Optimal. Leaf size=168 \[ -\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (c^2 x^2+1\right )}-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (c^2 x^2+1\right )}-\frac {3 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {b c}{2 d^2 \sqrt {c^2 x^2+1}}-\frac {b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{d^2}+\frac {3 i b c \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 i b c \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{2 d^2} \]
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Rubi [A] time = 0.19, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5747, 5690, 5693, 4180, 2279, 2391, 261, 266, 51, 63, 208} \[ \frac {3 i b c \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 i b c \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (c^2 x^2+1\right )}-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (c^2 x^2+1\right )}-\frac {3 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{d^2}-\frac {b c}{2 d^2 \sqrt {c^2 x^2+1}}-\frac {b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{d^2} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 208
Rule 261
Rule 266
Rule 2279
Rule 2391
Rule 4180
Rule 5690
Rule 5693
Rule 5747
Rubi steps
\begin {align*} \int \frac {a+b \sinh ^{-1}(c x)}{x^2 \left (d+c^2 d x^2\right )^2} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\left (3 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )^{3/2}} \, dx}{d^2}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{3/2}} \, dx,x,x^2\right )}{2 d^2}+\frac {\left (3 b c^3\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{2 d^2}-\frac {\left (3 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{2 d}\\ &=-\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {(3 c) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}+\frac {b \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{c^2}+\frac {x^2}{c^2}} \, dx,x,\sqrt {1+c^2 x^2}\right )}{c d^2}+\frac {(3 i b c) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2}-\frac {(3 i b c) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{2 d^2}\\ &=-\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac {b c \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^2}+\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {(3 i b c) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{2 d^2}\\ &=-\frac {b c}{2 d^2 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^2 x \left (1+c^2 x^2\right )}-\frac {3 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{2 d^2 \left (1+c^2 x^2\right )}-\frac {3 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{d^2}-\frac {b c \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^2}+\frac {3 i b c \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{2 d^2}-\frac {3 i b c \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{2 d^2}\\ \end {align*}
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Mathematica [C] time = 0.55, size = 253, normalized size = 1.51 \[ -\frac {-\frac {a}{c^2 x^3+x}+3 a c \tan ^{-1}(c x)+\frac {3 a}{x}+\frac {b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};c^2 x^2+1\right )}{\sqrt {c^2 x^2+1}}-3 b \sqrt {-c^2} \text {Li}_2\left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+3 b \sqrt {-c^2} \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )-\frac {b \sinh ^{-1}(c x)}{c^2 x^3+x}+3 b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )+3 b \sqrt {-c^2} \sinh ^{-1}(c x) \log \left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}+1\right )-3 b \sqrt {-c^2} \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )+\frac {3 b \sinh ^{-1}(c x)}{x}}{2 d^2} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (c x\right ) + a}{c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{2} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 267, normalized size = 1.59 \[ -\frac {a}{d^{2} x}-\frac {a \,c^{2} x}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {3 c a \arctan \left (c x \right )}{2 d^{2}}-\frac {b \arcsinh \left (c x \right )}{d^{2} x}-\frac {b \arcsinh \left (c x \right ) c^{2} x}{2 d^{2} \left (c^{2} x^{2}+1\right )}-\frac {3 c b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{2 d^{2}}-\frac {3 c b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}+\frac {3 c b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}+\frac {3 i c b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}-\frac {3 i c b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{2 d^{2}}-\frac {b c}{2 d^{2} \sqrt {c^{2} x^{2}+1}}-\frac {c b \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{d^{2}} \]
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 \[ -\frac {1}{2} \, a {\left (\frac {3 \, c^{2} x^{2} + 2}{c^{2} d^{2} x^{3} + d^{2} x} + \frac {3 \, c \arctan \left (c x\right )}{d^{2}}\right )} + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}\,{d x} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (d\,c^2\,x^2+d\right )}^2} \,d x \]
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 \[ \frac {\int \frac {a}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{4} x^{6} + 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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