Optimal. Leaf size=156 \[ \frac {2 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}-\frac {i b c^3 \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {i b c^3 \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {b c \sqrt {c^2 x^2+1}}{6 d x^2}+\frac {7 b c^3 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{6 d} \]
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Rubi [A] time = 0.25, antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5747, 5693, 4180, 2279, 2391, 266, 63, 208, 51} \[ -\frac {i b c^3 \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {i b c^3 \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {2 c^3 \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d}-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}-\frac {b c \sqrt {c^2 x^2+1}}{6 d x^2}+\frac {7 b c^3 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{6 d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 266
Rule 2279
Rule 2391
Rule 4180
Rule 5693
Rule 5747
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^4 \left (d+c^2 d x^2\right )} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}-c^2 \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )} \, dx+\frac {(b c) \int \frac {1}{x^3 \sqrt {1+c^2 x^2}} \, dx}{3 d}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+c^4 \int \frac {a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{6 d}-\frac {\left (b c^3\right ) \int \frac {1}{x \sqrt {1+c^2 x^2}} \, dx}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 d x^2}-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {c^3 \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{d}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{12 d}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 d x^2}-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{6 d}-\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{d}-\frac {\left (i b c^3\right ) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}+\frac {\left (i b c^3\right ) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 d x^2}-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {7 b c^3 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{6 d}-\frac {\left (i b c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {\left (i b c^3\right ) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{d}\\ &=-\frac {b c \sqrt {1+c^2 x^2}}{6 d x^2}-\frac {a+b \sinh ^{-1}(c x)}{3 d x^3}+\frac {c^2 \left (a+b \sinh ^{-1}(c x)\right )}{d x}+\frac {2 c^3 \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {7 b c^3 \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{6 d}-\frac {i b c^3 \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{d}+\frac {i b c^3 \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 247, normalized size = 1.58 \[ \frac {6 a c^3 x^3 \tan ^{-1}(c x)+6 a c^2 x^2-2 a+6 b \left (-c^2\right )^{3/2} x^3 \text {Li}_2\left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )-6 b \left (-c^2\right )^{3/2} x^3 \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-6 b \left (-c^2\right )^{3/2} x^3 \sinh ^{-1}(c x) \log \left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}+1\right )+6 b \left (-c^2\right )^{3/2} x^3 \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )-b c x \sqrt {c^2 x^2+1}+6 b c^2 x^2 \sinh ^{-1}(c x)+7 b c^3 x^3 \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )-2 b \sinh ^{-1}(c x)}{6 d x^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (c x\right ) + a}{c^{2} d x^{6} + d x^{4}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 261, normalized size = 1.67 \[ -\frac {a}{3 d \,x^{3}}+\frac {c^{2} a}{d x}+\frac {c^{3} a \arctan \left (c x \right )}{d}-\frac {b \arcsinh \left (c x \right )}{3 d \,x^{3}}+\frac {c^{2} b \arcsinh \left (c x \right )}{d x}+\frac {c^{3} b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{d}-\frac {b c \sqrt {c^{2} x^{2}+1}}{6 d \,x^{2}}+\frac {7 c^{3} b \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6 d}+\frac {c^{3} b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {c^{3} b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}-\frac {i c^{3} b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d}+\frac {i c^{3} b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{d} \]
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 \[ \frac {1}{3} \, {\left (\frac {3 \, c^{3} \arctan \left (c x\right )}{d} + \frac {3 \, c^{2} x^{2} - 1}{d x^{3}}\right )} a + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{2} d x^{6} + d x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,\left (d\,c^2\,x^2+d\right )} \,d x \]
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 \[ \frac {\int \frac {a}{c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{2} x^{6} + x^{4}}\, dx}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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