Optimal. Leaf size=80 \[ -\text {Int}\left (x^m \tanh (a+b x) \text {sech}(a+b x),x\right )+\frac {e^a x^m (-b x)^{-m} \Gamma (m+1,-b x)}{2 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (m+1,b x)}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx &=\int x^m \sinh (a+b x) \, dx-\int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac {1}{2} \int e^{-i (i a+i b x)} x^m \, dx-\frac {1}{2} \int e^{i (i a+i b x)} x^m \, dx-\int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx\\ &=\frac {e^a x^m (-b x)^{-m} \Gamma (1+m,-b x)}{2 b}+\frac {e^{-a} x^m (b x)^{-m} \Gamma (1+m,b x)}{2 b}-\int x^m \text {sech}(a+b x) \tanh (a+b x) \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 38.12, size = 0, normalized size = 0.00 \[ \int x^m \sinh (a+b x) \tanh ^2(a+b x) \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.42, size = 0, normalized size = 0.00 \[ \int x^{m} \mathrm {sech}\left (b x +a \right )^{2} \left (\sinh ^{3}\left (b x +a \right )\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {sech}\left (b x + a\right )^{2} \sinh \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^m\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{{\mathrm {cosh}\left (a+b\,x\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________