3.125 \(\int \sqrt{a+b \text{sech}(c+d x)} \tanh ^5(c+d x) \, dx\)
Optimal. Leaf size=169 \[ -\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 a \left (a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{9/2}}{9 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{d} \]
[Out]
(2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sech[c + d*x]])/d + (2*a*(a^2 - 2*b^2
)*(a + b*Sech[c + d*x])^(3/2))/(3*b^4*d) - (2*(3*a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d) + (6*a*(a
+ b*Sech[c + d*x])^(7/2))/(7*b^4*d) - (2*(a + b*Sech[c + d*x])^(9/2))/(9*b^4*d)
________________________________________________________________________________________
Rubi [A] time = 0.194028, antiderivative size = 169, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used =
{3885, 898, 1261, 207} \[ -\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 a \left (a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{9/2}}{9 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}+\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{d} \]
Antiderivative was successfully verified.
[In]
Int[Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x]^5,x]
[Out]
(2*Sqrt[a]*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/d - (2*Sqrt[a + b*Sech[c + d*x]])/d + (2*a*(a^2 - 2*b^2
)*(a + b*Sech[c + d*x])^(3/2))/(3*b^4*d) - (2*(3*a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d) + (6*a*(a
+ b*Sech[c + d*x])^(7/2))/(7*b^4*d) - (2*(a + b*Sech[c + d*x])^(9/2))/(9*b^4*d)
Rule 3885
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 898
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 + a*e^2)/e^2 - (2*c
*d*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]
Rule 1261
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] &&
NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \sqrt{a+b \text{sech}(c+d x)} \tanh ^5(c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+x} \left (b^2-x^2\right )^2}{x} \, dx,x,b \text{sech}(c+d x)\right )}{b^4 d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{x^2 \left (-a^2+b^2+2 a x^2-x^4\right )^2}{-a+x^2} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{b^4 d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \left (b^4-a \left (a^2-2 b^2\right ) x^2+\left (3 a^2-2 b^2\right ) x^4-3 a x^6+x^8+\frac{a b^4}{-a+x^2}\right ) \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{b^4 d}\\ &=-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}+\frac{2 a \left (a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}-\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{9/2}}{9 b^4 d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{d}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}+\frac{2 a \left (a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}-\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{9/2}}{9 b^4 d}\\ \end{align*}
Mathematica [A] time = 4.83465, size = 160, normalized size = 0.95 \[ \frac{2 \sqrt{a+b \text{sech}(c+d x)} \left (\left (\frac{6 a^2}{b^2}+126\right ) \text{sech}^2(c+d x)+\left (\frac{42 a}{b}-\frac{8 a^3}{b^3}\right ) \text{sech}(c+d x)+\frac{16 a^4}{b^4}-\frac{84 a^2}{b^2}-\frac{5 a \text{sech}^3(c+d x)}{b}+\frac{315 \sqrt{a \cosh (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a \cosh (c+d x)+b}}{\sqrt{a \cosh (c+d x)}}\right )}{\sqrt{a \cosh (c+d x)+b}}-35 \text{sech}^4(c+d x)-315\right )}{315 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x]^5,x]
[Out]
