Optimal. Leaf size=115 \[ d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \left (6 c^2 d-e\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{6 c^2 \sqrt {-c^2 x^2}}+\frac {b e x^2 \sqrt {-c^2 x^2-1}}{6 c \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {6292, 12, 388, 217, 203} \[ d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b x \left (6 c^2 d-e\right ) \tan ^{-1}\left (\frac {c x}{\sqrt {-c^2 x^2-1}}\right )}{6 c^2 \sqrt {-c^2 x^2}}+\frac {b e x^2 \sqrt {-c^2 x^2-1}}{6 c \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 203
Rule 217
Rule 388
Rule 6292
Rubi steps
\begin {align*} \int \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {3 d+e x^2}{3 \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {(b c x) \int \frac {3 d+e x^2}{\sqrt {-1-c^2 x^2}} \, dx}{3 \sqrt {-c^2 x^2}}\\ &=\frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c \left (6 d-\frac {e}{c^2}\right ) x\right ) \int \frac {1}{\sqrt {-1-c^2 x^2}} \, dx}{6 \sqrt {-c^2 x^2}}\\ &=\frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {\left (b c \left (6 d-\frac {e}{c^2}\right ) x\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x^2} \, dx,x,\frac {x}{\sqrt {-1-c^2 x^2}}\right )}{6 \sqrt {-c^2 x^2}}\\ &=\frac {b e x^2 \sqrt {-1-c^2 x^2}}{6 c \sqrt {-c^2 x^2}}+d x \left (a+b \text {csch}^{-1}(c x)\right )+\frac {1}{3} e x^3 \left (a+b \text {csch}^{-1}(c x)\right )-\frac {b \left (6 d-\frac {e}{c^2}\right ) x \tan ^{-1}\left (\frac {c x}{\sqrt {-1-c^2 x^2}}\right )}{6 \sqrt {-c^2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.26, size = 135, normalized size = 1.17 \[ a d x+\frac {1}{3} a e x^3+\frac {b d x \sqrt {\frac {1}{c^2 x^2}+1} \sinh ^{-1}(c x)}{\sqrt {c^2 x^2+1}}+\frac {b e x^2 \sqrt {\frac {c^2 x^2+1}{c^2 x^2}}}{6 c}-\frac {b e \log \left (x \left (\sqrt {\frac {c^2 x^2+1}{c^2 x^2}}+1\right )\right )}{6 c^3}+b d x \text {csch}^{-1}(c x)+\frac {1}{3} b e x^3 \text {csch}^{-1}(c x) \]
Antiderivative was successfully verified.
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fricas [B] time = 1.01, size = 245, normalized size = 2.13 \[ \frac {2 \, a c^{3} e x^{3} + b c^{2} e x^{2} \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 6 \, a c^{3} d x + 2 \, {\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x + 1\right ) - {\left (6 \, b c^{2} d - b e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x\right ) - 2 \, {\left (3 \, b c^{3} d + b c^{3} e\right )} \log \left (c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} - c x - 1\right ) + 2 \, {\left (b c^{3} e x^{3} + 3 \, b c^{3} d x - 3 \, b c^{3} d - b c^{3} e\right )} \log \left (\frac {c x \sqrt {\frac {c^{2} x^{2} + 1}{c^{2} x^{2}}} + 1}{c x}\right )}{6 \, c^{3}} \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 126, normalized size = 1.10 \[ \frac {\frac {a \left (\frac {1}{3} e \,c^{3} x^{3}+x \,c^{3} d \right )}{c^{2}}+\frac {b \left (\frac {\mathrm {arccsch}\left (c x \right ) e \,c^{3} x^{3}}{3}+\mathrm {arccsch}\left (c x \right ) c^{3} d x +\frac {\sqrt {c^{2} x^{2}+1}\, \left (6 c^{2} d \arcsinh \left (c x \right )+e c x \sqrt {c^{2} x^{2}+1}-e \arcsinh \left (c x \right )\right )}{6 \sqrt {\frac {c^{2} x^{2}+1}{c^{2} x^{2}}}\, c x}\right )}{c^{2}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 148, normalized size = 1.29 \[ \frac {1}{3} \, a e x^{3} + \frac {1}{12} \, {\left (4 \, x^{3} \operatorname {arcsch}\left (c x\right ) + \frac {\frac {2 \, \sqrt {\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} + 1\right )} - c^{2}} - \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} + \frac {\log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} b e + a d x + \frac {{\left (2 \, c x \operatorname {arcsch}\left (c x\right ) + \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} + 1\right ) - \log \left (\sqrt {\frac {1}{c^{2} x^{2}} + 1} - 1\right )\right )} b d}{2 \, c} \]
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 \[ \int \left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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