Integrand size = 23, antiderivative size = 75 \[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {(a+b) (a+3 b) \coth (c+d x)}{d}-\frac {(a+b)^2 \coth ^3(c+d x)}{3 d}+\frac {b (2 a+3 b) \tanh (c+d x)}{d}-\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
(a+b)*(a+3*b)*coth(d*x+c)/d-1/3*(a+b)^2*coth(d*x+c)^3/d+b*(2*a+3*b)*tanh(d *x+c)/d-1/3*b^2*tanh(d*x+c)^3/d
Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(75)=150\).
Time = 6.65 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.01 \[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {\text {csch}(2 c) \text {csch}^3(2 (c+d x)) \left (8 a (a+2 b) \sinh (2 c)-6 (a+2 b)^2 \sinh (2 d x)-3 a^2 \sinh (2 (c+d x))-6 a b \sinh (2 (c+d x))+a^2 \sinh (6 (c+d x))+2 a b \sinh (6 (c+d x))+3 a^2 \sinh (4 c+2 d x)+a^2 \sinh (4 c+6 d x)+8 a b \sinh (4 c+6 d x)+8 b^2 \sinh (4 c+6 d x)\right )}{6 d} \]
-1/6*(Csch[2*c]*Csch[2*(c + d*x)]^3*(8*a*(a + 2*b)*Sinh[2*c] - 6*(a + 2*b) ^2*Sinh[2*d*x] - 3*a^2*Sinh[2*(c + d*x)] - 6*a*b*Sinh[2*(c + d*x)] + a^2*S inh[6*(c + d*x)] + 2*a*b*Sinh[6*(c + d*x)] + 3*a^2*Sinh[4*c + 2*d*x] + a^2 *Sinh[4*c + 6*d*x] + 8*a*b*Sinh[4*c + 6*d*x] + 8*b^2*Sinh[4*c + 6*d*x]))/d
Time = 0.29 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.89, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3042, 4620, 355, 2009}
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.
\(\displaystyle \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a+b \sec (i c+i d x)^2\right )^2}{\sin (i c+i d x)^4}dx\) |
\(\Big \downarrow \) 4620 |
\(\displaystyle \frac {\int \coth ^4(c+d x) \left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 355 |
\(\displaystyle \frac {\int \left ((a+b)^2 \coth ^4(c+d x)+(-a-3 b) (a+b) \coth ^2(c+d x)-b^2 \tanh ^2(c+d x)+b (2 a+3 b)\right )d\tanh (c+d x)}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b (2 a+3 b) \tanh (c+d x)-\frac {1}{3} (a+b)^2 \coth ^3(c+d x)+(a+b) (a+3 b) \coth (c+d x)-\frac {1}{3} b^2 \tanh ^3(c+d x)}{d}\) |
((a + b)*(a + 3*b)*Coth[c + d*x] - ((a + b)^2*Coth[c + d*x]^3)/3 + b*(2*a + 3*b)*Tanh[c + d*x] - (b^2*Tanh[c + d*x]^3)/3)/d
3.1.16.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q _.), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^q, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[p, 0] & & IGtQ[q, 0]
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]
Time = 120.04 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.72
method | result | size |
risch | \(-\frac {4 \left (3 a^{2} {\mathrm e}^{8 d x +8 c}+8 a^{2} {\mathrm e}^{6 d x +6 c}+16 a b \,{\mathrm e}^{6 d x +6 c}+6 a^{2} {\mathrm e}^{4 d x +4 c}+24 a b \,{\mathrm e}^{4 d x +4 c}+24 \,{\mathrm e}^{4 d x +4 c} b^{2}-a^{2}-8 a b -8 b^{2}\right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3} \left ({\mathrm e}^{2 d x +2 c}-1\right )^{3}}\) | \(129\) |
derivativedivides | \(\frac {a^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+2 a b \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )}+\frac {4}{3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}+\frac {8 \tanh \left (d x +c \right )}{3}\right )+b^{2} \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )^{3}}+\frac {2}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}+8 \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )}{d}\) | \(138\) |
default | \(\frac {a^{2} \left (\frac {2}{3}-\frac {\operatorname {csch}\left (d x +c \right )^{2}}{3}\right ) \coth \left (d x +c \right )+2 a b \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )}+\frac {4}{3 \sinh \left (d x +c \right ) \cosh \left (d x +c \right )}+\frac {8 \tanh \left (d x +c \right )}{3}\right )+b^{2} \left (-\frac {1}{3 \sinh \left (d x +c \right )^{3} \cosh \left (d x +c \right )^{3}}+\frac {2}{\sinh \left (d x +c \right ) \cosh \left (d x +c \right )^{3}}+8 \left (\frac {2}{3}+\frac {\operatorname {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )\right )}{d}\) | \(138\) |
-4/3*(3*a^2*exp(8*d*x+8*c)+8*a^2*exp(6*d*x+6*c)+16*a*b*exp(6*d*x+6*c)+6*a^ 2*exp(4*d*x+4*c)+24*a*b*exp(4*d*x+4*c)+24*exp(4*d*x+4*c)*b^2-a^2-8*a*b-8*b ^2)/d/(exp(2*d*x+2*c)+1)^3/(exp(2*d*x+2*c)-1)^3
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (71) = 142\).
