3.91 \(\int \text {sech}^4(a+b x) \sqrt {\tanh (a+b x)} \, dx\)

Optimal. Leaf size=35 \[ \frac {2 \tanh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \tanh ^{\frac {7}{2}}(a+b x)}{7 b} \]

[Out]

2/3*tanh(b*x+a)^(3/2)/b-2/7*tanh(b*x+a)^(7/2)/b

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Rubi [A]  time = 0.04, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2607, 14} \[ \frac {2 \tanh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \tanh ^{\frac {7}{2}}(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Sech[a + b*x]^4*Sqrt[Tanh[a + b*x]],x]

[Out]

(2*Tanh[a + b*x]^(3/2))/(3*b) - (2*Tanh[a + b*x]^(7/2))/(7*b)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rubi steps

\begin {align*} \int \text {sech}^4(a+b x) \sqrt {\tanh (a+b x)} \, dx &=-\frac {i \operatorname {Subst}\left (\int \sqrt {-i x} \left (1+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac {i \operatorname {Subst}\left (\int \left (\sqrt {-i x}-(-i x)^{5/2}\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {2 \tanh ^{\frac {3}{2}}(a+b x)}{3 b}-\frac {2 \tanh ^{\frac {7}{2}}(a+b x)}{7 b}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 29, normalized size = 0.83 \[ \frac {2 \tanh ^{\frac {3}{2}}(a+b x) \left (3 \text {sech}^2(a+b x)+4\right )}{21 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[a + b*x]^4*Sqrt[Tanh[a + b*x]],x]

[Out]

(2*(4 + 3*Sech[a + b*x]^2)*Tanh[a + b*x]^(3/2))/(21*b)

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fricas [B]  time = 0.44, size = 551, normalized size = 15.74 \[ \frac {8 \, {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{4} + 3 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, {\left (5 \, \cosh \left (b x + a\right )^{4} + 6 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right )^{2} + 3 \, \cosh \left (b x + a\right )^{2} + 6 \, {\left (\cosh \left (b x + a\right )^{5} + 2 \, \cosh \left (b x + a\right )^{3} + \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + {\left (\cosh \left (b x + a\right )^{6} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + \sinh \left (b x + a\right )^{6} + {\left (15 \, \cosh \left (b x + a\right )^{2} + 4\right )} \sinh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right )^{4} + 4 \, {\left (5 \, \cosh \left (b x + a\right )^{3} + 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + {\left (15 \, \cosh \left (b x + a\right )^{4} + 24 \, \cosh \left (b x + a\right )^{2} - 4\right )} \sinh \left (b x + a\right )^{2} - 4 \, \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{5} + 8 \, \cosh \left (b x + a\right )^{3} - 4 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) - 1\right )} \sqrt {\frac {\sinh \left (b x + a\right )}{\cosh \left (b x + a\right )}} + 1\right )}}{21 \, {\left (b \cosh \left (b x + a\right )^{6} + 6 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{5} + b \sinh \left (b x + a\right )^{6} + 3 \, b \cosh \left (b x + a\right )^{4} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{4} + 4 \, {\left (5 \, b \cosh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{3} + 3 \, b \cosh \left (b x + a\right )^{2} + 3 \, {\left (5 \, b \cosh \left (b x + a\right )^{4} + 6 \, b \cosh \left (b x + a\right )^{2} + b\right )} \sinh \left (b x + a\right )^{2} + 6 \, {\left (b \cosh \left (b x + a\right )^{5} + 2 \, b \cosh \left (b x + a\right )^{3} + b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^4*tanh(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

