3.1.30 \(\int \frac {1}{\text {sech}^2(a+b x)^{5/2}} \, dx\) [30]

Optimal. Leaf size=76 \[ \frac {\tanh (a+b x)}{5 b \text {sech}^2(a+b x)^{5/2}}+\frac {4 \tanh (a+b x)}{15 b \text {sech}^2(a+b x)^{3/2}}+\frac {8 \tanh (a+b x)}{15 b \sqrt {\text {sech}^2(a+b x)}} \]

[Out]

1/5*tanh(b*x+a)/b/(sech(b*x+a)^2)^(5/2)+4/15*tanh(b*x+a)/b/(sech(b*x+a)^2)^(3/2)+8/15*tanh(b*x+a)/b/(sech(b*x+
a)^2)^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4207, 198, 197} \begin {gather*} \frac {8 \tanh (a+b x)}{15 b \sqrt {\text {sech}^2(a+b x)}}+\frac {4 \tanh (a+b x)}{15 b \text {sech}^2(a+b x)^{3/2}}+\frac {\tanh (a+b x)}{5 b \text {sech}^2(a+b x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sech[a + b*x]^2)^(-5/2),x]

[Out]

Tanh[a + b*x]/(5*b*(Sech[a + b*x]^2)^(5/2)) + (4*Tanh[a + b*x])/(15*b*(Sech[a + b*x]^2)^(3/2)) + (8*Tanh[a + b
*x])/(15*b*Sqrt[Sech[a + b*x]^2])

Rule 197

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1)/a), x] /; FreeQ[{a, b, n, p}, x] &
& EqQ[1/n + p + 1, 0]

Rule 198

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p +
1], 0] && NeQ[p, -1]

Rule 4207

Int[((b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Dist[b*(ff/
f), Subst[Int[(b + b*ff^2*x^2)^(p - 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{b, e, f, p}, x] &&  !IntegerQ[p
]

Rubi steps

\begin {align*} \int \frac {1}{\text {sech}^2(a+b x)^{5/2}} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{7/2}} \, dx,x,\tanh (a+b x)\right )}{b}\\ &=\frac {\tanh (a+b x)}{5 b \text {sech}^2(a+b x)^{5/2}}+\frac {4 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{5/2}} \, dx,x,\tanh (a+b x)\right )}{5 b}\\ &=\frac {\tanh (a+b x)}{5 b \text {sech}^2(a+b x)^{5/2}}+\frac {4 \tanh (a+b x)}{15 b \text {sech}^2(a+b x)^{3/2}}+\frac {8 \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^{3/2}} \, dx,x,\tanh (a+b x)\right )}{15 b}\\ &=\frac {\tanh (a+b x)}{5 b \text {sech}^2(a+b x)^{5/2}}+\frac {4 \tanh (a+b x)}{15 b \text {sech}^2(a+b x)^{3/2}}+\frac {8 \tanh (a+b x)}{15 b \sqrt {\text {sech}^2(a+b x)}}\\ \end {align*}

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Mathematica [A]
time = 0.07, size = 54, normalized size = 0.71 \begin {gather*} \frac {\text {sech}(a+b x) (150 \sinh (a+b x)+25 \sinh (3 (a+b x))+3 \sinh (5 (a+b x)))}{240 b \sqrt {\text {sech}^2(a+b x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sech[a + b*x]^2)^(-5/2),x]

[Out]

(Sech[a + b*x]*(150*Sinh[a + b*x] + 25*Sinh[3*(a + b*x)] + 3*Sinh[5*(a + b*x)]))/(240*b*Sqrt[Sech[a + b*x]^2])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(304\) vs. \(2(64)=128\).
time = 2.50, size = 305, normalized size = 4.01

method	result	size
risch	\(\frac {{\mathrm e}^{6 b x +6 a}}{160 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{4 b x +4 a}}{96 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}+\frac {5 \,{\mathrm e}^{2 b x +2 a}}{16 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}-\frac {5}{16 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}-\frac {5 \,{\mathrm e}^{-2 b x -2 a}}{96 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}-\frac {{\mathrm e}^{-4 b x -4 a}}{160 b \left ({\mathrm e}^{2 b x +2 a}+1\right ) \sqrt {\frac {{\mathrm e}^{2 b x +2 a}}{\left ({\mathrm e}^{2 b x +2 a}+1\right )^{2}}}}\)	\(305\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sech(b*x+a)^2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/160/b/(exp(2*b*x+2*a)+1)/(1/(exp(2*b*x+2*a)+1)^2*exp(2*b*x+2*a))^(1/2)*exp(6*b*x+6*a)+5/96/b/(exp(2*b*x+2*a)
+1)/(1/(exp(2*b*x+2*a)+1)^2*exp(2*b*x+2*a))^(1/2)*exp(4*b*x+4*a)+5/16/b/(exp(2*b*x+2*a)+1)/(1/(exp(2*b*x+2*a)+
1)^2*exp(2*b*x+2*a))^(1/2)*exp(2*b*x+2*a)-5/16/b/(exp(2*b*x+2*a)+1)/(1/(exp(2*b*x+2*a)+1)^2*exp(2*b*x+2*a))^(1
/2)-5/96/b/(exp(2*b*x+2*a)+1)/(1/(exp(2*b*x+2*a)+1)^2*exp(2*b*x+2*a))^(1/2)*exp(-2*b*x-2*a)-1/160/b/(exp(2*b*x
+2*a)+1)/(1/(exp(2*b*x+2*a)+1)^2*exp(2*b*x+2*a))^(1/2)*exp(-4*b*x-4*a)

