3.4.76 \(\int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx\) [376]

Optimal. Leaf size=85 \[ \frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}-\text {Int}\left (x^m \tanh (a+b x),x\right ) \]

[Out]

2^(-3-m)*exp(2*a)*x^m*GAMMA(1+m,-2*b*x)/b/((-b*x)^m)+2^(-3-m)*x^m*GAMMA(1+m,2*b*x)/b/exp(2*a)/((b*x)^m)-Uninte
grable(x^m*tanh(b*x+a),x)

________________________________________________________________________________________

Rubi [A]
time = 0.12, 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 = {} \begin {gather*} \int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[x^m*Sinh[a + b*x]^2*Tanh[a + b*x],x]

[Out]

(2^(-3 - m)*E^(2*a)*x^m*Gamma[1 + m, -2*b*x])/(b*(-(b*x))^m) + (2^(-3 - m)*x^m*Gamma[1 + m, 2*b*x])/(b*E^(2*a)
*(b*x)^m) - Defer[Int][x^m*Tanh[a + b*x], x]

Rubi steps

\begin {align*} \int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx &=\int x^m \cosh (a+b x) \sinh (a+b x) \, dx-\int x^m \tanh (a+b x) \, dx\\ &=\int \frac {1}{2} x^m \sinh (2 a+2 b x) \, dx-\int x^m \tanh (a+b x) \, dx\\ &=\frac {1}{2} \int x^m \sinh (2 a+2 b x) \, dx-\int x^m \tanh (a+b x) \, dx\\ &=\frac {1}{4} \int e^{-i (2 i a+2 i b x)} x^m \, dx-\frac {1}{4} \int e^{i (2 i a+2 i b x)} x^m \, dx-\int x^m \tanh (a+b x) \, dx\\ &=\frac {2^{-3-m} e^{2 a} x^m (-b x)^{-m} \Gamma (1+m,-2 b x)}{b}+\frac {2^{-3-m} e^{-2 a} x^m (b x)^{-m} \Gamma (1+m,2 b x)}{b}-\int x^m \tanh (a+b x) \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 16.75, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \sinh ^2(a+b x) \tanh (a+b x) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^m*Sinh[a + b*x]^2*Tanh[a + b*x],x]

[Out]

Integrate[x^m*Sinh[a + b*x]^2*Tanh[a + b*x], x]

________________________________________________________________________________________

Maple [A]
time = 1.69, size = 0, normalized size = 0.00 \[\int x^{m} \mathrm {sech}\left (b x +a \right ) \left (\sinh ^{3}\left (b x +a \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*sech(b*x+a)*sinh(b*x+a)^3,x)

[Out]

int(x^m*sech(b*x+a)*sinh(b*x+a)^3,x)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*sech(b*x+a)*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate(x^m*sech(b*x + a)*sinh(b*x + a)^3, x)

________________________________________________________________________________________

Fricas [A]
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.

[In]

integrate(x^m*sech(b*x+a)*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

integral(x^m*sech(b*x + a)*sinh(b*x + a)^3, x)

________________________________________________________________________________________

Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{m} \sinh ^{3}{\left (a + b x \right )} \operatorname {sech}{\left (a + b x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*sech(b*x+a)*sinh(b*x+a)**3,x)

[Out]

Integral(x**m*sinh(a + b*x)**3*sech(a + b*x), x)

________________________________________________________________________________________

Giac [A]
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.

[In]

integrate(x^m*sech(b*x+a)*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate(x^m*sech(b*x + a)*sinh(b*x + a)^3, x)

________________________________________________________________________________________

Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^m\,{\mathrm {sinh}\left (a+b\,x\right )}^3}{\mathrm {cosh}\left (a+b\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^m*sinh(a + b*x)^3)/cosh(a + b*x),x)

[Out]

int((x^m*sinh(a + b*x)^3)/cosh(a + b*x), x)

________________________________________________________________________________________