Optimal. Leaf size=477 \[ \frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}} \]
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Rubi [A] time = 0.85, antiderivative size = 477, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {6294, 5706, 5799, 5561, 2190, 2279, 2391} \[ -\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {PolyLog}\left (2,-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {PolyLog}\left (2,\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {e-c^2 d}}+1\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e-c^2 d}+\sqrt {e}}+1\right )}{2 \sqrt {-d} \sqrt {e}} \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2279
Rule 2391
Rule 5561
Rule 5706
Rule 5799
Rule 6294
Rubi steps
\begin {align*} \int \frac {a+b \text {csch}^{-1}(c x)}{d+e x^2} \, dx &=-\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{e+d x^2} \, dx,x,\frac {1}{x}\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}-\sqrt {-d} x\right )}+\frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{2 \sqrt {e} \left (\sqrt {e}+\sqrt {-d} x\right )}\right ) \, dx,x,\frac {1}{x}\right )\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}-\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {a+b \sinh ^{-1}\left (\frac {x}{c}\right )}{\sqrt {e}+\sqrt {-d} x} \, dx,x,\frac {1}{x}\right )}{2 \sqrt {e}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}-\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {(a+b x) \cosh (x)}{\frac {\sqrt {e}}{c}+\sqrt {-d} \sinh (x)} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {e}}\\ &=-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}-\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {e}}-\frac {\operatorname {Subst}\left (\int \frac {e^x (a+b x)}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}+\sqrt {-d} e^x} \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {e}}\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {Subst}\left (\int \log \left (1-\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int \log \left (1+\frac {\sqrt {-d} e^x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right ) \, dx,x,\text {csch}^{-1}(c x)\right )}{2 \sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}-\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt {-d} x}{\frac {\sqrt {e}}{c}+\frac {\sqrt {-c^2 d+e}}{c}}\right )}{x} \, dx,x,e^{\text {csch}^{-1}(c x)}\right )}{2 \sqrt {-d} \sqrt {e}}\\ &=\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\left (a+b \text {csch}^{-1}(c x)\right ) \log \left (1+\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}-\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {b \text {Li}_2\left (-\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {b \text {Li}_2\left (\frac {c \sqrt {-d} e^{\text {csch}^{-1}(c x)}}{\sqrt {e}+\sqrt {-c^2 d+e}}\right )}{2 \sqrt {-d} \sqrt {e}}\\ \end {align*}
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Mathematica [C] time = 0.51, size = 1055, normalized size = 2.21 \[ \frac {4 a \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )+8 i b \sin ^{-1}\left (\frac {\sqrt {\frac {\sqrt {e}}{c \sqrt {d}}+1}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {\left (c \sqrt {d}-\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (2 i \text {csch}^{-1}(c x)+\pi \right )\right )}{\sqrt {e-c^2 d}}\right )+8 i b \sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \tan ^{-1}\left (\frac {\left (\sqrt {d} c+\sqrt {e}\right ) \cot \left (\frac {1}{4} \left (2 i \text {csch}^{-1}(c x)+\pi \right )\right )}{\sqrt {e-c^2 d}}\right )+2 i b \text {csch}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e-c^2 d}-\sqrt {e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-4 b \sin ^{-1}\left (\frac {\sqrt {\frac {\sqrt {e}}{c \sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e-c^2 d}-\sqrt {e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-b \pi \log \left (1-\frac {i \left (\sqrt {e-c^2 d}-\sqrt {e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b \text {csch}^{-1}(c x) \log \left (\frac {i e^{\text {csch}^{-1}(c x)} \left (\sqrt {e-c^2 d}-\sqrt {e}\right )}{c \sqrt {d}}+1\right )+4 b \sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (\frac {i e^{\text {csch}^{-1}(c x)} \left (\sqrt {e-c^2 d}-\sqrt {e}\right )}{c \sqrt {d}}+1\right )+b \pi \log \left (\frac {i e^{\text {csch}^{-1}(c x)} \left (\sqrt {e-c^2 d}-\sqrt {e}\right )}{c \sqrt {d}}+1\right )-2 i b \text {csch}^{-1}(c x) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {e-c^2 d}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-4 b \sin ^{-1}\left (\frac {\sqrt {1-\frac {\sqrt {e}}{c \sqrt {d}}}}{\sqrt {2}}\right ) \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {e-c^2 d}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+b \pi \log \left (1-\frac {i \left (\sqrt {e}+\sqrt {e-c^2 d}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i b \text {csch}^{-1}(c x) \log \left (\frac {i e^{\text {csch}^{-1}(c x)} \left (\sqrt {e}+\sqrt {e-c^2 d}\right )}{c \sqrt {d}}+1\right )+4 b \sin ^{-1}\left (\frac {\sqrt {\frac {\sqrt {e}}{c \sqrt {d}}+1}}{\sqrt {2}}\right ) \log \left (\frac {i e^{\text {csch}^{-1}(c x)} \left (\sqrt {e}+\sqrt {e-c^2 d}\right )}{c \sqrt {d}}+1\right )-b \pi \log \left (\frac {i e^{\text {csch}^{-1}(c x)} \left (\sqrt {e}+\sqrt {e-c^2 d}\right )}{c \sqrt {d}}+1\right )-b \pi \log \left (\sqrt {e}-\frac {i \sqrt {d}}{x}\right )+b \pi \log \left (\frac {i \sqrt {d}}{x}+\sqrt {e}\right )-2 i b \text {Li}_2\left (-\frac {i \left (\sqrt {e-c^2 d}-\sqrt {e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i b \text {Li}_2\left (\frac {i \left (\sqrt {e-c^2 d}-\sqrt {e}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )+2 i b \text {Li}_2\left (-\frac {i \left (\sqrt {e}+\sqrt {e-c^2 d}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )-2 i b \text {Li}_2\left (\frac {i \left (\sqrt {e}+\sqrt {e-c^2 d}\right ) e^{\text {csch}^{-1}(c x)}}{c \sqrt {d}}\right )}{4 \sqrt {d} \sqrt {e}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.80, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \operatorname {arcsch}\left (c x\right ) + a}{e x^{2} + d}, x\right ) \]
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 \[ \int \frac {b \operatorname {arcsch}\left (c x\right ) + a}{e x^{2} + d}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.45, size = 0, normalized size = 0.00 \[ \int \frac {a +b \,\mathrm {arccsch}\left (c x \right )}{e \,x^{2}+d}\, dx \]
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 \[ b \int \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + \frac {1}{c x}\right )}{e x^{2} + d}\,{d x} + \frac {a \arctan \left (\frac {e x}{\sqrt {d e}}\right )}{\sqrt {d e}} \]
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 \[ \int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{e\,x^2+d} \,d x \]
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 \[ \int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{d + e x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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