Optimal. Leaf size=220 \[ \frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}+\frac {b x \sqrt {c^2 x^2+1} (f x)^{m+1} \left (e (m+1)^2-c^2 d (m+2) (m+3)\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};-c^2 x^2\right )}{c f (m+1)^2 (m+2) (m+3) \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}+\frac {b e x \sqrt {-c^2 x^2-1} (f x)^{m+1}}{c f \left (m^2+5 m+6\right ) \sqrt {-c^2 x^2}} \]
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Rubi [A] time = 0.22, antiderivative size = 208, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {14, 6302, 12, 459, 365, 364} \[ \frac {d (f x)^{m+1} \left (a+b \text {csch}^{-1}(c x)\right )}{f (m+1)}+\frac {e (f x)^{m+3} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (m+3)}-\frac {b c x \sqrt {c^2 x^2+1} (f x)^{m+1} \left (\frac {d}{(m+1)^2}-\frac {e}{c^2 (m+2) (m+3)}\right ) \, _2F_1\left (\frac {1}{2},\frac {m+1}{2};\frac {m+3}{2};-c^2 x^2\right )}{f \sqrt {-c^2 x^2} \sqrt {-c^2 x^2-1}}+\frac {b e x \sqrt {-c^2 x^2-1} (f x)^{m+1}}{c f \left (m^2+5 m+6\right ) \sqrt {-c^2 x^2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 364
Rule 365
Rule 459
Rule 6302
Rubi steps
\begin {align*} \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx &=\frac {d (f x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (3+m)}-\frac {(b c x) \int \frac {(f x)^m \left (d (3+m)+e (1+m) x^2\right )}{(1+m) (3+m) \sqrt {-1-c^2 x^2}} \, dx}{\sqrt {-c^2 x^2}}\\ &=\frac {d (f x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (3+m)}-\frac {(b c x) \int \frac {(f x)^m \left (d (3+m)+e (1+m) x^2\right )}{\sqrt {-1-c^2 x^2}} \, dx}{\left (3+4 m+m^2\right ) \sqrt {-c^2 x^2}}\\ &=\frac {b e x (f x)^{1+m} \sqrt {-1-c^2 x^2}}{c f \left (6+5 m+m^2\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (3+m)}-\frac {\left (b c \left (-\frac {e (1+m)^2}{c^2 (2+m)}+d (3+m)\right ) x\right ) \int \frac {(f x)^m}{\sqrt {-1-c^2 x^2}} \, dx}{\left (3+4 m+m^2\right ) \sqrt {-c^2 x^2}}\\ &=\frac {b e x (f x)^{1+m} \sqrt {-1-c^2 x^2}}{c f \left (6+5 m+m^2\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (3+m)}-\frac {\left (b c \left (-\frac {e (1+m)^2}{c^2 (2+m)}+d (3+m)\right ) x \sqrt {1+c^2 x^2}\right ) \int \frac {(f x)^m}{\sqrt {1+c^2 x^2}} \, dx}{\left (3+4 m+m^2\right ) \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}\\ &=\frac {b e x (f x)^{1+m} \sqrt {-1-c^2 x^2}}{c f \left (6+5 m+m^2\right ) \sqrt {-c^2 x^2}}+\frac {d (f x)^{1+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f (1+m)}+\frac {e (f x)^{3+m} \left (a+b \text {csch}^{-1}(c x)\right )}{f^3 (3+m)}+\frac {b c \left (\frac {e (1+m)^2}{c^2 (2+m)}-d (3+m)\right ) x (f x)^{1+m} \sqrt {1+c^2 x^2} \, _2F_1\left (\frac {1}{2},\frac {1+m}{2};\frac {3+m}{2};-c^2 x^2\right )}{f (1+m) \left (3+4 m+m^2\right ) \sqrt {-c^2 x^2} \sqrt {-1-c^2 x^2}}\\ \end {align*}
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Mathematica [F] time = 0.11, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d+e x^2\right ) \left (a+b \text {csch}^{-1}(c x)\right ) \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a e x^{2} + a d + {\left (b e x^{2} + b d\right )} \operatorname {arcsch}\left (c x\right )\right )} \left (f x\right )^{m}, x\right ) \]
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 \[ \int {\left (e x^{2} + d\right )} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )} \left (f x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.06, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (e \,x^{2}+d \right ) \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )\, dx \]
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 \[ \frac {a e f^{m} x^{3} x^{m}}{m + 3} + \frac {\left (f x\right )^{m + 1} a d}{f {\left (m + 1\right )}} - \frac {{\left (b e f^{m} {\left (m + 1\right )} x^{3} + b d f^{m} {\left (m + 3\right )} x\right )} x^{m} \log \relax (x) - {\left (b e f^{m} {\left (m + 1\right )} x^{3} + b d f^{m} {\left (m + 3\right )} x\right )} x^{m} \log \left (\sqrt {c^{2} x^{2} + 1} + 1\right )}{m^{2} + 4 \, m + 3} + \int \frac {{\left (b c^{2} e f^{m} {\left (m + 1\right )} x^{4} + b c^{2} d f^{m} {\left (m + 3\right )} x^{2}\right )} x^{m}}{{\left (m^{2} + 4 \, m + 3\right )} c^{2} x^{2} + m^{2} + {\left ({\left (m^{2} + 4 \, m + 3\right )} c^{2} x^{2} + m^{2} + 4 \, m + 3\right )} \sqrt {c^{2} x^{2} + 1} + 4 \, m + 3}\,{d x} - \int \frac {{\left ({\left (m^{2} + 4 \, m + 3\right )} b c^{2} e f^{m} x^{4} \log \relax (c) + {\left ({\left (m^{2} + 4 \, m + 3\right )} c^{2} d f^{m} \log \relax (c) + {\left (m^{2} + 4 \, m + 3\right )} e f^{m} \log \relax (c) - e f^{m} {\left (m + 1\right )}\right )} b x^{2} + {\left ({\left (m^{2} + 4 \, m + 3\right )} d f^{m} \log \relax (c) - d f^{m} {\left (m + 3\right )}\right )} b\right )} x^{m}}{{\left (m^{2} + 4 \, m + 3\right )} c^{2} x^{2} + m^{2} + 4 \, m + 3}\,{d x} \]
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 \[ \int {\left (f\,x\right )}^m\,\left (e\,x^2+d\right )\,\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right ) \,d x \]
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 \[ \int \left (f x\right )^{m} \left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \left (d + e x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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