3.82 \(\int \frac{1}{(a+a \text{sech}(c+d x))^{3/2}} \, dx\)
Optimal. Leaf size=114 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a \text{sech}(c+d x)+a}}\right )}{a^{3/2} d}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{2} \sqrt{a \text{sech}(c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\tanh (c+d x)}{2 d (a \text{sech}(c+d x)+a)^{3/2}} \]
[Out]
(2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Tanh[c + d*x]
)/(Sqrt[2]*Sqrt[a + a*Sech[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tanh[c + d*x]/(2*d*(a + a*Sech[c + d*x])^(3/2)
)
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Rubi [A] time = 0.127655, antiderivative size = 114, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used =
{3777, 3920, 3774, 203, 3795} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a \text{sech}(c+d x)+a}}\right )}{a^{3/2} d}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{2} \sqrt{a \text{sech}(c+d x)+a}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\tanh (c+d x)}{2 d (a \text{sech}(c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sech[c + d*x])^(-3/2),x]
[Out]
(2*ArcTanh[(Sqrt[a]*Tanh[c + d*x])/Sqrt[a + a*Sech[c + d*x]]])/(a^(3/2)*d) - (5*ArcTanh[(Sqrt[a]*Tanh[c + d*x]
)/(Sqrt[2]*Sqrt[a + a*Sech[c + d*x]])])/(2*Sqrt[2]*a^(3/2)*d) - Tanh[c + d*x]/(2*d*(a + a*Sech[c + d*x])^(3/2)
)
Rule 3777
Int[(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Simp[(Cot[c + d*x]*(a + b*Csc[c + d*x])^n)/(d*(
2*n + 1)), x] + Dist[1/(a^2*(2*n + 1)), Int[(a + b*Csc[c + d*x])^(n + 1)*(a*(2*n + 1) - b*(n + 1)*Csc[c + d*x]
), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0] && LeQ[n, -1] && IntegerQ[2*n]
Rule 3920
Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[c/a,
Int[Sqrt[a + b*Csc[e + f*x]], x], x] - Dist[(b*c - a*d)/a, Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /
; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0]
Rule 3774
Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[(-2*b)/d, Subst[Int[1/(a + x^2), x], x, (b*C
ot[c + d*x])/Sqrt[a + b*Csc[c + d*x]]], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 - b^2, 0]
Rule 203
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 3795
Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Dist[-2/f, Subst[Int[1/(2
*a + x^2), x], x, (b*Cot[e + f*x])/Sqrt[a + b*Csc[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 - b^2, 0
]
Rubi steps
\begin{align*} \int \frac{1}{(a+a \text{sech}(c+d x))^{3/2}} \, dx &=-\frac{\tanh (c+d x)}{2 d (a+a \text{sech}(c+d x))^{3/2}}-\frac{\int \frac{-2 a+\frac{1}{2} a \text{sech}(c+d x)}{\sqrt{a+a \text{sech}(c+d x)}} \, dx}{2 a^2}\\ &=-\frac{\tanh (c+d x)}{2 d (a+a \text{sech}(c+d x))^{3/2}}+\frac{\int \sqrt{a+a \text{sech}(c+d x)} \, dx}{a^2}-\frac{5 \int \frac{\text{sech}(c+d x)}{\sqrt{a+a \text{sech}(c+d x)}} \, dx}{4 a}\\ &=-\frac{\tanh (c+d x)}{2 d (a+a \text{sech}(c+d x))^{3/2}}+\frac{(2 i) \operatorname{Subst}\left (\int \frac{1}{a+x^2} \, dx,x,-\frac{i a \tanh (c+d x)}{\sqrt{a+a \text{sech}(c+d x)}}\right )}{a d}-\frac{(5 i) \operatorname{Subst}\left (\int \frac{1}{2 a+x^2} \, dx,x,-\frac{i a \tanh (c+d x)}{\sqrt{a+a \text{sech}(c+d x)}}\right )}{2 a d}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{a+a \text{sech}(c+d x)}}\right )}{a^{3/2} d}-\frac{5 \tanh ^{-1}\left (\frac{\sqrt{a} \tanh (c+d x)}{\sqrt{2} \sqrt{a+a \text{sech}(c+d x)}}\right )}{2 \sqrt{2} a^{3/2} d}-\frac{\tanh (c+d x)}{2 d (a+a \text{sech}(c+d x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 4.62702, size = 177, normalized size = 1.55 \[ \frac{\cosh ^2\left (\frac{1}{2} (c+d x)\right ) \text{sech}(c+d x) \left (4 \left (e^{c+d x}+1\right ) \sinh ^{-1}\left (e^{c+d x}\right )+5 \sqrt{2} \left (e^{c+d x}+1\right ) \tanh ^{-1}\left (\frac{1-e^{c+d x}}{\sqrt{2} \sqrt{e^{2 (c+d x)}+1}}\right )-4 \left (e^{c+d x}+1\right ) \tanh ^{-1}\left (\sqrt{e^{2 (c+d x)}+1}\right )-2 \sqrt{e^{2 (c+d x)}+1} \tanh \left (\frac{1}{2} (c+d x)\right )\right )}{2 d \sqrt{e^{2 (c+d x)}+1} (a (\text{sech}(c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sech[c + d*x])^(-3/2),x]
