Optimal. Leaf size=452 \[ \frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {c^2 d x^2+d}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {c^2 d x^2+d}}+\frac {8 c^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {c^2 d x^2+d}}-\frac {16 b c^3 \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt {c^2 d x^2+d}}+\frac {20 b c^3 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt {c^2 d x^2+d}}-\frac {b^2 c^3 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {5 b^2 c^3 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt {c^2 d x^2+d}} \]
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Rubi [A] time = 0.84, antiderivative size = 452, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {5747, 5687, 5714, 3718, 2190, 2279, 2391, 5720, 5461, 4182, 264} \[ -\frac {b^2 c^3 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt {c^2 d x^2+d}}-\frac {5 b^2 c^3 \sqrt {c^2 x^2+1} \text {PolyLog}\left (2,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt {c^2 d x^2+d}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {c^2 d x^2+d}}+\frac {8 c^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {c^2 d x^2+d}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {c^2 d x^2+d}}-\frac {b c \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {c^2 d x^2+d}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {c^2 d x^2+d}}-\frac {16 b c^3 \sqrt {c^2 x^2+1} \log \left (e^{2 \sinh ^{-1}(c x)}+1\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt {c^2 d x^2+d}}+\frac {20 b c^3 \sqrt {c^2 x^2+1} \tanh ^{-1}\left (e^{2 \sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{3 d \sqrt {c^2 d x^2+d}}-\frac {b^2 c^2 \left (c^2 x^2+1\right )}{3 d x \sqrt {c^2 d x^2+d}} \]
Antiderivative was successfully verified.
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Rule 264
Rule 2190
Rule 2279
Rule 2391
Rule 3718
Rule 4182
Rule 5461
Rule 5687
Rule 5714
Rule 5720
Rule 5747
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 )^{3/2}} \, dx &=-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}-\frac {1}{3} \left (4 c^2\right ) \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{x^2 \left (d+c^2 d x^2\right )^{3/2}} \, 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 )} \, dx}{3 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {1}{3} \left (8 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+\frac {\left (b^2 c^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{x^2 \sqrt {1+c^2 x^2}} \, dx}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 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 \sqrt {d+c^2 d x^2}}-\frac {\left (8 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 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (2 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 \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}(x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \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 )}{1+c^2 x^2} \, dx}{3 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \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}(2 x) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {\left (16 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 \sqrt {d+c^2 d x^2}}-\frac {\left (16 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 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 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 \sqrt {d+c^2 d x^2}}-\frac {\left (32 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 \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 \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \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 \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (8 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 \sqrt {d+c^2 d x^2}}-\frac {\left (8 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 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 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 \sqrt {d+c^2 d x^2}}-\frac {16 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 \sqrt {d+c^2 d x^2}}+\frac {\left (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 \sqrt {d+c^2 d x^2}}-\frac {\left (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 \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 \frac {\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \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 \frac {\log (1+x)}{x} \, dx,x,e^{2 \sinh ^{-1}(c x)}\right )}{3 d \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 \log \left (1+e^{2 x}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{3 d \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 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 \sqrt {d+c^2 d x^2}}-\frac {16 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 \sqrt {d+c^2 d x^2}}+\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}+\frac {\left (8 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 \sqrt {d+c^2 d x^2}}\\ &=-\frac {b^2 c^2 \left (1+c^2 x^2\right )}{3 d x \sqrt {d+c^2 d x^2}}-\frac {b c \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 d x^2 \sqrt {d+c^2 d x^2}}-\frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x^3 \sqrt {d+c^2 d x^2}}+\frac {4 c^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d x \sqrt {d+c^2 d x^2}}+\frac {8 c^4 x \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {8 c^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{3 d \sqrt {d+c^2 d x^2}}+\frac {20 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 \sqrt {d+c^2 d x^2}}-\frac {16 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 \sqrt {d+c^2 d x^2}}-\frac {b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (-e^{2 \sinh ^{-1}(c x)}\right )}{d \sqrt {d+c^2 d x^2}}-\frac {5 b^2 c^3 \sqrt {1+c^2 x^2} \text {Li}_2\left (e^{2 \sinh ^{-1}(c x)}\right )}{3 d \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 438, normalized size = 0.97 \[ \frac {8 a^2 c^4 x^4+4 a^2 c^2 x^2-a^2+16 a b c^4 x^4 \sinh ^{-1}(c x)-a b c x \sqrt {c^2 x^2+1}+8 a b c^2 x^2 \sinh ^{-1}(c x)-10 a b c^3 x^3 \sqrt {c^2 x^2+1} \log (c x)-3 a b c^3 x^3 \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )-2 a b \sinh ^{-1}(c x)-b^2 c^4 x^4+8 b^2 c^4 x^4 \sinh ^{-1}(c x)^2-b^2 c^2 x^2+4 b^2 c^2 x^2 \sinh ^{-1}(c x)^2-b^2 c x \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)+3 b^2 c^3 x^3 \sqrt {c^2 x^2+1} \text {Li}_2\left (-e^{-2 \sinh ^{-1}(c x)}\right )+5 b^2 c^3 x^3 \sqrt {c^2 x^2+1} \text {Li}_2\left (e^{-2 \sinh ^{-1}(c x)}\right )-8 b^2 c^3 x^3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x)^2-10 b^2 c^3 x^3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (1-e^{-2 \sinh ^{-1}(c x)}\right )-6 b^2 c^3 x^3 \sqrt {c^2 x^2+1} \sinh ^{-1}(c x) \log \left (e^{-2 \sinh ^{-1}(c x)}+1\right )-b^2 \sinh ^{-1}(c x)^2}{3 d x^3 \sqrt {c^2 d x^2+d}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {c^{2} d x^{2} + d} {\left (b^{2} \operatorname {arsinh}\left (c x\right )^{2} + 2 \, a b \operatorname {arsinh}\left (c x\right ) + a^{2}\right )}}{c^{4} d^{2} x^{8} + 2 \, c^{2} d^{2} x^{6} + d^{2} x^{4}}, x\right ) \]
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 \[ \int \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.50, size = 2609, normalized size = 5.77 \[ \text {result too large to display} \]
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 \[ \frac {1}{3} \, {\left (\frac {8 \, c^{4} x}{\sqrt {c^{2} d x^{2} + d} d} + \frac {4 \, c^{2}}{\sqrt {c^{2} d x^{2} + d} d x} - \frac {1}{\sqrt {c^{2} d x^{2} + d} d x^{3}}\right )} a^{2} + \int \frac {b^{2} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}} + \frac {2 \, a b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{x^4\,{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{4} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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