\(\int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx\) [130]

Optimal result

Integrand size = 23, antiderivative size = 344 \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=-\frac {2 a (a-b) \sqrt {a+b} \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^2 d}-\frac {2 \sqrt {a+b} (a+2 b) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{d}-\frac {2 \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 d} \] Output:

-2/3*a*(a-b)*(a+b)^(1/2)*coth(d*x+c)*EllipticE((a+b*sech(d*x+c))^(1/2)/(a+ 
b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech( 
d*x+c))/(a-b))^(1/2)/b^2/d-2/3*(a+b)^(1/2)*(a+2*b)*coth(d*x+c)*EllipticF(( 
a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sech(d*x+c)) 
/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/b/d+2*(a+b)^(1/2)*coth(d*x+ 
c)*EllipticPi((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1 
/2))*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/d-2/ 
3*(a+b*sech(d*x+c))^(1/2)*tanh(d*x+c)/d
 

Mathematica [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx \] Input:

Integrate[Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x]^2,x]
 

Output:

Integrate[Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x]^2, x]
 

Rubi [A] (verified)

Time = 2.22 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.609, Rules used = {3042, 25, 4382, 3042, 4545, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \tanh ^2(c+d x) \sqrt {a+b \text {sech}(c+d x)} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int -\cot \left (i c+i d x+\frac {\pi }{2}\right )^2 \sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}dx\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \cot \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2 \sqrt {a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )}dx\)

\(\Big \downarrow \) 4382

\(\displaystyle -\int \sqrt {a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )} \left (\csc ^2\left (\frac {1}{2} (2 i c+\pi )+i d x\right )-1\right )dx\)

\(\Big \downarrow \) 3042

\(\displaystyle -\int \sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )} \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )^2-1\right )dx\)

\(\Big \downarrow \) 4545

\(\displaystyle -\frac {2}{3} \int -\frac {-a \text {sech}^2(c+d x)+2 b \text {sech}(c+d x)+3 a}{2 \sqrt {a+b \text {sech}(c+d x)}}dx-\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \int \frac {-a \text {sech}^2(c+d x)+2 b \text {sech}(c+d x)+3 a}{\sqrt {a+b \text {sech}(c+d x)}}dx-\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}+\frac {1}{3} \int \frac {-a \csc \left (i c+i d x+\frac {\pi }{2}\right )^2+2 b \csc \left (i c+i d x+\frac {\pi }{2}\right )+3 a}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {1}{3} \left (\int \frac {3 a+(a+2 b) \text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}dx-a \int \frac {\text {sech}(c+d x) (\text {sech}(c+d x)+1)}{\sqrt {a+b \text {sech}(c+d x)}}dx\right )-\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}+\frac {1}{3} \left (\int \frac {3 a+(a+2 b) \csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx-a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx\right )\)

\(\Big \downarrow \) 4409

\(\displaystyle -\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}+\frac {1}{3} \left (-a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+3 a \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}}dx+(a+2 b) \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}dx\right )\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}+\frac {1}{3} \left (3 a \int \frac {1}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+(a+2 b) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx-a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx\right )\)

\(\Big \downarrow \) 4271

\(\displaystyle -\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}+\frac {1}{3} \left ((a+2 b) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx-a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\frac {6 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )\)

\(\Big \downarrow \) 4319

\(\displaystyle -\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}+\frac {1}{3} \left (-a \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx-\frac {2 \sqrt {a+b} (a+2 b) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {6 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {1}{3} \left (-\frac {2 a (a-b) \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}-\frac {2 \sqrt {a+b} (a+2 b) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {6 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}\right )-\frac {2 \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{3 d}\)

Input:

Int[Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x]^2,x]
 

Output:

((-2*a*(a - b)*Sqrt[a + b]*Coth[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sech[ 
c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + 
 b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b^2*d) - (2*Sqrt[a + b]*(a 
 + 2*b)*Coth[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + 
b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + 
 Sech[c + d*x]))/(a - b))])/(b*d) + (6*Sqrt[a + b]*Coth[c + d*x]*EllipticP 
i[(a + b)/a, ArcSin[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b 
)]*Sqrt[(b*(1 - Sech[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sech[c + d*x]))/(a 
 - b))])/d)/3 - (2*Sqrt[a + b*Sech[c + d*x]]*Tanh[c + d*x])/(3*d)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4382

Int[cot[(c_.) + (d_.)*(x_)]^2*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), 
x_Symbol] :> Int[(-1 + Csc[c + d*x]^2)*(a + b*Csc[c + d*x])^n, x] /; FreeQ[ 
{a, b, c, d, n}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   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] && NeQ[a^2 - b^2, 0]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4545

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_. 
) + (a_))^(m_.), x_Symbol] :> Simp[(-C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^ 
m/(f*(m + 1))), x] + Simp[1/(m + 1)   Int[(a + b*Csc[e + f*x])^(m - 1)*Simp 
[a*A*(m + 1) + (A*b*(m + 1) + b*C*m)*Csc[e + f*x] + a*C*m*Csc[e + f*x]^2, x 
], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IGtQ[2*m, 
 0]
 

rule 4546

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C 
)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C   Int[Csc[e + f*x]*(( 
1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A 
, B, C}, x] && NeQ[a^2 - b^2, 0]
 

Maple [F]

\[\int \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}\, \tanh \left (d x +c \right )^{2}d x\]

Input:

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

Output:

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

Fricas [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integral(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^2, x)
 

Sympy [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int \sqrt {a + b \operatorname {sech}{\left (c + d x \right )}} \tanh ^{2}{\left (c + d x \right )}\, dx \] Input:

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

Output:

Integral(sqrt(a + b*sech(c + d*x))*tanh(c + d*x)**2, x)
 

Maxima [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^2, x)
 

Giac [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int { \sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{2} \,d x } \] Input:

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

Output:

integrate(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^2, x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int {\mathrm {tanh}\left (c+d\,x\right )}^2\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b \text {sech}(c+d x)} \tanh ^2(c+d x) \, dx=\int \sqrt {\mathrm {sech}\left (d x +c \right ) b +a}\, \tanh \left (d x +c \right )^{2}d x \] Input:

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

Output:

int(sqrt(sech(c + d*x)*b + a)*tanh(c + d*x)**2,x)