[Out]
(Cosh[(c + d*x)/2]^2*Sech[c + d*x]*(4*(1 + E^(c + d*x))*ArcSinh[E^(c + d*x)] + 5*Sqrt[2]*(1 + E^(c + d*x))*Arc
Tanh[(1 - E^(c + d*x))/(Sqrt[2]*Sqrt[1 + E^(2*(c + d*x))])] - 4*(1 + E^(c + d*x))*ArcTanh[Sqrt[1 + E^(2*(c + d
*x))]] - 2*Sqrt[1 + E^(2*(c + d*x))]*Tanh[(c + d*x)/2]))/(2*d*Sqrt[1 + E^(2*(c + d*x))]*(a*(1 + Sech[c + d*x])
)^(3/2))
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Maple [F] time = 0.13, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a{\rm sech} \left (dx+c\right ) \right ) ^{-{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(a+a*sech(d*x+c))^(3/2),x)
[Out]
int(1/(a+a*sech(d*x+c))^(3/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (a \operatorname{sech}\left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sech(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
integrate((a*sech(d*x + c) + a)^(-3/2), x)
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Fricas [B] time = 2.91463, size = 3401, normalized size = 29.83 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sech(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
1/8*(5*sqrt(2)*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)
*sqrt(a)*log(-(3*a*cosh(d*x + c)^2 + 3*a*sinh(d*x + c)^2 - 2*sqrt(2)*(cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*
sinh(d*x + c)^2 + sinh(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c)
+ cosh(d*x + c) - 1)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) -
2*a*cosh(d*x + c) + 2*(3*a*cosh(d*x + c) - a)*sinh(d*x + c) + 3*a)/(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*s
inh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)) + 4*(cosh(d*x + c)^2 + 2*(cosh(d*x + c) + 1)*sinh(d*x +
c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sqrt(a)*log(-(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 - 3*a*cosh(d*
x + c)^3 + (4*a*cosh(d*x + c) - 3*a)*sinh(d*x + c)^3 + 5*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 - 9*a*cosh(d
*x + c) + 5*a)*sinh(d*x + c)^2 + (cosh(d*x + c)^5 + (5*cosh(d*x + c) - 3)*sinh(d*x + c)^4 + sinh(d*x + c)^5 -
3*cosh(d*x + c)^4 + (10*cosh(d*x + c)^2 - 12*cosh(d*x + c) + 5)*sinh(d*x + c)^3 + 5*cosh(d*x + c)^3 + (10*cosh
(d*x + c)^3 - 18*cosh(d*x + c)^2 + 15*cosh(d*x + c) - 7)*sinh(d*x + c)^2 - 7*cosh(d*x + c)^2 + (5*cosh(d*x + c
)^4 - 12*cosh(d*x + c)^3 + 15*cosh(d*x + c)^2 - 14*cosh(d*x + c) + 4)*sinh(d*x + c) + 4*cosh(d*x + c) - 4)*sqr
t(a)*sqrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) - 4*a*cosh(d*x + c) + (4*
a*cosh(d*x + c)^3 - 9*a*cosh(d*x + c)^2 + 10*a*cosh(d*x + c) - 4*a)*sinh(d*x + c) + 4*a)/(cosh(d*x + c)^3 + 3*
cosh(d*x + c)^2*sinh(d*x + c) + 3*cosh(d*x + c)*sinh(d*x + c)^2 + sinh(d*x + c)^3)) + 4*(cosh(d*x + c)^2 + 2*(
cosh(d*x + c) + 1)*sinh(d*x + c) + sinh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sqrt(a)*log((a*cosh(d*x + c)^2 + a*s
inh(d*x + c)^2 + (cosh(d*x + c)^3 + (3*cosh(d*x + c) + 1)*sinh(d*x + c)^2 + sinh(d*x + c)^3 + cosh(d*x + c)^2
+ (3*cosh(d*x + c)^2 + 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) + 1)*sqrt(a)*sqrt(a/(cosh(d*x + c)^2
+ 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)) + a*cosh(d*x + c) + (2*a*cosh(d*x + c) + a)*sinh(d*x
+ c) + a)/(cosh(d*x + c) + sinh(d*x + c))) - 4*(cosh(d*x + c)^3 + (3*cosh(d*x + c) - 1)*sinh(d*x + c)^2 + sinh
(d*x + c)^3 - cosh(d*x + c)^2 + (3*cosh(d*x + c)^2 - 2*cosh(d*x + c) + 1)*sinh(d*x + c) + cosh(d*x + c) - 1)*s
qrt(a/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2 + 1)))/(a^2*d*cosh(d*x + c)^2 + a^2*d
*sinh(d*x + c)^2 + 2*a^2*d*cosh(d*x + c) + a^2*d + 2*(a^2*d*cosh(d*x + c) + a^2*d)*sinh(d*x + c))
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \operatorname{sech}{\left (c + d x \right )} + a\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sech(d*x+c))**(3/2),x)
[Out]
Integral((a*sech(c + d*x) + a)**(-3/2), x)
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(a+a*sech(d*x+c))^(3/2),x, algorithm="giac")
[Out]
Exception raised: TypeError