Integrand size = 14, antiderivative size = 207 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\frac {x}{a^4}-\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{16 a^4 (a+b)^{7/2} d}-\frac {b \tanh (c+d x)}{6 a (a+b) d \left (a+b-b \tanh ^2(c+d x)\right )^3}-\frac {b (11 a+6 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b-b \tanh ^2(c+d x)\right )^2}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b-b \tanh ^2(c+d x)\right )} \]
x/a^4-1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)*arctanh(b^(1/2)*tanh(d*x+c)/( a+b)^(1/2))*b^(1/2)/a^4/(a+b)^(7/2)/d-1/6*b*tanh(d*x+c)/a/(a+b)/d/(a+b-b*t anh(d*x+c)^2)^3-1/24*b*(11*a+6*b)*tanh(d*x+c)/a^2/(a+b)^2/d/(a+b-b*tanh(d* x+c)^2)^2-1/16*b*(19*a^2+22*a*b+8*b^2)*tanh(d*x+c)/a^3/(a+b)^3/d/(a+b-b*ta nh(d*x+c)^2)
Result contains complex when optimal does not.
Time = 8.74 (sec) , antiderivative size = 1405, normalized size of antiderivative = 6.79 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\frac {\left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) (a+2 b+a \cosh (2 c+2 d x))^4 \text {sech}^8(c+d x) \left (\frac {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)}{256 a^4 \sqrt {a+b} d \sqrt {b \cosh (4 c)-b \sinh (4 c)}}-\frac {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)}{256 a^4 \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 )^4}+\frac {(a+2 b+a \cosh (2 c+2 d x)) \text {sech}(2 c) \text {sech}^8(c+d x) \left (480 a^6 d x \cosh (2 c)+3168 a^5 b d x \cosh (2 c)+8928 a^4 b^2 d x \cosh (2 c)+14112 a^3 b^3 d x \cosh (2 c)+13248 a^2 b^4 d x \cosh (2 c)+6912 a b^5 d x \cosh (2 c)+1536 b^6 d x \cosh (2 c)+360 a^6 d x \cosh (2 d x)+2232 a^5 b d x \cosh (2 d x)+5688 a^4 b^2 d x \cosh (2 d x)+7272 a^3 b^3 d x \cosh (2 d x)+4608 a^2 b^4 d x \cosh (2 d x)+1152 a b^5 d x \cosh (2 d x)+360 a^6 d x \cosh (4 c+2 d x)+2232 a^5 b d x \cosh (4 c+2 d x)+5688 a^4 b^2 d x \cosh (4 c+2 d x)+7272 a^3 b^3 d x \cosh (4 c+2 d x)+4608 a^2 b^4 d x \cosh (4 c+2 d x)+1152 a b^5 d x \cosh (4 c+2 d x)+144 a^6 d x \cosh (2 c+4 d x)+720 a^5 b d x \cosh (2 c+4 d x)+1296 a^4 b^2 d x \cosh (2 c+4 d x)+1008 a^3 b^3 d x \cosh (2 c+4 d x)+288 a^2 b^4 d x \cosh (2 c+4 d x)+144 a^6 d x \cosh (6 c+4 d x)+720 a^5 b d x \cosh (6 c+4 d x)+1296 a^4 b^2 d x \cosh (6 c+4 d x)+1008 a^3 b^3 d x \cosh (6 c+4 d x)+288 a^2 b^4 d x \cosh (6 