8/21*(cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + 3*(5*cosh(b*x + a)^2 + 1)*sinh(b*x
+ a)^4 + 3*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + 3*cosh(b*x + a))*sinh(b*x + a)^3 + 3*(5*cosh(b*x + a)^4 +
6*cosh(b*x + a)^2 + 1)*sinh(b*x + a)^2 + 3*cosh(b*x + a)^2 + 6*(cosh(b*x + a)^5 + 2*cosh(b*x + a)^3 + cosh(b*
x + a))*sinh(b*x + a) + (cosh(b*x + a)^6 + 6*cosh(b*x + a)*sinh(b*x + a)^5 + sinh(b*x + a)^6 + (15*cosh(b*x +
a)^2 + 4)*sinh(b*x + a)^4 + 4*cosh(b*x + a)^4 + 4*(5*cosh(b*x + a)^3 + 4*cosh(b*x + a))*sinh(b*x + a)^3 + (15*
cosh(b*x + a)^4 + 24*cosh(b*x + a)^2 - 4)*sinh(b*x + a)^2 - 4*cosh(b*x + a)^2 + 2*(3*cosh(b*x + a)^5 + 8*cosh(
b*x + a)^3 - 4*cosh(b*x + a))*sinh(b*x + a) - 1)*sqrt(sinh(b*x + a)/cosh(b*x + a)) + 1)/(b*cosh(b*x + a)^6 + 6
*b*cosh(b*x + a)*sinh(b*x + a)^5 + b*sinh(b*x + a)^6 + 3*b*cosh(b*x + a)^4 + 3*(5*b*cosh(b*x + a)^2 + b)*sinh(
b*x + a)^4 + 4*(5*b*cosh(b*x + a)^3 + 3*b*cosh(b*x + a))*sinh(b*x + a)^3 + 3*b*cosh(b*x + a)^2 + 3*(5*b*cosh(b
*x + a)^4 + 6*b*cosh(b*x + a)^2 + b)*sinh(b*x + a)^2 + 6*(b*cosh(b*x + a)^5 + 2*b*cosh(b*x + a)^3 + b*cosh(b*x
+ a))*sinh(b*x + a) + b)

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giac [B]  time = 0.18, size = 148, normalized size = 4.23 \[ \frac {16 \, {\left (21 \, {\left (\sqrt {e^{\left (4 \, b x + 4 \, a\right )} - 1} - e^{\left (2 \, b x + 2 \, a\right )}\right )}^{5} - 7 \, {\left (\sqrt {e^{\left (4 \, b x + 4 \, a\right )} - 1} - e^{\left (2 \, b x + 2 \, a\right )}\right )}^{4} + 28 \, {\left (\sqrt {e^{\left (4 \, b x + 4 \, a\right )} - 1} - e^{\left (2 \, b x + 2 \, a\right )}\right )}^{3} + 7 \, \sqrt {e^{\left (4 \, b x + 4 \, a\right )} - 1} - 7 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}}{21 \, b {\left (\sqrt {e^{\left (4 \, b x + 4 \, a\right )} - 1} - e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^4*tanh(b*x+a)^(1/2),x, algorithm="giac")

[Out]

16/21*(21*(sqrt(e^(4*b*x + 4*a) - 1) - e^(2*b*x + 2*a))^5 - 7*(sqrt(e^(4*b*x + 4*a) - 1) - e^(2*b*x + 2*a))^4
+ 28*(sqrt(e^(4*b*x + 4*a) - 1) - e^(2*b*x + 2*a))^3 + 7*sqrt(e^(4*b*x + 4*a) - 1) - 7*e^(2*b*x + 2*a) - 1)/(b
*(sqrt(e^(4*b*x + 4*a) - 1) - e^(2*b*x + 2*a) - 1)^7)

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maple [F]  time = 0.71, size = 0, normalized size = 0.00 \[ \int \mathrm {sech}\left (b x +a \right )^{4} \left (\sqrt {\tanh }\left (b x +a \right )\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(b*x+a)^4*tanh(b*x+a)^(1/2),x)

[Out]

int(sech(b*x+a)^4*tanh(b*x+a)^(1/2),x)

