Optimal. Leaf size=222 \[ -\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (c^2 x^2+1\right )}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}-\frac {15 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}-\frac {7 b c}{8 d^3 \sqrt {c^2 x^2+1}}-\frac {b c}{12 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac {b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{d^3}+\frac {15 i b c \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 i b c \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{8 d^3} \]
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Rubi [A] time = 0.24, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 16, 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} \[ \frac {15 i b c \text {PolyLog}\left (2,-i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 i b c \text {PolyLog}\left (2,i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (c^2 x^2+1\right )}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (c^2 x^2+1\right )^2}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (c^2 x^2+1\right )^2}-\frac {15 c \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right ) \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3}-\frac {7 b c}{8 d^3 \sqrt {c^2 x^2+1}}-\frac {b c}{12 d^3 \left (c^2 x^2+1\right )^{3/2}}-\frac {b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )}{d^3} \]
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 )^3} \, dx &=-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\left (5 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )^{5/2}} \, dx}{d^3}\\ &=-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )^{5/2}} \, dx,x,x^2\right )}{2 d^3}+\frac {\left (5 b c^3\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{5/2}} \, dx}{4 d^3}-\frac {\left (15 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\left (d+c^2 d x^2\right )^2} \, dx}{4 d}\\ &=-\frac {b c}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \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^3}+\frac {\left (15 b c^3\right ) \int \frac {x}{\left (1+c^2 x^2\right )^{3/2}} \, dx}{8 d^3}-\frac {\left (15 c^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{d+c^2 d x^2} \, dx}{8 d^2}\\ &=-\frac {b c}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {(15 c) \operatorname {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\sinh ^{-1}(c x)\right )}{8 d^3}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2 d^3}\\ &=-\frac {b c}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {15 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}+\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^3}+\frac {(15 i b c) \operatorname {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{8 d^3}-\frac {(15 i b c) \operatorname {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{8 d^3}\\ &=-\frac {b c}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {15 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac {b c \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^3}+\frac {(15 i b c) \operatorname {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac {(15 i b c) \operatorname {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\sinh ^{-1}(c x)}\right )}{8 d^3}\\ &=-\frac {b c}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {7 b c}{8 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \sinh ^{-1}(c x)}{d^3 x \left (1+c^2 x^2\right )^2}-\frac {5 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{4 d^3 \left (1+c^2 x^2\right )^2}-\frac {15 c^2 x \left (a+b \sinh ^{-1}(c x)\right )}{8 d^3 \left (1+c^2 x^2\right )}-\frac {15 c \left (a+b \sinh ^{-1}(c x)\right ) \tan ^{-1}\left (e^{\sinh ^{-1}(c x)}\right )}{4 d^3}-\frac {b c \tanh ^{-1}\left (\sqrt {1+c^2 x^2}\right )}{d^3}+\frac {15 i b c \text {Li}_2\left (-i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}-\frac {15 i b c \text {Li}_2\left (i e^{\sinh ^{-1}(c x)}\right )}{8 d^3}\\ \end {align*}
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Mathematica [C] time = 1.05, size = 298, normalized size = 1.34 \[ -\frac {-\frac {15 \left (a+b \sinh ^{-1}(c x)\right )}{c^2 x^3+x}-\frac {6 \left (a+b \sinh ^{-1}(c x)\right )}{x \left (c^2 x^2+1\right )^2}+\frac {45 \left (a+b \sinh ^{-1}(c x)\right )}{x}+45 a c \tan ^{-1}(c x)+\frac {2 b c \, _2F_1\left (-\frac {3}{2},1;-\frac {1}{2};c^2 x^2+1\right )}{\left (c^2 x^2+1\right )^{3/2}}+\frac {15 b c \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};c^2 x^2+1\right )}{\sqrt {c^2 x^2+1}}-45 b \sqrt {-c^2} \text {Li}_2\left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}\right )+45 b \sqrt {-c^2} \text {Li}_2\left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}\right )+45 b c \tanh ^{-1}\left (\sqrt {c^2 x^2+1}\right )+45 b \sqrt {-c^2} \sinh ^{-1}(c x) \log \left (\frac {c e^{\sinh ^{-1}(c x)}}{\sqrt {-c^2}}+1\right )-45 b \sqrt {-c^2} \sinh ^{-1}(c x) \log \left (\frac {\sqrt {-c^2} e^{\sinh ^{-1}(c x)}}{c}+1\right )}{24 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arsinh}\left (c x\right ) + a}{c^{6} d^{3} x^{8} + 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} + d^{3} x^{2}}, x\right ) \]
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 \[ \int \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 357, normalized size = 1.61 \[ -\frac {a}{d^{3} x}-\frac {7 a \,c^{4} x^{3}}{8 d^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 a \,c^{2} x}{8 d^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {15 c a \arctan \left (c x \right )}{8 d^{3}}-\frac {b \arcsinh \left (c x \right )}{d^{3} x}-\frac {7 b \arcsinh \left (c x \right ) c^{4} x^{3}}{8 d^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {9 b \arcsinh \left (c x \right ) c^{2} x}{8 d^{3} \left (c^{2} x^{2}+1\right )^{2}}-\frac {15 c b \arcsinh \left (c x \right ) \arctan \left (c x \right )}{8 d^{3}}-\frac {15 c b \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 d^{3}}+\frac {15 c b \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 d^{3}}+\frac {15 i c b \dilog \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 d^{3}}-\frac {15 i c b \dilog \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8 d^{3}}-\frac {15 b \,c^{3} x^{2}}{8 d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {47 b c}{24 d^{3} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {b c}{d^{3} \sqrt {c^{2} x^{2}+1}}-\frac {c b \arctanh \left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{d^{3}} \]
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 \[ -\frac {1}{8} \, a {\left (\frac {15 \, c^{4} x^{4} + 25 \, c^{2} x^{2} + 8}{c^{4} d^{3} x^{5} + 2 \, c^{2} d^{3} x^{3} + d^{3} x} + \frac {15 \, c \arctan \left (c x\right )}{d^{3}}\right )} + b \int \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )}{c^{6} d^{3} x^{8} + 3 \, c^{4} d^{3} x^{6} + 3 \, c^{2} d^{3} x^{4} + d^{3} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^2\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]
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 \[ \frac {\int \frac {a}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{8} + 3 c^{4} x^{6} + 3 c^{2} x^{4} + x^{2}}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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