c+4 d x)+24 a^6 d x \cosh (4 c+6 d x)+72 a^5 b d x \cosh (4 c+6 d x)+72 a^4 b^2 d x \cosh (4 c+6 d x)+24 a^3 b^3 d x \cosh (4 c+6 d x)+24 a^6 d x \cosh (8 c+6 d x)+72 a^5 b d x \cosh (8 c+6 d x)+72 a^4 b^2 d x \cosh (8 c+6 d x)+24 a^3 b^3 d x \cosh (8 c+6 d x)+870 a^5 b \sinh (2 c)+4292 a^4 b^2 \sinh (2 c)+8792 a^3 b^3 \sinh (2 c)+9936 a^2 b^4 \sinh (2 c)+5824 a b^5 \sinh (2 c)+1408 b^6 \sinh (2 c)-870 a^5 b \sinh (2 d x)-3792 a^4 b^2 \sinh (2 d x)-6432 a^3 b^3 \sinh (2 d x)-4608 a^2 b^4 \sinh (2 d x)-1248 a b^5 \sinh (2 d x)+435 a^5 b \sinh (4 c+2 d x)+2124 a^4 b^2 \sinh (4 c+2 d x)+3972 a^3 b^3 \sinh (4 c+2 d x)+3072 a^2 b^4 \sinh (4 c+2 d x)+864 a b^5 \sinh (4 c+2 d x)-435 a^5 b \sinh (2 c+4 d x)-1374 a^4 b^2 \sinh (2 c+4 d x)-1248 a^3 b^3 \sinh (2 c+4 d x)-384 a^2 b^4 \sinh (2 c+4 d x)+87 a^5 b \sinh (6 c+4 d x)+366 a^4 b^2 \sinh (6 c+4 d x)+408 a^3 b^3 \sinh (6 c+4 d x)+144 a^2 b^4 \sinh (6 c+4 d x)-87 a^5 b \sinh (4 c+6 d x)-116 a^4 b^2 \sinh (4 c+6 d x)-44 a^3 b^3 \sinh (4 c+6 d x)\right )}{3072 a^4 (a+b)^3 d \left (a+b \text {sech}^2(c+d x)\right )^4} \]
((35*a^3 + 70*a^2*b + 56*a*b^2 + 16*b^3)*(a + 2*b + a*Cosh[2*c + 2*d*x])^4 *Sech[c + d*x]^8*(((I/256)*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]*S qrt[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])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]] ) - ((I/256)*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])]*S inh[2*c])/(a^4*Sqrt[a + b]*d*Sqrt[b*Cosh[4*c] - b*Sinh[4*c]])))/((a + b)^3 *(a + b*Sech[c + d*x]^2)^4) + ((a + 2*b + a*Cosh[2*c + 2*d*x])*Sech[2*c]*S ech[c + d*x]^8*(480*a^6*d*x*Cosh[2*c] + 3168*a^5*b*d*x*Cosh[2*c] + 8928*a^ 4*b^2*d*x*Cosh[2*c] + 14112*a^3*b^3*d*x*Cosh[2*c] + 13248*a^2*b^4*d*x*Cosh [2*c] + 6912*a*b^5*d*x*Cosh[2*c] + 1536*b^6*d*x*Cosh[2*c] + 360*a^6*d*x*Co sh[2*d*x] + 2232*a^5*b*d*x*Cosh[2*d*x] + 5688*a^4*b^2*d*x*Cosh[2*d*x] + 72 72*a^3*b^3*d*x*Cosh[2*d*x] + 4608*a^2*b^4*d*x*Cosh[2*d*x] + 1152*a*b^5*d*x *Cosh[2*d*x] + 360*a^6*d*x*Cosh[4*c + 2*d*x] + 2232*a^5*b*d*x*Cosh[4*c + 2 *d*x] + 5688*a^4*b^2*d*x*Cosh[4*c + 2*d*x] + 7272*a^3*b^3*d*x*Cosh[4*c + 2 *d*x] + 4608*a^2*b^4*d*x*Cosh[4*c + 2*d*x] + 1152*a*b^5*d*x*Cosh[4*c + 2*d *x] + 144*a^6*d*x*Cosh[2*c + 4*d*x] + 720*a^5*b*d*x*Cosh[2*c + 4*d*x] + 12 96*a^4*b^2*d*x*Cosh[2*c + 4*d*x] + 1008*a^3*b^3*d*x*Cosh[2*c + 4*d*x] +...
