Optimal. Leaf size=506 \[ -\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (c^2 d x^2+d\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {16 c^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {32 b c^3 \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {32 b c^3 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2}{3 d^2 x \sqrt {c^2 d x^2+d}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {8 b^2 c^3 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {8 b^2 c^3 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 1.09, antiderivative size = 506, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {5747, 5690, 5687, 5714, 3718, 2190, 2279, 2391, 5717, 191, 5755, 5720, 5461, 4182, 271} \[ -\frac {8 b^2 c^3 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {8 b^2 c^3 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {16 c^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}}-\frac {32 b c^3 \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {32 b c^3 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (c^2 d x^2+d\right )^{3/2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (c^2 d x^2+d\right )^{3/2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2}{3 d^2 x \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 271
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 4182
Rule 5461
Rule 5687
Rule 5690
Rule 5714
Rule 5717
Rule 5720
Rule 5747
Rule 5755
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^4 \left (d+c^2 d x^2\right )^{5/2}} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}-\left (2 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (2 b c \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x^3 \left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\left (8 c^4\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x^2 \left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )^2} \, dx}{d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {8 b c^3 \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\left (16 c^4\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx}{3 d}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \sinh ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 c^4 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (16 b c^5 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\left (1+c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}+\frac {2 b^2 c^4 x}{d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b^2 c^4 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (32 b c^5 \sqrt {1+c^2 x^2}\right ) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (8 b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \text {csch}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (32 b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \tanh (x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {16 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (64 b c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} (a+b x)}{1+e^{2 x}} \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (4 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (4 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {16 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (2 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}-\frac {\left (2 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (32 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {16 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {\left (16 b^2 c^3 \sqrt {1+c^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {16 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 3.05, size = 417, normalized size = 0.82 \[ \frac {\frac {a^2 \left (16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right )}{x^3}-\frac {a b \left (c x \sqrt {c^2 x^2+1} \left (16 \left (c^4 x^4+c^2 x^2\right ) \log (c x)+8 \left (c^4 x^4+c^2 x^2\right ) \log \left (c^2 x^2+1\right )+1\right )-2 \left (16 c^6 x^6+24 c^4 x^4+6 c^2 x^2-1\right ) \sinh ^{-1}(c x)\right )}{x^3}+b^2 c^3 \left (c^2 x^2+1\right )^{3/2} \left (-\frac {\sqrt {c^2 x^2+1}}{c x}-\frac {c x}{\sqrt {c^2 x^2+1}}+\frac {8 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2}{c x}+\frac {8 c x \sinh ^{-1}(c x)^2}{\sqrt {c^2 x^2+1}}+\frac {c x \sinh ^{-1}(c x)^2}{\left (c^2 x^2+1\right )^{3/2}}+\frac {\sinh ^{-1}(c x)}{c^2 x^2+1}-\frac {\sinh ^{-1}(c x)}{c^2 x^2}-\frac {\sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2}{c^3 x^3}+8 \text {Li}_2\left (-e^{-2 \sinh ^{-1}(c x)}\right )+8 \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-16 \sinh ^{-1}(c x)^2-16 \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-16 \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )\right )}{3 d \left (c^2 d x^2+d\right )^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{c^{6} d^{3} x^{10} + 3 \, c^{4} d^{3} x^{8} + 3 \, c^{2} d^{3} x^{6} + d^{3} x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.53, size = 4955, normalized size = 9.79 \[ \text {output too large to display} \]
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 \[ -\frac {1}{3} \, a b c {\left (\frac {8 \, c^{2} \log \left (c^{2} x^{2} + 1\right )}{d^{\frac {5}{2}}} + \frac {16 \, c^{2} \log \relax (x)}{d^{\frac {5}{2}}} + \frac {1}{c^{2} d^{\frac {5}{2}} x^{4} + d^{\frac {5}{2}} x^{2}}\right )} + \frac {2}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} + \frac {6 \, c^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a b \operatorname {arsinh}\left (c x\right ) + \frac {1}{3} \, {\left (\frac {16 \, c^{4} x}{\sqrt {c^{2} d x^{2} + d} d^{2}} + \frac {8 \, c^{4} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d} + \frac {6 \, c^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x} - \frac {1}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} d x^{3}}\right )} a^{2} + b^{2} \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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