Integrand size = 23, antiderivative size = 126 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {15 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{8 (a+b)^{7/2} d}-\frac {15 \coth (c+d x)}{8 (a+b)^3 d}+\frac {\coth (c+d x)}{4 (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^2}+\frac {5 \coth (c+d x)}{8 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
-15/8*coth(d*x+c)/(a+b)^3/d+15/8*arctanh(b^(1/2)*tanh(d*x+c)/(a+b)^(1/2))* b^(1/2)/(a+b)^(7/2)/d+1/4*coth(d*x+c)/(a+b)/d/(a+b-b*tanh(d*x+c)^2)^2+5/8* coth(d*x+c)/(a+b)^2/d/(a+b-b*tanh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 8.35 (sec) , antiderivative size = 981, normalized size of antiderivative = 7.79 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\frac {(a+2 b+a \cosh (2 c+2 d x))^3 \text {sech}^6(c+d x) \left (-\frac {15 i b \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \cosh (2 c)}{64 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {15 i b \arctan \left (\text {sech}(d x) \left (-\frac {i \cosh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}+\frac {i \sinh (2 c)}{2 \sqrt {a+b} \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right ) (-a \sinh (d x)-2 b \sinh (d x)+a \sinh (2 c+d x))\right ) \sinh (2 c)}{64 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}\right )}{(a+b)^3 \left (a+b \text {sech}^2(c+d x)\right )^3}+\frac {(a+2 b+a \cosh (2 c+2 d x)) \text {csch}(c) \text {csch}(c+d x) \text {sech}(2 c) \text {sech}^6(c+d x) \left (-32 a^4 \sinh (d x)-64 a^3 b \sinh (d x)+22 a^2 b^2 \sinh (d x)+80 a b^3 \sinh (d x)+16 b^4 \sinh (d x)+32 a^4 \sinh (3 d x)+46 a^3 b \sinh (3 d x)-54 a^2 b^2 \sinh (3 d x)-8 a b^3 \sinh (3 d x)-48 a^4 \sinh (2 c-d x)-128 a^3 b \sinh (2 c-d x)-106 a^2 b^2 \sinh (2 c-d x)+80 a b^3 \sinh (2 c-d x)+16 b^4 \sinh (2 c-d x)+48 a^4 \sinh (2 c+d x)+146 a^3 b \sinh (2 c+d x)+182 a^2 b^2 \sinh (2 c+d x)+80 a b^3 \sinh (2 c+d x)+16 b^4 \sinh (2 c+d x)-32 a^4 \sinh (4 c+d x)-82 a^3 b \sinh (4 c+d x)-54 a^2 b^2 \sinh (4 c+d x)-80 a b^3 \sinh (4 c+d x)-16 b^4 \sinh (4 c+d x)-8 a^4 \sinh (2 c+3 d x)+18 a^3 b \sinh (2 c+3 d x)+54 a^2 b^2 \sinh (2 c+3 d x)+8 a b^3 \sinh (2 c+3 d x)+32 a^4 \sinh (4 c+3 d x)+73 a^3 b \sinh (4 c+3 d x)+24 a^2 b^2 \sinh (4 c+3 d x)+8 a b^3 \sinh (4 c+3 d x)-8 a^4 \sinh (6 c+3 d x)-9 a^3 b \sinh (6 c+3 d x)-24 a^2 b^2 \sinh (6 c+3 d x)-8 a b^3 \sinh (6 c+3 d x)+8 a^4 \sinh (2 c+5 d x)-9 a^3 b \sinh (2 c+5 d x)-2 a^2 b^2 \sinh (2 c+5 d x)+9 a^3 b \sinh (4 c+5 d x)+2 a^2 b^2 \sinh (4 c+5 d x)+8 a^4 \sinh (6 c+5 d x)\right )}{512 a^2 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^3} \]
((a + 2*b + a*Cosh[2*c + 2*d*x])^3*Sech[c + d*x]^6*((((-15*I)/64)*b*ArcTan [Sech[d*x]*(((-1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4* c]]) + ((I/2)*Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(- (a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Cosh[2*c])/(Sqrt[a + b ]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + (((15*I)/64)*b*ArcTan[Sech[d*x]*((( -1/2*I)*Cosh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]) + ((I/2)* Sinh[2*c])/(Sqrt[a + b]*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]]))*(-(a*Sinh[d*x]) - 2*b*Sinh[d*x] + a*Sinh[2*c + d*x])]*Sinh[2*c])/(Sqrt[a + b]*d*Sqrt[b*Cos h[4*c] - b*Sinh[4*c]])))/((a + b)^3*(a + b*Sech[c + d*x]^2)^3) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Csch[c]*Csch[c + d*x]*Sech[2*c]*Sech[c + d*x]^6*(- 32*a^4*Sinh[d*x] - 64*a^3*b*Sinh[d*x] + 22*a^2*b^2*Sinh[d*x] + 80*a*b^3*Si nh[d*x] + 16*b^4*Sinh[d*x] + 32*a^4*Sinh[3*d*x] + 46*a^3*b*Sinh[3*d*x] - 5 4*a^2*b^2*Sinh[3*d*x] - 8*a*b^3*Sinh[3*d*x] - 48*a^4*Sinh[2*c - d*x] - 128 *a^3*b*Sinh[2*c - d*x] - 106*a^2*b^2*Sinh[2*c - d*x] + 80*a*b^3*Sinh[2*c - d*x] + 16*b^4*Sinh[2*c - d*x] + 48*a^4*Sinh[2*c + d*x] + 146*a^3*b*Sinh[2 *c + d*x] + 182*a^2*b^2*Sinh[2*c + d*x] + 80*a*b^3*Sinh[2*c + d*x] + 16*b^ 4*Sinh[2*c + d*x] - 32*a^4*Sinh[4*c + d*x] - 82*a^3*b*Sinh[4*c + d*x] - 54 *a^2*b^2*Sinh[4*c + d*x] - 80*a*b^3*Sinh[4*c + d*x] - 16*b^4*Sinh[4*c + d* x] - 8*a^4*Sinh[2*c + 3*d*x] + 18*a^3*b*Sinh[2*c + 3*d*x] + 54*a^2*b^2*Sin h[2*c + 3*d*x] + 8*a*b^3*Sinh[2*c + 3*d*x] + 32*a^4*Sinh[4*c + 3*d*x] +...