Time = 0.46 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {3042, 4616, 316, 25, 402, 27, 402, 25, 397, 219, 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.
| \(\displaystyle \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a+b \sec (i c+i d x)^2\right )^4}dx\) |
| \(\Big \downarrow \) 4616 |
| \(\displaystyle \frac {\int \frac {1}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^4}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 316 |
| \(\displaystyle \frac {-\frac {\int -\frac {5 b \tanh ^2(c+d x)+6 a+b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {5 b \tanh ^2(c+d x)+6 a+b}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^3}d\tanh (c+d x)}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {3 \left (8 a^2+5 b a+2 b^2+b (11 a+6 b) \tanh ^2(c+d x)\right )}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {8 a^2+5 b a+2 b^2+b (11 a+6 b) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )^2}d\tanh (c+d x)}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\int -\frac {16 a^3+29 b a^2+26 b^2 a+8 b^3+b \left (19 a^2+22 b a+8 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\int \frac {16 a^3+29 b a^2+26 b^2 a+8 b^3+b \left (19 a^2+22 b a+8 b^2\right ) \tanh ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (-b \tanh ^2(c+d x)+a+b\right )}d\tanh (c+d x)}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 397 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{a}-\frac {b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {b \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \int \frac {1}{-b \tanh ^2(c+d x)+a+b}d\tanh (c+d x)}{a}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {3 \left (\frac {\frac {16 (a+b)^3 \text {arctanh}(\tanh (c+d x))}{a}-\frac {\sqrt {b} \left (35 a^3+70 a^2 b+56 a b^2+16 b^3\right ) \text {arctanh}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a \sqrt {a+b}}}{2 a (a+b)}-\frac {b \left (19 a^2+22 a b+8 b^2\right ) \tanh (c+d x)}{2 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )}\right )}{4 a (a+b)}-\frac {b (11 a+6 b) \tanh (c+d x)}{4 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^2}}{6 a (a+b)}-\frac {b \tanh (c+d x)}{6 a (a+b) \left (a-b \tanh ^2(c+d x)+b\right )^3}}{d}\) |
(-1/6*(b*Tanh[c + d*x])/(a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^3) + (-1/4* (b*(11*a + 6*b)*Tanh[c + d*x])/(a*(a + b)*(a + b - b*Tanh[c + d*x]^2)^2) + (3*(((16*(a + b)^3*ArcTanh[Tanh[c + d*x]])/a - (Sqrt[b]*(35*a^3 + 70*a^2* b + 56*a*b^2 + 16*b^3)*ArcTanh[(Sqrt[b]*Tanh[c + d*x])/Sqrt[a + b]])/(a*Sq rt[a + b]))/(2*a*(a + b)) - (b*(19*a^2 + 22*a*b + 8*b^2)*Tanh[c + d*x])/(2 *a*(a + b)*(a + b - b*Tanh[c + d*x]^2))))/(4*a*(a + b)))/(6*a*(a + b)))/d
3.2.69.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*a*(p + 1)*(b*c - a*d)) ), x] + Simp[1/(2*a*(p + 1)*(b*c - a*d)) Int[(a + b*x^2)^(p + 1)*(c + d*x ^2)^q*Simp[b*c + 2*(p + 1)*(b*c - a*d) + d*b*(2*(p + q + 2) + 1)*x^2, x], x ], x] /; FreeQ[{a, b, c, d, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && ! ( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*((c_) + (d_.)*(x_)^2)), x_ Symbol] :> Simp[(b*e - a*f)/(b*c - a*d) Int[1/(a + b*x^2), x], x] - Simp[ (d*e - c*f)/(b*c - a*d) Int[1/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e , f}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x _)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ (q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[((a_) + (b_.)*sec[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[ff/f Subst[Int[(a + b + b*ff^2*x^2)^p /(1 + ff^2*x^2), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f, p}, x] && NeQ[a + b, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(544\) vs. \(2(191)=382\).