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maxima [B]  time = 0.44, size = 352, normalized size = 10.06 \[ \frac {32 \, \sqrt {e^{\left (-b x - a\right )} + 1} \sqrt {-e^{\left (-b x - a\right )} + 1} e^{\left (-2 \, b x - 2 \, a\right )}}{21 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt {e^{\left (-2 \, b x - 2 \, a\right )} + 1}} - \frac {32 \, \sqrt {e^{\left (-b x - a\right )} + 1} \sqrt {-e^{\left (-b x - a\right )} + 1} e^{\left (-4 \, b x - 4 \, a\right )}}{21 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt {e^{\left (-2 \, b x - 2 \, a\right )} + 1}} - \frac {8 \, \sqrt {e^{\left (-b x - a\right )} + 1} \sqrt {-e^{\left (-b x - a\right )} + 1} e^{\left (-6 \, b x - 6 \, a\right )}}{21 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt {e^{\left (-2 \, b x - 2 \, a\right )} + 1}} + \frac {8 \, \sqrt {e^{\left (-b x - a\right )} + 1} \sqrt {-e^{\left (-b x - a\right )} + 1}}{21 \, b {\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt {e^{\left (-2 \, b x - 2 \, a\right )} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)^4*tanh(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

32/21*sqrt(e^(-b*x - a) + 1)*sqrt(-e^(-b*x - a) + 1)*e^(-2*b*x - 2*a)/(b*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4
*a) + e^(-6*b*x - 6*a) + 1)*sqrt(e^(-2*b*x - 2*a) + 1)) - 32/21*sqrt(e^(-b*x - a) + 1)*sqrt(-e^(-b*x - a) + 1)
*e^(-4*b*x - 4*a)/(b*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) + 1)*sqrt(e^(-2*b*x - 2*a) +
1)) - 8/21*sqrt(e^(-b*x - a) + 1)*sqrt(-e^(-b*x - a) + 1)*e^(-6*b*x - 6*a)/(b*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*
x - 4*a) + e^(-6*b*x - 6*a) + 1)*sqrt(e^(-2*b*x - 2*a) + 1)) + 8/21*sqrt(e^(-b*x - a) + 1)*sqrt(-e^(-b*x - a)
+ 1)/(b*(3*e^(-2*b*x - 2*a) + 3*e^(-4*b*x - 4*a) + e^(-6*b*x - 6*a) + 1)*sqrt(e^(-2*b*x - 2*a) + 1))

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mupad [B]  time = 1.63, size = 168, normalized size = 4.80 \[ \frac {8\,\sqrt {\frac {{\mathrm {e}}^{2\,a+2\,b\,x}-1}{{\mathrm {e}}^{2\,a+2\,b\,x}+1}}}{21\,b}+\frac {8\,\sqrt {\frac {{\mathrm {e}}^{2\,a+2\,b\,x}-1}{{\mathrm {e}}^{2\,a+2\,b\,x}+1}}}{21\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {24\,\sqrt {\frac {{\mathrm {e}}^{2\,a+2\,b\,x}-1}{{\mathrm {e}}^{2\,a+2\,b\,x}+1}}}{7\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^2}+\frac {16\,\sqrt {\frac {{\mathrm {e}}^{2\,a+2\,b\,x}-1}{{\mathrm {e}}^{2\,a+2\,b\,x}+1}}}{7\,b\,{\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(a + b*x)^(1/2)/cosh(a + b*x)^4,x)

[Out]

(8*((exp(2*a + 2*b*x) - 1)/(exp(2*a + 2*b*x) + 1))^(1/2))/(21*b) + (8*((exp(2*a + 2*b*x) - 1)/(exp(2*a + 2*b*x
) + 1))^(1/2))/(21*b*(exp(2*a + 2*b*x) + 1)) - (24*((exp(2*a + 2*b*x) - 1)/(exp(2*a + 2*b*x) + 1))^(1/2))/(7*b
*(exp(2*a + 2*b*x) + 1)^2) + (16*((exp(2*a + 2*b*x) - 1)/(exp(2*a + 2*b*x) + 1))^(1/2))/(7*b*(exp(2*a + 2*b*x)
+ 1)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\tanh {\left (a + b x \right )}} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(b*x+a)**4*tanh(b*x+a)**(1/2),x)

[Out]

Integral(sqrt(tanh(a + b*x))*sech(a + b*x)**4, x)

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