sqrt {1+c^2 x^2}}+\frac {2 a b d^3 \sqrt {1+c^2 x^2} \left (30 c x-5 c^3 x^3-9 c^5 x^5+15 \sqrt {1+c^2 x^2} \left (-2+c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)\right )}{225 \sqrt {d+c^2 d x^2}}-\frac {b^2 d^3 \sqrt {1+c^2 x^2} \left (480 c x \left (-30+5 c^2 x^2+9 c^4 x^4\right ) \text {arcsinh}(c x)+6750 \sqrt {1+c^2 x^2} \left (2+\text {arcsinh}(c x)^2\right )+125 \left (2+9 \text {arcsinh}(c x)^2\right ) \cosh (3 \text {arcsinh}(c x))-27 \left (2+25 \text {arcsinh}(c x)^2\right ) \cosh (5 \text {arcsinh}(c x))\right )}{54000 \sqrt {d+c^2 d x^2}}+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^3 \sqrt {1+c^2 x^2} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {d+c^2 d x^2}}+\frac {b^2 d^3 \sqrt {1+c^2 x^2} \left (2 \sqrt {1+c^2 x^2}-2 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2+\text {arcsinh}(c x)^2 \left (\log \left (1-e^{-\text {arcsinh}(c x)}\right )-\log \left (1+e^{-\text {arcsinh}(c x)}\right )\right )+2 \text {arcsinh}(c x) \left (\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )+2 \left (\operatorname {PolyLog}\left (3,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (3,e^{-\text {arcsinh}(c x)}\right )\right )\right )}{\sqrt {d+c^2 d x^2}}+\frac {b^2 d^3 \sqrt {1+c^2 x^2} \left (27 \sqrt {1+c^2 x^2} \left (2+\text {arcsinh}(c x)^2\right )+\left (2+9 \text {arcsinh}(c x)^2\right ) \cosh (3 \text {arcsinh}(c x))-6 \text {arcsinh}(c x) (9 c x+\sinh (3 \text {arcsinh}(c x)))\right )}{54 \sqrt {d+c^2 d x^2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1320\) vs. \(2(614)=1228\).
Time = 0.32 (sec) , antiderivative size = 1321, normalized size of antiderivative = 2.08
method | result | size |
default | \(\text {Expression too large to display}\) | \(1321\) |
parts | \(\text {Expression too large to display}\) | \(1321\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x}\, dx \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x} \,d x \]
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