Time = 3.26 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.63
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (\frac {-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{16 \left (a +b \right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{16 a^{3}+48 a^{2} b +48 a \,b^{2}+16 b^{3}}\right )}{a^{4}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(545\) |
| default | \(\frac {\frac {2 b \left (\frac {-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{16 \left (a +b \right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{8 \left (a +b \right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (145 a^{4}+148 a^{3} b +37 a^{2} b^{2}+2 a \,b^{3}+8 b^{4}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {\left (435 a^{3}+281 a^{2} b -66 a \,b^{2}-72 b^{3}\right ) a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{48 \left (a^{2}+2 a b +b^{2}\right )}-\frac {a \left (29 a^{2}+26 a b +8 b^{2}\right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{16 \left (a +b \right )}}{\left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} a +\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{4} b +2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} b +a +b \right )^{3}}+\frac {\left (35 a^{3}+70 a^{2} b +56 a \,b^{2}+16 b^{3}\right ) \left (-\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}-2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b}+\sqrt {a +b}\right )}{4 \sqrt {b}\, \sqrt {a +b}}\right )}{16 a^{3}+48 a^{2} b +48 a \,b^{2}+16 b^{3}}\right )}{a^{4}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{4}}+\frac {\ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{4}}}{d}\) | \(545\) |
| risch | \(\frac {x}{a^{4}}+\frac {b \left (435 a^{5} {\mathrm e}^{8 d x +8 c}+1408 b^{5} {\mathrm e}^{6 d x +6 c}+870 a^{5} {\mathrm e}^{4 d x +4 c}+87 a^{5}+4292 a^{4} b \,{\mathrm e}^{6 d x +6 c}+8792 a^{3} b^{2} {\mathrm e}^{6 d x +6 c}+9936 a^{2} b^{3} {\mathrm e}^{6 d x +6 c}+5824 a \,b^{4} {\mathrm e}^{6 d x +6 c}+3792 a^{4} b \,{\mathrm e}^{4 d x +4 c}+6432 a^{3} b^{2} {\mathrm e}^{4 d x +4 c}+4608 a^{2} b^{3} {\mathrm e}^{4 d x +4 c}+366 a^{4} b \,{\mathrm e}^{10 d x +10 c}+408 a^{3} b^{2} {\mathrm e}^{10 d x +10 c}+144 a^{2} b^{3} {\mathrm e}^{10 d x +10 c}+116 a^{4} b +44 a^{3} b^{2}+2124 a^{4} b \,{\mathrm e}^{8 d x +8 c}+1248 a \,b^{4} {\mathrm e}^{4 d x +4 c}+1374 a^{4} b \,{\mathrm e}^{2 d x +2 c}+1248 a^{3} b^{2} {\mathrm e}^{2 d x +2 c}+384 a^{2} b^{3} {\mathrm e}^{2 d x +2 c}+3972 a^{3} b^{2} {\mathrm e}^{8 d x +8 c}+3072 a^{2} b^{3} {\mathrm e}^{8 d x +8 c}+864 a \,b^{4} {\mathrm e}^{8 d x +8 c}+435 a^{5} {\mathrm e}^{2 d x +2 c}+87 a^{5} {\mathrm e}^{10 d x +10 c}+870 a^{5} {\mathrm e}^{6 d x +6 c}\right )}{24 a^{4} \left (a +b \right )^{3} d \left (a \,{\mathrm e}^{4 d x +4 c}+2 \,{\mathrm