Time = 0.25 (sec) , antiderivative size = 408, normalized size of antiderivative = 5.44 \[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {8 \, {\left ({\left (a^{2} - 4 \, a b - 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{4} + 8 \, {\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} - 4 \, a b - 4 \, b^{2}\right )} \sinh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - 4 \, a b - 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} + 4 \, a b\right )} \sinh \left (d x + c\right )^{2} + 3 \, a^{2} + 12 \, a b + 12 \, b^{2} + 8 \, {\left ({\left (a^{2} + 2 \, a b + 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}{3 \, {\left (d \cosh \left (d x + c\right )^{8} + 56 \, d \cosh \left (d x + c\right )^{3} \sinh \left (d x + c\right )^{5} + 28 \, d \cosh \left (d x + c\right )^{2} \sinh \left (d x + c\right )^{6} + 8 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{7} + d \sinh \left (d x + c\right )^{8} - 4 \, d \cosh \left (d x + c\right )^{4} + 2 \, {\left (35 \, d \cosh \left (d x + c\right )^{4} - 2 \, d\right )} \sinh \left (d x + c\right )^{4} + 8 \, {\left (7 \, d \cosh \left (d x + c\right )^{5} - d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left (7 \, d \cosh \left (d x + c\right )^{6} - 6 \, d \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left (d \cosh \left (d x + c\right )^{7} - d \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + 3 \, d\right )}} \]
-8/3*((a^2 - 4*a*b - 4*b^2)*cosh(d*x + c)^4 + 8*(a^2 + 2*a*b + 2*b^2)*cosh (d*x + c)*sinh(d*x + c)^3 + (a^2 - 4*a*b - 4*b^2)*sinh(d*x + c)^4 + 4*(a^2 + 2*a*b)*cosh(d*x + c)^2 + 2*(3*(a^2 - 4*a*b - 4*b^2)*cosh(d*x + c)^2 + 2 *a^2 + 4*a*b)*sinh(d*x + c)^2 + 3*a^2 + 12*a*b + 12*b^2 + 8*((a^2 + 2*a*b + 2*b^2)*cosh(d*x + c)^3 + (a^2 + 2*a*b)*cosh(d*x + c))*sinh(d*x + c))/(d* cosh(d*x + c)^8 + 56*d*cosh(d*x + c)^3*sinh(d*x + c)^5 + 28*d*cosh(d*x + c )^2*sinh(d*x + c)^6 + 8*d*cosh(d*x + c)*sinh(d*x + c)^7 + d*sinh(d*x + c)^ 8 - 4*d*cosh(d*x + c)^4 + 2*(35*d*cosh(d*x + c)^4 - 2*d)*sinh(d*x + c)^4 + 8*(7*d*cosh(d*x + c)^5 - d*cosh(d*x + c))*sinh(d*x + c)^3 + 4*(7*d*cosh(d *x + c)^6 - 6*d*cosh(d*x + c)^2)*sinh(d*x + c)^2 + 8*(d*cosh(d*x + c)^7 - d*cosh(d*x + c)^3)*sinh(d*x + c) + 3*d)
\[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}^{4}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 285 vs. \(2 (71) = 142\).
Time = 0.21 (sec) , antiderivative size = 285, normalized size of antiderivative = 3.80 \[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=\frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {32}{3} \, a b {\left (\frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}} - \frac {1}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} + \frac {32}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}} - \frac {1}{d {\left (3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 3 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-12 \, d x - 12 \, c\right )} - 1\right )}}\right )} \]
4/3*a^2*(3*e^(-2*d*x - 2*c)/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e^(-6*d*x - 6*c) - 1)) - 1/(d*(3*e^(-2*d*x - 2*c) - 3*e^(-4*d*x - 4*c) + e ^(-6*d*x - 6*c) - 1))) + 32/3*a*b*(2*e^(-2*d*x - 2*c)/(d*(2*e^(-2*d*x - 2* c) - 2*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) - 1)) - 1/(d*(2*e^(-2*d*x - 2*c ) - 2*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c) - 1))) + 32/3*b^2*(3*e^(-4*d*x - 4*c)/(d*(3*e^(-4*d*x - 4*c) - 3*e^(-8*d*x - 8*c) + e^(-12*d*x - 12*c) - 1 )) - 1/(d*(3*e^(-4*d*x - 4*c) - 3*e^(-8*d*x - 8*c) + e^(-12*d*x - 12*c) - 1)))
Time = 0.29 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.53 \[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {4 \, {\left (3 \, a^{2} e^{\left (8 \, d x + 8 \, c\right )} + 8 \, a^{2} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a b e^{\left (6 \, d x + 6 \, c\right )} + 6 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 24 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 24 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - a^{2} - 8 \, a b - 8 \, b^{2}\right )}}{3 \, d {\left (e^{\left (4 \, d x + 4 \, c\right )} - 1\right )}^{3}} \]
-4/3*(3*a^2*e^(8*d*x + 8*c) + 8*a^2*e^(6*d*x + 6*c) + 16*a*b*e^(6*d*x + 6* c) + 6*a^2*e^(4*d*x + 4*c) + 24*a*b*e^(4*d*x + 4*c) + 24*b^2*e^(4*d*x + 4* c) - a^2 - 8*a*b - 8*b^2)/(d*(e^(4*d*x + 4*c) - 1)^3)
Time = 2.16 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.53 \[ \int \text {csch}^4(c+d x) \left (a+b \text {sech}^2(c+d x)\right )^2 \, dx=-\frac {4\,\left (6\,a^2\,{\mathrm {e}}^{4\,c+4\,d\,x}-a^2-8\,b^2-8\,a\,b+8\,a^2\,{\mathrm {e}}^{6\,c+6\,d\,x}+3\,a^2\,{\mathrm {e}}^{8\,c+8\,d\,x}+24\,b^2\,{\mathrm {e}}^{4\,c+4\,d\,x}+24\,a\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}+16\,a\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}\right )}{3\,d\,{\left ({\mathrm {e}}^{4\,c+4\,d\,x}-1\right )}^3} \]