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Maxima [A]
time = 0.27, size = 82, normalized size = 1.08 \begin {gather*} \frac {e^{\left (5 \, b x + 5 \, a\right )}}{160 \, b} + \frac {5 \, e^{\left (3 \, b x + 3 \, a\right )}}{96 \, b} + \frac {5 \, e^{\left (b x + a\right )}}{16 \, b} - \frac {5 \, e^{\left (-b x - a\right )}}{16 \, b} - \frac {5 \, e^{\left (-3 \, b x - 3 \, a\right )}}{96 \, b} - \frac {e^{\left (-5 \, b x - 5 \, a\right )}}{160 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(b*x+a)^2)^(5/2),x, algorithm="maxima")

[Out]

1/160*e^(5*b*x + 5*a)/b + 5/96*e^(3*b*x + 3*a)/b + 5/16*e^(b*x + a)/b - 5/16*e^(-b*x - a)/b - 5/96*e^(-3*b*x -
3*a)/b - 1/160*e^(-5*b*x - 5*a)/b

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Fricas [A]
time = 0.38, size = 66, normalized size = 0.87 \begin {gather*} \frac {3 \, \sinh \left (b x + a\right )^{5} + 5 \, {\left (6 \, \cosh \left (b x + a\right )^{2} + 5\right )} \sinh \left (b x + a\right )^{3} + 15 \, {\left (\cosh \left (b x + a\right )^{4} + 5 \, \cosh \left (b x + a\right )^{2} + 10\right )} \sinh \left (b x + a\right )}{240 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(b*x+a)^2)^(5/2),x, algorithm="fricas")

[Out]

1/240*(3*sinh(b*x + a)^5 + 5*(6*cosh(b*x + a)^2 + 5)*sinh(b*x + a)^3 + 15*(cosh(b*x + a)^4 + 5*cosh(b*x + a)^2
+ 10)*sinh(b*x + a))/b

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Sympy [A]
time = 14.51, size = 80, normalized size = 1.05 \begin {gather*} \begin {cases} \frac {8 \tanh ^{5}{\left (a + b x \right )}}{15 b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {5}{2}}} - \frac {4 \tanh ^{3}{\left (a + b x \right )}}{3 b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {5}{2}}} + \frac {\tanh {\left (a + b x \right )}}{b \left (\operatorname {sech}^{2}{\left (a + b x \right )}\right )^{\frac {5}{2}}} & \text {for}\: b \neq 0 \\\frac {x}{\left (\operatorname {sech}^{2}{\left (a \right )}\right )^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(b*x+a)**2)**(5/2),x)

[Out]

Piecewise((8*tanh(a + b*x)**5/(15*b*(sech(a + b*x)**2)**(5/2)) - 4*tanh(a + b*x)**3/(3*b*(sech(a + b*x)**2)**(
5/2)) + tanh(a + b*x)/(b*(sech(a + b*x)**2)**(5/2)), Ne(b, 0)), (x/(sech(a)**2)**(5/2), True))

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Giac [A]
time = 0.39, size = 70, normalized size = 0.92 \begin {gather*} -\frac {{\left (150 \, e^{\left (4 \, b x + 4 \, a\right )} + 25 \, e^{\left (2 \, b x + 2 \, a\right )} + 3\right )} e^{\left (-5 \, b x - 5 \, a\right )} - 3 \, e^{\left (5 \, b x + 5 \, a\right )} - 25 \, e^{\left (3 \, b x + 3 \, a\right )} - 150 \, e^{\left (b x + a\right )}}{480 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(sech(b*x+a)^2)^(5/2),x, algorithm="giac")

[Out]

-1/480*((150*e^(4*b*x + 4*a) + 25*e^(2*b*x + 2*a) + 3)*e^(-5*b*x - 5*a) - 3*e^(5*b*x + 5*a) - 25*e^(3*b*x + 3*
a) - 150*e^(b*x + a))/b

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (\frac {1}{{\mathrm {cosh}\left (a+b\,x\right )}^2}\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1/cosh(a + b*x)^2)^(5/2),x)

[Out]

int(1/(1/cosh(a + b*x)^2)^(5/2), x)

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