e}^{2 d x +2 c} a +4 b \,{\mathrm e}^{2 d x +2 c}+a \right )^{3}}+\frac {35 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right )}{32 \left (a +b \right )^{4} d a}+\frac {35 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right ) b}{16 \left (a +b \right )^{4} d \,a^{2}}+\frac {7 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right ) b^{2}}{4 \left (a +b \right )^{4} d \,a^{3}}+\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {\left (a +b \right ) b}+a +2 b}{a}\right ) b^{3}}{2 \left (a +b \right )^{4} d \,a^{4}}-\frac {35 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right )}{32 \left (a +b \right )^{4} d a}-\frac {35 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right ) b}{16 \left (a +b \right )^{4} d \,a^{2}}-\frac {7 \sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right ) b^{2}}{4 \left (a +b \right )^{4} d \,a^{3}}-\frac {\sqrt {\left (a +b \right ) b}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {\left (a +b \right ) b}-a -2 b}{a}\right ) b^{3}}{2 \left (a +b \right )^{4} d \,a^{4}}\) | \(872\) |
1/d*(2*b/a^4*((-1/16*a*(29*a^2+26*a*b+8*b^2)/(a+b)*tanh(1/2*d*x+1/2*c)^11- 1/48*(435*a^3+281*a^2*b-66*a*b^2-72*b^3)*a/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/ 2*c)^9-1/8*a*(145*a^4+148*a^3*b+37*a^2*b^2+2*a*b^3+8*b^4)/(a+b)/(a^2+2*a*b +b^2)*tanh(1/2*d*x+1/2*c)^7-1/8*a*(145*a^4+148*a^3*b+37*a^2*b^2+2*a*b^3+8* b^4)/(a^3+3*a^2*b+3*a*b^2+b^3)*tanh(1/2*d*x+1/2*c)^5-1/48*(435*a^3+281*a^2 *b-66*a*b^2-72*b^3)*a/(a^2+2*a*b+b^2)*tanh(1/2*d*x+1/2*c)^3-1/16*a*(29*a^2 +26*a*b+8*b^2)/(a+b)*tanh(1/2*d*x+1/2*c))/(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)^ 3+1/16*(35*a^3+70*a^2*b+56*a*b^2+16*b^3)/(a^3+3*a^2*b+3*a*b^2+b^3)*(-1/4/b ^(1/2)/(a+b)^(1/2)*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))+1/4/b^(1/2)/(a+b)^(1/2)*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))))-1/a^4*ln(tanh(1/2 *d*x+1/2*c)-1)+1/a^4*ln(1+tanh(1/2*d*x+1/2*c)))
Leaf count of result is larger than twice the leaf count of optimal. 8503 vs. \(2 (200) = 400\).
Time = 0.45 (sec) , antiderivative size = 17283, normalized size of antiderivative = 83.49 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 718 vs. \(2 (200) = 400\).
Time = 0.34 (sec) , antiderivative size = 718, normalized size of antiderivative = 3.47 \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\frac {{\left (35 \, a^{3} b + 70 \, a^{2} b^{2} + 56 \, a b^{3} + 16 \, b^{4}\right )} \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 )}{32 \, {\left (a^{7} + 3 \, a^{6} b + 3 \, a^{5} b^{2} + a^{4} b^{3}\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {87 \, a^{5} b + 116 \, a^{4} b^{2} + 44 \, a^{3} b^{3} + 3 \, {\left (145 \, a^{5} b + 458 \, a^{4} b^{2} + 416 \, a^{3} b^{3} + 128 \, a^{2} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 