Integrand size = 21, antiderivative size = 54 \[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2} d}-\frac {2}{a d \sqrt {a+b \text {sech}(c+d x)}} \] Output:
2*arctanh((a+b*sech(d*x+c))^(1/2)/a^(1/2))/a^(3/2)/d-2/a/d/(a+b*sech(d*x+c ))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.16 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.80 \[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=-\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1+\frac {b \text {sech}(c+d x)}{a}\right )}{a d \sqrt {a+b \text {sech}(c+d x)}} \] Input:
Integrate[Tanh[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]
Output:
(-2*Hypergeometric2F1[-1/2, 1, 1/2, 1 + (b*Sech[c + d*x])/a])/(a*d*Sqrt[a + b*Sech[c + d*x]])
Time = 0.27 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {3042, 26, 4373, 61, 73, 220}
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.
\(\displaystyle \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {i \cot \left (i c+i d x+\frac {\pi }{2}\right )}{\left (a+b \csc \left (i c+i d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int \frac {\cot \left (\frac {1}{2} (2 i c+\pi )+i d x\right )}{\left (a+b \csc \left (\frac {1}{2} (2 i c+\pi )+i d x\right )\right )^{3/2}}dx\) |
\(\Big \downarrow \) 4373 |
\(\displaystyle -\frac {\int \frac {\cosh (c+d x)}{b (a+b \text {sech}(c+d x))^{3/2}}d(b \text {sech}(c+d x))}{d}\) |
\(\Big \downarrow \) 61 |
\(\displaystyle -\frac {\frac {\int \frac {\cosh (c+d x)}{b \sqrt {a+b \text {sech}(c+d x)}}d(b \text {sech}(c+d x))}{a}+\frac {2}{a \sqrt {a+b \text {sech}(c+d x)}}}{d}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle -\frac {\frac {2 \int \frac {1}{b^2 \text {sech}^2(c+d x)-a}d\sqrt {a+b \text {sech}(c+d x)}}{a}+\frac {2}{a \sqrt {a+b \text {sech}(c+d x)}}}{d}\) |
\(\Big \downarrow \) 220 |
\(\displaystyle -\frac {\frac {2}{a \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a}}\right )}{a^{3/2}}}{d}\) |
Input:
Int[Tanh[c + d*x]/(a + b*Sech[c + d*x])^(3/2),x]
Output:
-(((-2*ArcTanh[Sqrt[a + b*Sech[c + d*x]]/Sqrt[a]])/a^(3/2) + 2/(a*Sqrt[a + b*Sech[c + d*x]]))/d)
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(- 1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n _), x_Symbol] :> Simp[-(-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]
Time = 0.13 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.85
method | result | size |
derivativedivides | \(-\frac {\frac {2}{a \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}}{d}\) | \(46\) |
default | \(-\frac {\frac {2}{a \sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}-\frac {2 \,\operatorname {arctanh}\left (\frac {\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}}{d}\) | \(46\) |
Input:
int(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/d*(2/a/(a+b*sech(d*x+c))^(1/2)-2/a^(3/2)*arctanh((a+b*sech(d*x+c))^(1/2 )/a^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 329 vs. \(2 (46) = 92\).
Time = 0.50 (sec) , antiderivative size = 917, normalized size of antiderivative = 16.98 \[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\text {Too large to display} \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="fricas")
Output:
[1/2*((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*co sh(d*x + c) + b)*sinh(d*x + c) + a)*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)*s inh(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)) - 4*(a*cosh(d*x + c)^2 + 2*a*cosh(d*x + c)*sinh(d*x + c) + a*sinh(d*x + c)^2 + a)*sqrt((a*cosh(d*x + c) + b)/cosh(d*x + c)))/(a^ 3*d*cosh(d*x + c)^2 + a^3*d*sinh(d*x + c)^2 + 2*a^2*b*d*cosh(d*x + c) + a^ 3*d + 2*(a^3*d*cosh(d*x + c) + a^2*b*d)*sinh(d*x + c)), -((a*cosh(d*x + c) ^2 + a*sinh(d*x + c)^2 + 2*b*cosh(d*x + c) + 2*(a*cosh(d*x + c) + b)*sinh( d*x + c) + a)*sqrt(-a)*arctan((a*cosh(d*x + c)^2 + a*sinh(d*x + c)^2 + b*c osh(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 ...
\[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\tanh {\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c))**(3/2),x)
Output:
Integral(tanh(c + d*x)/(a + b*sech(c + d*x))**(3/2), x)
\[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="maxima")
Output:
integrate(tanh(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)
\[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int { \frac {\tanh \left (d x + c\right )}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x, algorithm="giac")
Output:
integrate(tanh(d*x + c)/(b*sech(d*x + c) + a)^(3/2), x)
Time = 2.82 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.93 \[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}}{\sqrt {a}}\right )}{a^{3/2}\,d}-\frac {2}{a\,d\,\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \] Input:
int(tanh(c + d*x)/(a + b/cosh(c + d*x))^(3/2),x)
Output:
(2*atanh((a + b/cosh(c + d*x))^(1/2)/a^(1/2)))/(a^(3/2)*d) - 2/(a*d*(a + b /cosh(c + d*x))^(1/2))
\[ \int \frac {\tanh (c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {sech}\left (d x +c \right ) b +a}\, \tanh \left (d x +c \right )}{\mathrm {sech}\left (d x +c \right )^{2} b^{2}+2 \,\mathrm {sech}\left (d x +c \right ) a b +a^{2}}d x \] Input:
int(tanh(d*x+c)/(a+b*sech(d*x+c))^(3/2),x)
Output:
int((sqrt(sech(c + d*x)*b + a)*tanh(c + d*x))/(sech(c + d*x)**2*b**2 + 2*s ech(c + d*x)*a*b + a**2),x)