(2*Sqrt[a + b*Sech[c + d*x]]*(-315 + (16*a^4)/b^4 - (84*a^2)/b^2 + (315*ArcTanh[Sqrt[b + a*Cosh[c + d*x]]/Sqrt
[a*Cosh[c + d*x]]]*Sqrt[a*Cosh[c + d*x]])/Sqrt[b + a*Cosh[c + d*x]] + ((-8*a^3)/b^3 + (42*a)/b)*Sech[c + d*x]
+ (126 + (6*a^2)/b^2)*Sech[c + d*x]^2 - (5*a*Sech[c + d*x]^3)/b - 35*Sech[c + d*x]^4))/(315*d)
________________________________________________________________________________________
Maple [F] time = 0.221, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b{\rm sech} \left (dx+c\right )} \left ( \tanh \left ( dx+c \right ) \right ) ^{5}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)^5,x)
[Out]
int((a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)^5,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)^5,x, algorithm="maxima")
[Out]
integrate(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^5, x)
________________________________________________________________________________________
Fricas [B] time = 9.43823, size = 10971, normalized size = 64.92 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)^5,x, algorithm="fricas")
[Out]
[1/630*(315*(b^4*cosh(d*x + c)^8 + 8*b^4*cosh(d*x + c)*sinh(d*x + c)^7 + b^4*sinh(d*x + c)^8 + 4*b^4*cosh(d*x
+ c)^6 + 6*b^4*cosh(d*x + c)^4 + 4*(7*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x + c)^6 + 4*b^4*cosh(d*x + c)^2 + 8*(
7*b^4*cosh(d*x + c)^3 + 3*b^4*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*b^4*cosh(d*x + c)^4 + 30*b^4*cosh(d*x + c
)^2 + 3*b^4)*sinh(d*x + c)^4 + b^4 + 8*(7*b^4*cosh(d*x + c)^5 + 10*b^4*cosh(d*x + c)^3 + 3*b^4*cosh(d*x + c))*
sinh(d*x + c)^3 + 4*(7*b^4*cosh(d*x + c)^6 + 15*b^4*cosh(d*x + c)^4 + 9*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x +
c)^2 + 8*(b^4*cosh(d*x + c)^7 + 3*b^4*cosh(d*x + c)^5 + 3*b^4*cosh(d*x + c)^3 + b^4*cosh(d*x + c))*sinh(d*x +
c))*sqrt(a)*log(-(2*a^2*cosh(d*x + c)^4 + 2*a^2*sinh(d*x + c)^4 + 4*a*b*cosh(d*x + c)^3 + 4*(2*a^2*cosh(d*x +
c) + a*b)*sinh(d*x + c)^3 + 4*a*b*cosh(d*x + c) + (4*a^2 + b^2)*cosh(d*x + c)^2 + (12*a^2*cosh(d*x + c)^2 + 12
*a*b*cosh(d*x + c) + 4*a^2 + b^2)*sinh(d*x + c)^2 + 2*a^2 + 2*(a*cosh(d*x + c)^4 + a*sinh(d*x + c)^4 + b*cosh(
d*x + c)^3 + (4*a*cosh(d*x + c) + b)*sinh(d*x + c)^3 + 2*a*cosh(d*x + c)^2 + (6*a*cosh(d*x + c)^2 + 3*b*cosh(d
*x + c) + 2*a)*sinh(d*x + c)^2 + b*cosh(d*x + c) + (4*a*cosh(d*x + c)^3 + 3*b*cosh(d*x + c)^2 + 4*a*cosh(d*x +
c) + b)*sinh(d*x + c) + a)*sqrt(a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)) + 2*(4*a^2*cosh(d*x + c)^3 + 6*a
*b*cosh(d*x + c)^2 + 2*a*b + (4*a^2 + b^2)*cosh(d*x + c))*sinh(d*x + c))/(cosh(d*x + c)^2 + 2*cosh(d*x + c)*si
nh(d*x + c) + sinh(d*x + c)^2)) + 4*((16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^8 + (16*a^4 - 84*a^2*b^2 -
315*b^4)*sinh(d*x + c)^8 - 4*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^7 - 4*(4*a^3*b - 21*a*b^3 - 2*(16*a^4 - 84*a^2
*b^2 - 315*b^4)*cosh(d*x + c))*sinh(d*x + c)^7 + 4*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^6 + 4*(16*a^4
- 78*a^2*b^2 - 189*b^4 + 7*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^2 - 7*(4*a^3*b - 21*a*b^3)*cosh(d*x
+ c))*sinh(d*x + c)^6 - 4*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^5 - 4*(12*a^3*b - 53*a*b^3 - 14*(16*a^4 - 84*a^2
*b^2 - 315*b^4)*cosh(d*x + c)^3 + 21*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^2 - 6*(16*a^4 - 78*a^2*b^2 - 189*b^4)*
cosh(d*x + c))*sinh(d*x + c)^5 + 2*(48*a^4 - 228*a^2*b^2 - 721*b^4)*cosh(d*x + c)^4 + 2*(35*(16*a^4 - 84*a^2*b