Time = 0.31 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 4620, 253, 253, 264, 221}
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.
| \(\displaystyle \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\sin (i c+i d x)^2 \left (a+b \sec (i c+i d x)^2\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{\left (b \sec (i c+i d x)^2+a\right )^3 \sin (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 4620 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {5 \int \frac {\coth ^2(c+d x)}{\left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 253 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \int \frac {\coth ^2(c+d x)}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 264 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {b \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a+b}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {5 \left (\frac {3 \left (\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {\coth (c+d x)}{a+b}\right )}{2 (a+b)}+\frac {\coth (c+d x)}{2 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 (a+b)}+\frac {\coth (c+d x)}{4 (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{d}\) |
(Coth[c + d*x]/(4*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + (5*((3*((Sqrt[b ]*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a + b)^(3/2) - Coth[c + d *x]/(a + b)))/(2*(a + b)) + Coth[c + d*x]/(2*(a + b)*(a + b - b*Tanh[c + d *x]^2))))/(4*(a + b)))/d
3.1.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-(c*x )^(m + 1))*((a + b*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Simp[(m + 2*p + 3)/( 2*a*(p + 1)) Int[(c*x)^m*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, m }, x] && LtQ[p, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(c*x)^( m + 1)*((a + b*x^2)^(p + 1)/(a*c*(m + 1))), x] - Simp[b*((m + 2*p + 3)/(a*c ^2*(m + 1))) Int[(c*x)^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, p }, x] && LtQ[m, -1] && IntBinomialQ[a, b, c, 2, m, p, x]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^(n_))^(p_.)*sin[(e_.) + (f_.)*(x_ )]^(m_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff^(m + 1)/f Subst[Int[x^m*(ExpandToSum[a + b*(1 + ff^2*x^2)^(n/2), x]^p/(1 + f f^2*x^2)^(m/2 + 1)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && IntegerQ[m/2] && IntegerQ[n/2]
Leaf count of result is larger than twice the leaf count of optimal. \(818\) vs. \(2(110)=220\).
Time = 0.26 (sec) , antiderivative size = 819, normalized size of antiderivative = 6.50
\[\text {Expression too large to display}\]
-1/2/d/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)-1/2/d/(a+b)^3/tanh(1/ 2*d*x+1/2*c)+9/4/d*b/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c)^ 4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d* x+1/2*c)^7*a+9/4/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c )^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2* d*x+1/2*c)^7+27/4/d*b/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c) ^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2*d *x+1/2*c)^5*a-1/4/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2* c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2 *d*x+1/2*c)^5+27/4/d*b/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c )^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2* d*x+1/2*c)^3*a-1/4/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2 *c)^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/ 2*d*x+1/2*c)^3+9/4/d*b/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c )^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2* d*x+1/2*c)*a+9/4/d*b^2/(a+b)^3/(tanh(1/2*d*x+1/2*c)^4*a+tanh(1/2*d*x+1/2*c )^4*b+2*tanh(1/2*d*x+1/2*c)^2*a-2*tanh(1/2*d*x+1/2*c)^2*b+a+b)^2*tanh(1/2* d*x+1/2*c)+15/16/d*b^(1/2)/(a+b)^(7/2)*ln((a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^ 2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)+(a+b)^(1/2))-15/16/d*b^(1/2)/(a+b)^(7/2)*l n(-(a+b)^(1/2)*tanh(1/2*d*x+1/2*c)^2+2*tanh(1/2*d*x+1/2*c)*b^(1/2)-(a+b...
