3.133 \(\int \frac{\tanh ^5(c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}+\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

[Out]

(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sech[c + d*x]])/(b^
4*d) - (2*(3*a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(3/2))/(3*b^4*d) + (6*a*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d)
- (2*(a + b*Sech[c + d*x])^(7/2))/(7*b^4*d)

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Rubi [A]  time = 0.164909, antiderivative size = 148, 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, 1153, 207} \[ -\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}+\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}+\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[c + d*x]^5/Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + (2*a*(a^2 - 2*b^2)*Sqrt[a + b*Sech[c + d*x]])/(b^
4*d) - (2*(3*a^2 - 2*b^2)*(a + b*Sech[c + d*x])^(3/2))/(3*b^4*d) + (6*a*(a + b*Sech[c + d*x])^(5/2))/(5*b^4*d)
- (2*(a + b*Sech[c + d*x])^(7/2))/(7*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 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 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 \frac{\tanh ^5(c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (b^2-x^2\right )^2}{x \sqrt{a+x}} \, dx,x,b \text{sech}(c+d x)\right )}{b^4 d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{\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 (-a^3+2 a b^2+\left (3 a^2-2 b^2\right ) x^2-3 a x^4+x^6+\frac{b^4}{-a+x^2}\right ) \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{b^4 d}\\ &=\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}-\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{d}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}+\frac{2 a \left (a^2-2 b^2\right ) \sqrt{a+b \text{sech}(c+d x)}}{b^4 d}-\frac{2 \left (3 a^2-2 b^2\right ) (a+b \text{sech}(c+d x))^{3/2}}{3 b^4 d}+\frac{6 a (a+b \text{sech}(c+d x))^{5/2}}{5 b^4 d}-\frac{2 (a+b \text{sech}(c+d x))^{7/2}}{7 b^4 d}\\ \end{align*}

Mathematica [A]  time = 4.43192, size = 167, normalized size = 1.13 \[ \frac{2 \left (\left (70 b^4-6 a^2 b^2\right ) \text{sech}^2(c+d x)+\left (24 a^3 b-70 a b^3\right ) \text{sech}(c+d x)-140 a^2 b^2+48 a^4+3 a b^3 \text{sech}^3(c+d x)+\frac{105 b^4 \sqrt{a \cosh (c+d x)+b} \tanh ^{-1}\left (\frac{\sqrt{a \cosh (c+d x)+b}}{\sqrt{a \cosh (c+d x)}}\right )}{\sqrt{a \cosh (c+d x)}}-15 b^4 \text{sech}^4(c+d x)\right )}{105 b^4 d \sqrt{a+b \text{sech}(c+d x)}} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[c + d*x]^5/Sqrt[a + b*Sech[c + d*x]],x]

[Out]

(2*(48*a^4 - 140*a^2*b^2 + (105*b^4*ArcTanh[Sqrt[b + a*Cosh[c + d*x]]/Sqrt[a*Cosh[c + d*x]]]*Sqrt[b + a*Cosh[c
+ d*x]])/Sqrt[a*Cosh[c + d*x]] + (24*a^3*b - 70*a*b^3)*Sech[c + d*x] + (-6*a^2*b^2 + 70*b^4)*Sech[c + d*x]^2
+ 3*a*b^3*Sech[c + d*x]^3 - 15*b^4*Sech[c + d*x]^4))/(105*b^4*d*Sqrt[a + b*Sech[c + d*x]])

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Maple [F]  time = 0.22, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \tanh \left ( dx+c \right ) \right ) ^{5}{\frac{1}{\sqrt{a+b{\rm sech} \left (dx+c\right )}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(d*x+c)^5/(a+b*sech(d*x+c))^(1/2),x)

[Out]

int(tanh(d*x+c)^5/(a+b*sech(d*x+c))^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (d x + c\right )^{5}}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(d*x + c)^5/sqrt(b*sech(d*x + c) + a), x)

