Optimal. Leaf size=506 \[ -\frac {b^2 c^2}{3 d^2 x \sqrt {d+c^2 d x^2}}-\frac {2 b^2 c^4 x}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {b c \left (a+b \sinh ^{-1}(c x)\right )}{3 d^2 x^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \left (d+c^2 d x^2\right )^{3/2}}+\frac {2 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{d x \left (d+c^2 d x^2\right )^{3/2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {16 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {16 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d^2 \sqrt {d+c^2 d x^2}}+\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {32 b c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \log \left (1+e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}}-\frac {8 b^2 c^3 \sqrt {1+c^2 x^2} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d^2 \sqrt {d+c^2 d x^2}} \]
[Out]
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Rubi [A]
time = 0.75, 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 =
{5809, 5788, 5787, 5797, 3799, 2221, 2317, 2438, 5798, 197, 5811, 5799, 5569, 4267, 277}
\begin {gather*} -\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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 197
Rule 277
Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 4267
Rule 5569
Rule 5787
Rule 5788
Rule 5797
Rule 5798
Rule 5799
Rule 5809
Rule 5811
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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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 = 2.09, size = 417, normalized size = 0.82 \begin {gather*} \frac {\frac {a^2 \left (-1+6 c^2 x^2+24 c^4 x^4+16 c^6 x^6\right )}{x^3}-\frac {a b \left (-2 \left (-1+6 c^2 x^2+24 c^4 x^4+16 c^6 x^6\right ) \sinh ^{-1}(c x)+c x \sqrt {1+c^2 x^2} \left (1+16 \left (c^2 x^2+c^4 x^4\right ) \log (c x)+8 \left (c^2 x^2+c^4 x^4\right ) \log \left (1+c^2 x^2\right )\right )\right )}{x^3}+b^2 c^3 \left (1+c^2 x^2\right )^{3/2} \left (-\frac {c x}{\sqrt {1+c^2 x^2}}-\frac {\sqrt {1+c^2 x^2}}{c x}-\frac {\sinh ^{-1}(c x)}{c^2 x^2}+\frac {\sinh ^{-1}(c x)}{1+c^2 x^2}-16 \sinh ^{-1}(c x)^2+\frac {c x \sinh ^{-1}(c x)^2}{\left (1+c^2 x^2\right )^{3/2}}+\frac {8 c x \sinh ^{-1}(c x)^2}{\sqrt {1+c^2 x^2}}-\frac {\sqrt {1+c^2 x^2} \sinh ^{-1}(c x)^2}{c^3 x^3}+\frac {8 \sqrt {1+c^2 x^2} \sinh ^{-1}(c x)^2}{c x}-16 \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-16 \sinh ^{-1}(c x) \log \left (1+e^{-2 \sinh ^{-1}(c x)}\right )+8 \text {PolyLog}\left (2,-e^{-2 \sinh ^{-1}(c x)}\right )+8 \text {PolyLog}\left (2,e^{-2 \sinh ^{-1}(c x)}\right )\right )}{3 d \left (d+c^2 d x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4950\) vs.
\(2(484)=968\).
time = 5.00, size = 4951, normalized size = 9.78
method | result | size |
default | \(\text {Expression too large to display}\) | \(4951\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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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.
[In]
[Out]
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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.
[In]
[Out]
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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