Leaf count of result is larger than twice the leaf count of optimal. 3499 vs. \(2 (116) = 232\).
Time = 0.34 (sec) , antiderivative size = 7275, normalized size of antiderivative = 57.74 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\text {Too large to display} \]
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {\operatorname {csch}^{2}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{3}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (116) = 232\).
Time = 0.35 (sec) , antiderivative size = 533, normalized size of antiderivative = 4.23 \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=-\frac {15 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {8 \, a^{4} - 9 \, a^{3} b - 2 \, a^{2} b^{2} + 2 \, {\left (16 \, a^{4} + 23 \, a^{3} b - 27 \, a^{2} b^{2} - 4 \, a b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (24 \, a^{4} + 64 \, a^{3} b + 53 \, a^{2} b^{2} - 40 \, a b^{3} - 8 \, b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (16 \, a^{4} + 41 \, a^{3} b + 27 \, a^{2} b^{2} + 40 \, a b^{3} + 8 \, b^{4}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (8 \, a^{4} + 9 \, a^{3} b + 24 \, a^{2} b^{2} + 8 \, a b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}}{4 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3} + {\left (3 \, a^{7} + 17 \, a^{6} b + 33 \, a^{5} b^{2} + 27 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{7} + 7 \, a^{6} b + 23 \, a^{5} b^{2} + 37 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, {\left (a^{7} + 7 \, a^{6} b + 23 \, a^{5} b^{2} + 37 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 8 \, a^{2} b^{5}\right )} e^{\left (-6 \, d x - 6 \, c\right )} - {\left (3 \, a^{7} + 17 \, a^{6} b + 33 \, a^{5} b^{2} + 27 \, a^{4} b^{3} + 8 \, a^{3} b^{4}\right )} e^{\left (-8 \, d x - 8 \, c\right )} - {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} e^{\left (-10 \, d x - 10 \, c\right )}\right )} d} \]
-15/16*b*log((a*e^(-2*d*x - 2*c) + a + 2*b - 2*sqrt((a + b)*b))/(a*e^(-2*d *x - 2*c) + a + 2*b + 2*sqrt((a + b)*b)))/((a^3 + 3*a^2*b + 3*a*b^2 + b^3) *sqrt((a + b)*b)*d) - 1/4*(8*a^4 - 9*a^3*b - 2*a^2*b^2 + 2*(16*a^4 + 23*a^ 3*b - 27*a^2*b^2 - 4*a*b^3)*e^(-2*d*x - 2*c) + 2*(24*a^4 + 64*a^3*b + 53*a ^2*b^2 - 40*a*b^3 - 8*b^4)*e^(-4*d*x - 4*c) + 2*(16*a^4 + 41*a^3*b + 27*a^ 2*b^2 + 40*a*b^3 + 8*b^4)*e^(-6*d*x - 6*c) + (8*a^4 + 9*a^3*b + 24*a^2*b^2 + 8*a*b^3)*e^(-8*d*x - 8*c))/((a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3 + (3*a ^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8*a^3*b^4)*e^(-2*d*x - 2*c) + 2* (a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^2*b^5)*e^(-4*d *x - 4*c) - 2*(a^7 + 7*a^6*b + 23*a^5*b^2 + 37*a^4*b^3 + 28*a^3*b^4 + 8*a^ 2*b^5)*e^(-6*d*x - 6*c) - (3*a^7 + 17*a^6*b + 33*a^5*b^2 + 27*a^4*b^3 + 8* a^3*b^4)*e^(-8*d*x - 8*c) - (a^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*e^(-10*d *x - 10*c))*d)
\[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int { \frac {\operatorname {csch}\left (d x + c\right )^{2}}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\text {csch}^2(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^3} \, dx=\int \frac {{\mathrm {cosh}\left (c+d\,x\right )}^6}{{\mathrm {sinh}\left (c+d\,x\right )}^2\,{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^3} \,d x \]