^2 - 315*b^4)*cosh(d*x + c)^4 + 48*a^4 - 228*a^2*b^2 - 721*b^4 - 70*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^3 + 30*
(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^2 - 10*(12*a^3*b - 53*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 16
*a^4 - 84*a^2*b^2 - 315*b^4 - 4*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^3 + 4*(14*(16*a^4 - 84*a^2*b^2 - 315*b^4)*
cosh(d*x + c)^5 - 35*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^4 - 12*a^3*b + 53*a*b^3 + 20*(16*a^4 - 78*a^2*b^2 - 18
9*b^4)*cosh(d*x + c)^3 - 10*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^2 + 2*(48*a^4 - 228*a^2*b^2 - 721*b^4)*cosh(d*
x + c))*sinh(d*x + c)^3 + 4*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^2 + 4*(7*(16*a^4 - 84*a^2*b^2 - 315*
b^4)*cosh(d*x + c)^6 - 21*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^5 + 15*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x +
c)^4 + 16*a^4 - 78*a^2*b^2 - 189*b^4 - 10*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^3 + 3*(48*a^4 - 228*a^2*b^2 - 7
21*b^4)*cosh(d*x + c)^2 - 3*(12*a^3*b - 53*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(4*a^3*b - 21*a*b^3)*cosh
(d*x + c) + 4*(2*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^7 - 7*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^6 + 6*
(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^5 - 5*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^4 - 4*a^3*b + 21*a*b^3
+ 2*(48*a^4 - 228*a^2*b^2 - 721*b^4)*cosh(d*x + c)^3 - 3*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^2 + 2*(16*a^4 -
78*a^2*b^2 - 189*b^4)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)))/(b^4*d*cosh(d*x
+ c)^8 + 8*b^4*d*cosh(d*x + c)*sinh(d*x + c)^7 + b^4*d*sinh(d*x + c)^8 + 4*b^4*d*cosh(d*x + c)^6 + 6*b^4*d*co
sh(d*x + c)^4 + 4*b^4*d*cosh(d*x + c)^2 + 4*(7*b^4*d*cosh(d*x + c)^2 + b^4*d)*sinh(d*x + c)^6 + 8*(7*b^4*d*cos
h(d*x + c)^3 + 3*b^4*d*cosh(d*x + c))*sinh(d*x + c)^5 + b^4*d + 2*(35*b^4*d*cosh(d*x + c)^4 + 30*b^4*d*cosh(d*
x + c)^2 + 3*b^4*d)*sinh(d*x + c)^4 + 8*(7*b^4*d*cosh(d*x + c)^5 + 10*b^4*d*cosh(d*x + c)^3 + 3*b^4*d*cosh(d*x
+ c))*sinh(d*x + c)^3 + 4*(7*b^4*d*cosh(d*x + c)^6 + 15*b^4*d*cosh(d*x + c)^4 + 9*b^4*d*cosh(d*x + c)^2 + b^4
*d)*sinh(d*x + c)^2 + 8*(b^4*d*cosh(d*x + c)^7 + 3*b^4*d*cosh(d*x + c)^5 + 3*b^4*d*cosh(d*x + c)^3 + b^4*d*cos
h(d*x + c))*sinh(d*x + c)), -1/315*(315*(b^4*cosh(d*x + c)^8 + 8*b^4*cosh(d*x + c)*sinh(d*x + c)^7 + b^4*sinh(
d*x + c)^8 + 4*b^4*cosh(d*x + c)^6 + 6*b^4*cosh(d*x + c)^4 + 4*(7*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x + c)^6 +
4*b^4*cosh(d*x + c)^2 + 8*(7*b^4*cosh(d*x + c)^3 + 3*b^4*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(35*b^4*cosh(d*x
+ c)^4 + 30*b^4*cosh(d*x + c)^2 + 3*b^4)*sinh(d*x + c)^4 + b^4 + 8*(7*b^4*cosh(d*x + c)^5 + 10*b^4*cosh(d*x +
c)^3 + 3*b^4*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^4*cosh(d*x + c)^6 + 15*b^4*cosh(d*x + c)^4 + 9*b^4*cosh(d
*x + c)^2 + b^4)*sinh(d*x + c)^2 + 8*(b^4*cosh(d*x + c)^7 + 3*b^4*cosh(d*x + c)^5 + 3*b^4*cosh(d*x + c)^3 + b^
4*cosh(d*x + c))*sinh(d*x + c))*sqrt(-a)*arctan((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + b*cosh(d*x + c) + (2*
a*cosh(d*x + c) + b)*sinh(d*x + c) + a)*sqrt(-a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c))/(a^2*cosh(d*x + c)^
2 + a^2*sinh(d*x + c)^2 + 2*a*b*cosh(d*x + c) + a^2 + 2*(a^2*cosh(d*x + c) + a*b)*sinh(d*x + c))) - 2*((16*a^4
- 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^8 + (16*a^4 - 84*a^2*b^2 - 315*b^4)*sinh(d*x + c)^8 - 4*(4*a^3*b - 21*a