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Fricas [B]  time = 10.5067, size = 7063, normalized size = 47.72 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/210*(105*(b^4*cosh(d*x + c)^6 + 6*b^4*cosh(d*x + c)*sinh(d*x + c)^5 + b^4*sinh(d*x + c)^6 + 3*b^4*cosh(d*x
+ c)^4 + 3*b^4*cosh(d*x + c)^2 + 3*(5*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x + c)^4 + b^4 + 4*(5*b^4*cosh(d*x + c
)^3 + 3*b^4*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^4*cosh(d*x + c)^4 + 6*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x
+ c)^2 + 6*(b^4*cosh(d*x + c)^5 + 2*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)*sinh(d*x + c) + sinh(d*x
+ c)^2)) + 16*((12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^6 + (12*a^4 - 35*a^2*b^2)*sinh(d*x + c)^6 - (12*a^3*b - 35
*a*b^3)*cosh(d*x + c)^5 - (12*a^3*b - 35*a*b^3 - 6*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(1
2*a^4 - 29*a^2*b^2)*cosh(d*x + c)^4 + (36*a^4 - 87*a^2*b^2 + 15*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^2 - 5*(12*
a^3*b - 35*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 12*a^4 - 35*a^2*b^2 - 8*(3*a^3*b - 5*a*b^3)*cosh(d*x + c)^3
- 2*(12*a^3*b - 20*a*b^3 - 10*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^3 + 5*(12*a^3*b - 35*a*b^3)*cosh(d*x + c)^2
- 6*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c)^2 + (15*(12*
a^4 - 35*a^2*b^2)*cosh(d*x + c)^4 + 36*a^4 - 87*a^2*b^2 - 10*(12*a^3*b - 35*a*b^3)*cosh(d*x + c)^3 + 18*(12*a^
4 - 29*a^2*b^2)*cosh(d*x + c)^2 - 24*(3*a^3*b - 5*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^2 - (12*a^3*b - 35*a*b^3
)*cosh(d*x + c) + (6*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^5 - 5*(12*a^3*b - 35*a*b^3)*cosh(d*x + c)^4 - 12*a^3*
b + 35*a*b^3 + 12*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c)^3 - 24*(3*a^3*b - 5*a*b^3)*cosh(d*x + c)^2 + 6*(12*a^4 -
29*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)))/(a*b^4*d*cosh(d*x + c)^6
+ 6*a*b^4*d*cosh(d*x + c)*sinh(d*x + c)^5 + a*b^4*d*sinh(d*x + c)^6 + 3*a*b^4*d*cosh(d*x + c)^4 + 3*a*b^4*d*c
osh(d*x + c)^2 + a*b^4*d + 3*(5*a*b^4*d*cosh(d*x + c)^2 + a*b^4*d)*sinh(d*x + c)^4 + 4*(5*a*b^4*d*cosh(d*x + c
)^3 + 3*a*b^4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*a*b^4*d*cosh(d*x + c)^4 + 6*a*b^4*d*cosh(d*x + c)^2 + a*
b^4*d)*sinh(d*x + c)^2 + 6*(a*b^4*d*cosh(d*x + c)^5 + 2*a*b^4*d*cosh(d*x + c)^3 + a*b^4*d*cosh(d*x + c))*sinh(
d*x + c)), -1/105*(105*(b^4*cosh(d*x + c)^6 + 6*b^4*cosh(d*x + c)*sinh(d*x + c)^5 + b^4*sinh(d*x + c)^6 + 3*b^
4*cosh(d*x + c)^4 + 3*b^4*cosh(d*x + c)^2 + 3*(5*b^4*cosh(d*x + c)^2 + b^4)*sinh(d*x + c)^4 + b^4 + 4*(5*b^4*c
osh(d*x + c)^3 + 3*b^4*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*b^4*cosh(d*x + c)^4 + 6*b^4*cosh(d*x + c)^2 + b^4
)*sinh(d*x + c)^2 + 6*(b^4*cosh(d*x + c)^5 + 2*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))) - 8*((12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^6 + (12*a^
4 - 35*a^2*b^2)*sinh(d*x + c)^6 - (12*a^3*b - 35*a*b^3)*cosh(d*x + c)^5 - (12*a^3*b - 35*a*b^3 - 6*(12*a^4 - 3
5*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^5 + 3*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c)^4 + (36*a^4 - 87*a^2*b^2 + 1
5*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^2 - 5*(12*a^3*b - 35*a*b^3)*cosh(d*x + c))*sinh(d*x + c)^4 + 12*a^4 - 35
*a^2*b^2 - 8*(3*a^3*b - 5*a*b^3)*cosh(d*x + c)^3 - 2*(12*a^3*b - 20*a*b^3 - 10*(12*a^4 - 35*a^2*b^2)*cosh(d*x
+ c)^3 + 5*(12*a^3*b - 35*a*b^3)*cosh(d*x + c)^2 - 6*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c)^3 + 3*
(12*a^4 - 29*a^2*b^2)*cosh(d*x + c)^2 + (15*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^4 + 36*a^4 - 87*a^2*b^2 - 10*(
12*a^3*b - 35*a*b^3)*cosh(d*x + c)^3 + 18*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c)^2 - 24*(3*a^3*b - 5*a*b^3)*cosh(
d*x + c))*sinh(d*x + c)^2 - (12*a^3*b - 35*a*b^3)*cosh(d*x + c) + (6*(12*a^4 - 35*a^2*b^2)*cosh(d*x + c)^5 - 5
*(12*a^3*b - 35*a*b^3)*cosh(d*x + c)^4 - 12*a^3*b + 35*a*b^3 + 12*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c)^3 - 24*(
3*a^3*b - 5*a*b^3)*cosh(d*x + c)^2 + 6*(12*a^4 - 29*a^2*b^2)*cosh(d*x + c))*sinh(d*x + c))*sqrt((a*cosh(d*x +
c) + b)/cosh(d*x + c)))/(a*b^4*d*cosh(d*x + c)^6 + 6*a*b^4*d*cosh(d*x + c)*sinh(d*x + c)^5 + a*b^4*d*sinh(d*x
+ c)^6 + 3*a*b^4*d*cosh(d*x + c)^4 + 3*a*b^4*d*cosh(d*x + c)^2 + a*b^4*d + 3*(5*a*b^4*d*cosh(d*x + c)^2 + a*b^
4*d)*sinh(d*x + c)^4 + 4*(5*a*b^4*d*cosh(d*x + c)^3 + 3*a*b^4*d*cosh(d*x + c))*sinh(d*x + c)^3 + 3*(5*a*b^4*d*
cosh(d*x + c)^4 + 6*a*b^4*d*cosh(d*x + c)^2 + a*b^4*d)*sinh(d*x + c)^2 + 6*(a*b^4*d*cosh(d*x + c)^5 + 2*a*b^4*
d*cosh(d*x + c)^3 + a*b^4*d*cosh(d*x + c))*sinh(d*x + c))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh ^{5}{\left (c + d x \right )}}{\sqrt{a + b \operatorname{sech}{\left (c + d x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(tanh(c + d*x)**5/sqrt(a + b*sech(c + d*x)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tanh \left (d x + c\right )^{5}}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(tanh(d*x + c)^5/sqrt(b*sech(d*x + c) + a), x)