Integrand size = 23, antiderivative size = 344 \[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 (a-b) \sqrt {a+b} \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a b^2 d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a b d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a^2 d}-\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}} \]
2*(a-b)*coth(d*x+c)*EllipticE((a+b*sech(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/( a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c ))/(a-b))^(1/2)/a/b^2/d+2*coth(d*x+c)*EllipticF((a+b*sech(d*x+c))^(1/2)/(a +b)^(1/2),((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b*(1-sech(d*x+c))/(a+b))^(1/2) *(-b*(1+sech(d*x+c))/(a-b))^(1/2)/a/b/d+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))*(a+b)^(1/2)*(b*(1- sech(d*x+c))/(a+b))^(1/2)*(-b*(1+sech(d*x+c))/(a-b))^(1/2)/a^2/d-2*tanh(d* x+c)/a/d/(a+b*sech(d*x+c))^(1/2)
\[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx \]
Time = 1.49 (sec) , antiderivative size = 381, normalized size of antiderivative = 1.11, 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, 4549, 27, 3042, 4547, 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.
| \(\displaystyle \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cot \left (i c+i d x+\frac {\pi }{2}\right )^2}{\left (a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cot \left (\frac {1}{2} (2 i c+\pi )+i d x\right )^2}{\left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4382 |
| \(\displaystyle -\int \frac {\csc ^2\left (\frac {1}{2} (2 i c+\pi )+i d x\right )-1}{\left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )^2-1}{\left (a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4549 |
| \(\displaystyle \frac {2 \int \frac {a^2-b^2+\left (a^2-b^2\right ) \text {sech}^2(c+d x)}{2 \sqrt {a+b \text {sech}(c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {a^2-b^2+\left (a^2-b^2\right ) \text {sech}^2(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}+\frac {\int \frac {a^2-b^2+\left (a^2-b^2\right ) \csc \left (i c+i d x+\frac {\pi }{2}\right )^2}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4547 |
| \(\displaystyle \frac {\left (a^2-b^2\right ) \int \frac {\text {sech}(c+d x) (\text {sech}(c+d x)+1)}{\sqrt {a+b \text {sech}(c+d x)}}dx+\int \frac {a^2-b^2-\left (a^2-b^2\right ) \text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}+\frac {\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\int \frac {a^2-b^2+\left (b^2-a^2\right ) \csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4409 |
| \(\displaystyle -\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}+\frac {\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}}dx-\left (a^2-b^2\right ) \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}+\frac {\left (a^2-b^2\right ) \int \frac {1}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx-\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4271 |
| \(\displaystyle -\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}+\frac {-\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4319 |
| \(\displaystyle -\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}+\frac {\left (a^2-b^2\right ) \int \frac {\csc \left (i c+i d x+\frac {\pi }{2}\right ) \left (\csc \left (i c+i d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )}}dx+\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 4492 |
| \(\displaystyle \frac {\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 (a-b) \sqrt {a+b} \left (a^2-b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}+\frac {2 \sqrt {a+b} \left (a^2-b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 \tanh (c+d x)}{a d \sqrt {a+b \text {sech}(c+d x)}}\) |
((2*(a - b)*Sqrt[a + b]*(a^2 - b^2)*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^2)*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)]*Sqr t[-((b*(1 + Sech[c + d*x]))/(a - b))])/(b*d) + (2*Sqrt[a + b]*(a^2 - b^2)* Coth[c + d*x]*EllipticPi[(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))])/(a*d))/(a*(a^2 - b^2)) - (2*Tanh[c + d*x]) /(a*d*Sqrt[a + b*Sech[c + d*x]])
3.2.48.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
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]
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]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))/Sqrt[csc[(e_.) + (f_.)*(x_)]* (b_.) + (a_)], x_Symbol] :> Int[(A - 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, C}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_.))*(csc[(e_.) + (f_.)*(x_)]*(b_. ) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 + a^2*C)*Cot[e + f*x]*((a + b*Csc[ e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2)) Int[(a + b*Csc[e + f*x])^(m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*b* (A + C)*(m + 1)*Csc[e + f*x] + (A*b^2 + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, C}, x] && NeQ[a^2 - b^2, 0] && IntegerQ[2*m ] && LtQ[m, -1]
\[\int \frac {\tanh \left (d x +c \right )^{2}}{\left (a +b \,\operatorname {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}d x\]
\[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
integral(sqrt(b*sech(d*x + c) + a)*tanh(d*x + c)^2/(b^2*sech(d*x + c)^2 + 2*a*b*sech(d*x + c) + a^2), x)
\[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\tanh ^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {\tanh ^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^2}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]