*b^3)*cosh(d*x + c)^7 - 4*(4*a^3*b - 21*a*b^3 - 2*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c))*sinh(d*x + c)
^7 + 4*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^6 + 4*(16*a^4 - 78*a^2*b^2 - 189*b^4 + 7*(16*a^4 - 84*a^2
*b^2 - 315*b^4)*cosh(d*x + c)^2 - 7*(4*a^3*b - 21*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^6 - 4*(12*a^3*b - 53*a*b
^3)*cosh(d*x + c)^5 - 4*(12*a^3*b - 53*a*b^3 - 14*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^3 + 21*(4*a^3*
b - 21*a*b^3)*cosh(d*x + c)^2 - 6*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c))*sinh(d*x + c)^5 + 2*(48*a^4 -
228*a^2*b^2 - 721*b^4)*cosh(d*x + c)^4 + 2*(35*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^4 + 48*a^4 - 228
*a^2*b^2 - 721*b^4 - 70*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^3 + 30*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c
)^2 - 10*(12*a^3*b - 53*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 16*a^4 - 84*a^2*b^2 - 315*b^4 - 4*(12*a^3*b -
53*a*b^3)*cosh(d*x + c)^3 + 4*(14*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^5 - 35*(4*a^3*b - 21*a*b^3)*co
sh(d*x + c)^4 - 12*a^3*b + 53*a*b^3 + 20*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^3 - 10*(12*a^3*b - 53*a
*b^3)*cosh(d*x + c)^2 + 2*(48*a^4 - 228*a^2*b^2 - 721*b^4)*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(16*a^4 - 78*a^2
*b^2 - 189*b^4)*cosh(d*x + c)^2 + 4*(7*(16*a^4 - 84*a^2*b^2 - 315*b^4)*cosh(d*x + c)^6 - 21*(4*a^3*b - 21*a*b^
3)*cosh(d*x + c)^5 + 15*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c)^4 + 16*a^4 - 78*a^2*b^2 - 189*b^4 - 10*(
12*a^3*b - 53*a*b^3)*cosh(d*x + c)^3 + 3*(48*a^4 - 228*a^2*b^2 - 721*b^4)*cosh(d*x + c)^2 - 3*(12*a^3*b - 53*a
*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - 4*(4*a^3*b - 21*a*b^3)*cosh(d*x + c) + 4*(2*(16*a^4 - 84*a^2*b^2 - 315*
b^4)*cosh(d*x + c)^7 - 7*(4*a^3*b - 21*a*b^3)*cosh(d*x + c)^6 + 6*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c
)^5 - 5*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^4 - 4*a^3*b + 21*a*b^3 + 2*(48*a^4 - 228*a^2*b^2 - 721*b^4)*cosh(d
*x + c)^3 - 3*(12*a^3*b - 53*a*b^3)*cosh(d*x + c)^2 + 2*(16*a^4 - 78*a^2*b^2 - 189*b^4)*cosh(d*x + c))*sinh(d*
x + c))*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)))/(b^4*d*cosh(d*x + c)^8 + 8*b^4*d*cosh(d*x + c)*sinh(d*x + c
)^7 + b^4*d*sinh(d*x + c)^8 + 4*b^4*d*cosh(d*x + c)^6 + 6*b^4*d*cosh(d*x + c)^4 + 4*b^4*d*cosh(d*x + c)^2 + 4*
(7*b^4*d*cosh(d*x + c)^2 + b^4*d)*sinh(d*x + c)^6 + 8*(7*b^4*d*cosh(d*x + c)^3 + 3*b^4*d*cosh(d*x + c))*sinh(d
*x + c)^5 + b^4*d + 2*(35*b^4*d*cosh(d*x + c)^4 + 30*b^4*d*cosh(d*x + c)^2 + 3*b^4*d)*sinh(d*x + c)^4 + 8*(7*b
^4*d*cosh(d*x + c)^5 + 10*b^4*d*cosh(d*x + c)^3 + 3*b^4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*b^4*d*cosh(d*x
+ c)^6 + 15*b^4*d*cosh(d*x + c)^4 + 9*b^4*d*cosh(d*x + c)^2 + b^4*d)*sinh(d*x + c)^2 + 8*(b^4*d*cosh(d*x + c)
^7 + 3*b^4*d*cosh(d*x + c)^5 + 3*b^4*d*cosh(d*x + c)^3 + b^4*d*cosh(d*x + c))*sinh(d*x + c))]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a + b \operatorname{sech}{\left (c + d x \right )}} \tanh ^{5}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c))**(1/2)*tanh(d*x+c)**5,x)
[Out]
Integral(sqrt(a + b*sech(c + d*x))*tanh(c + d*x)**5, x)
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)^5,x, algorithm="giac")
[Out]
integrate(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^5, x)