Optimal. Leaf size=80 \[ \frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \]
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Rubi [A]
time = 0.07, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {5549, 5545,
3858, 3855, 3852, 8} \begin {gather*} \frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 3852
Rule 3855
Rule 3858
Rule 5545
Rule 5549
Rubi steps
\begin {align*} \int (e x)^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx &=\frac {\left (x^{-n} (e x)^n\right ) \int x^{-1+n} \left (a+b \text {csch}\left (c+d x^n\right )\right )^2 \, dx}{e}\\ &=\frac {\left (x^{-n} (e x)^n\right ) \text {Subst}\left (\int (a+b \text {csch}(c+d x))^2 \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^n}{e n}+\frac {\left (2 a b x^{-n} (e x)^n\right ) \text {Subst}\left (\int \text {csch}(c+d x) \, dx,x,x^n\right )}{e n}+\frac {\left (b^2 x^{-n} (e x)^n\right ) \text {Subst}\left (\int \text {csch}^2(c+d x) \, dx,x,x^n\right )}{e n}\\ &=\frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {\left (i b^2 x^{-n} (e x)^n\right ) \text {Subst}\left (\int 1 \, dx,x,-i \coth \left (c+d x^n\right )\right )}{d e n}\\ &=\frac {a^2 (e x)^n}{e n}-\frac {2 a b x^{-n} (e x)^n \tanh ^{-1}\left (\cosh \left (c+d x^n\right )\right )}{d e n}-\frac {b^2 x^{-n} (e x)^n \coth \left (c+d x^n\right )}{d e n}\\ \end {align*}
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Mathematica [A]
time = 0.26, size = 87, normalized size = 1.09 \begin {gather*} \frac {x^{-n} (e x)^n \left (-b^2 \coth \left (\frac {1}{2} \left (c+d x^n\right )\right )+2 a \left (a c+a d x^n+2 b \log \left (\tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )\right )-b^2 \tanh \left (\frac {1}{2} \left (c+d x^n\right )\right )\right )}{2 d e n} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 4.50, size = 271, normalized size = 3.39
method | result | size |
risch | \(\frac {a^{2} x \,{\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{n}-\frac {2 x \,x^{-n} b^{2} {\mathrm e}^{\frac {\left (-1+n \right ) \left (-i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )+i \pi \,\mathrm {csgn}\left (i e \right ) \mathrm {csgn}\left (i e x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i e x \right )^{2}-i \pi \mathrm {csgn}\left (i e x \right )^{3}+2 \ln \left (x \right )+2 \ln \left (e \right )\right )}{2}}}{d n \left ({\mathrm e}^{2 c +2 d \,x^{n}}-1\right )}-\frac {4 \arctanh \left ({\mathrm e}^{c +d \,x^{n}}\right ) e^{n} a b \,{\mathrm e}^{\frac {i \pi \,\mathrm {csgn}\left (i e x \right ) \left (-1+n \right ) \left (\mathrm {csgn}\left (i e x \right )-\mathrm {csgn}\left (i x \right )\right ) \left (-\mathrm {csgn}\left (i e x \right )+\mathrm {csgn}\left (i e \right )\right )}{2}}}{d e n}\) | \(271\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 109, normalized size = 1.36 \begin {gather*} -2 \, a b {\left (\frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} + 1\right )} e^{\left (-c\right )}\right )}{d n} - \frac {e^{n - 1} \log \left ({\left (e^{\left (d x^{n} + c\right )} - 1\right )} e^{\left (-c\right )}\right )}{d n}\right )} - \frac {2 \, b^{2} e^{n}}{d e n e^{\left (2 \, d x^{n} + 2 \, c\right )} - d e n} + \frac {\left (e x\right )^{n} a^{2}}{e n} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 744 vs.
\(2 (80) = 160\).
time = 0.46, size = 744, normalized size = 9.30 \begin {gather*} \frac {{\left ({\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - 2 \, b^{2} \cosh \left (n - 1\right ) + 2 \, {\left ({\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left ({\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) + {\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - 2 \, b^{2} \sinh \left (n - 1\right ) - {\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \cosh \left (n \log \left (x\right )\right ) - 2 \, {\left ({\left (a b \cosh \left (n - 1\right ) + a b \sinh \left (n - 1\right )\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - a b \cosh \left (n - 1\right ) + 2 \, {\left (a b \cosh \left (n - 1\right ) + a b \sinh \left (n - 1\right )\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left (a b \cosh \left (n - 1\right ) + a b \sinh \left (n - 1\right )\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - a b \sinh \left (n - 1\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + 1\right ) + 2 \, {\left ({\left (a b \cosh \left (n - 1\right ) + a b \sinh \left (n - 1\right )\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - a b \cosh \left (n - 1\right ) + 2 \, {\left (a b \cosh \left (n - 1\right ) + a b \sinh \left (n - 1\right )\right )} \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + {\left (a b \cosh \left (n - 1\right ) + a b \sinh \left (n - 1\right )\right )} \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - a b \sinh \left (n - 1\right )\right )} \log \left (\cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) - 1\right ) - {\left (a^{2} d \cosh \left (n - 1\right ) + a^{2} d \sinh \left (n - 1\right )\right )} \sinh \left (n \log \left (x\right )\right )}{d n \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} + 2 \, d n \cosh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right ) + d n \sinh \left (d \cosh \left (n \log \left (x\right )\right ) + d \sinh \left (n \log \left (x\right )\right ) + c\right )^{2} - d n} \end {gather*}
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 \begin {gather*} \int \left (e x\right )^{n - 1} \left (a + b \operatorname {csch}{\left (c + d x^{n} \right )}\right )^{2}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.61, size = 160, normalized size = 2.00 \begin {gather*} \frac {a^2\,x\,{\left (e\,x\right )}^{n-1}}{n}-\frac {4\,\mathrm {atan}\left (\frac {a\,b\,x\,{\mathrm {e}}^{d\,x^n}\,{\mathrm {e}}^c\,{\left (e\,x\right )}^{n-1}\,\sqrt {-d^2\,n^2\,x^{2\,n}}}{d\,n\,x^n\,\sqrt {a^2\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}\right )\,\sqrt {a^2\,b^2\,x^2\,{\left (e\,x\right )}^{2\,n-2}}}{\sqrt {-d^2\,n^2\,x^{2\,n}}}-\frac {2\,b^2\,x\,{\left (e\,x\right )}^{n-1}}{d\,n\,x^n\,\left ({\mathrm {e}}^{2\,c+2\,d\,x^n}-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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