6 \, {\left (145 \, a^{5} b + 632 \, a^{4} b^{2} + 1072 \, a^{3} b^{3} + 768 \, a^{2} b^{4} + 208 \, a b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 2 \, {\left (435 \, a^{5} b + 2146 \, a^{4} b^{2} + 4396 \, a^{3} b^{3} + 4968 \, a^{2} b^{4} + 2912 \, a b^{5} + 704 \, b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (145 \, a^{5} b + 708 \, a^{4} b^{2} + 1324 \, a^{3} b^{3} + 1024 \, a^{2} b^{4} + 288 \, a b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 3 \, {\left (29 \, a^{5} b + 122 \, a^{4} b^{2} + 136 \, a^{3} b^{3} + 48 \, a^{2} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )}}{24 \, {\left (a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3} + 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 7 \, a^{7} b^{3} + 2 \, a^{6} b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, {\left (5 \, a^{10} + 31 \, a^{9} b + 79 \, a^{8} b^{2} + 101 \, a^{7} b^{3} + 64 \, a^{6} b^{4} + 16 \, a^{5} b^{5}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, {\left (5 \, a^{10} + 33 \, a^{9} b + 93 \, a^{8} b^{2} + 147 \, a^{7} b^{3} + 138 \, a^{6} b^{4} + 72 \, a^{5} b^{5} + 16 \, a^{4} b^{6}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + 3 \, {\left (5 \, a^{10} + 31 \, a^{9} b + 79 \, a^{8} b^{2} + 101 \, a^{7} b^{3} + 64 \, a^{6} b^{4} + 16 \, a^{5} b^{5}\right )} e^{\left (-8 \, d x - 8 \, c\right )} + 6 \, {\left (a^{10} + 5 \, a^{9} b + 9 \, a^{8} b^{2} + 7 \, a^{7} b^{3} + 2 \, a^{6} b^{4}\right )} e^{\left (-10 \, d x - 10 \, c\right )} + {\left (a^{10} + 3 \, a^{9} b + 3 \, a^{8} b^{2} + a^{7} b^{3}\right )} e^{\left (-12 \, d x - 12 \, c\right )}\right )} d} + \frac {d x + c}{a^{4} d} \]
1/32*(35*a^3*b + 70*a^2*b^2 + 56*a*b^3 + 16*b^4)*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^7 + 3*a^6*b + 3*a^5*b^2 + a^4*b^3)*sqrt((a + b)*b)*d) - 1/24*( 87*a^5*b + 116*a^4*b^2 + 44*a^3*b^3 + 3*(145*a^5*b + 458*a^4*b^2 + 416*a^3 *b^3 + 128*a^2*b^4)*e^(-2*d*x - 2*c) + 6*(145*a^5*b + 632*a^4*b^2 + 1072*a ^3*b^3 + 768*a^2*b^4 + 208*a*b^5)*e^(-4*d*x - 4*c) + 2*(435*a^5*b + 2146*a ^4*b^2 + 4396*a^3*b^3 + 4968*a^2*b^4 + 2912*a*b^5 + 704*b^6)*e^(-6*d*x - 6 *c) + 3*(145*a^5*b + 708*a^4*b^2 + 1324*a^3*b^3 + 1024*a^2*b^4 + 288*a*b^5 )*e^(-8*d*x - 8*c) + 3*(29*a^5*b + 122*a^4*b^2 + 136*a^3*b^3 + 48*a^2*b^4) *e^(-10*d*x - 10*c))/((a^10 + 3*a^9*b + 3*a^8*b^2 + a^7*b^3 + 6*(a^10 + 5* a^9*b + 9*a^8*b^2 + 7*a^7*b^3 + 2*a^6*b^4)*e^(-2*d*x - 2*c) + 3*(5*a^10 + 31*a^9*b + 79*a^8*b^2 + 101*a^7*b^3 + 64*a^6*b^4 + 16*a^5*b^5)*e^(-4*d*x - 4*c) + 4*(5*a^10 + 33*a^9*b + 93*a^8*b^2 + 147*a^7*b^3 + 138*a^6*b^4 + 72 *a^5*b^5 + 16*a^4*b^6)*e^(-6*d*x - 6*c) + 3*(5*a^10 + 31*a^9*b + 79*a^8*b^ 2 + 101*a^7*b^3 + 64*a^6*b^4 + 16*a^5*b^5)*e^(-8*d*x - 8*c) + 6*(a^10 + 5* a^9*b + 9*a^8*b^2 + 7*a^7*b^3 + 2*a^6*b^4)*e^(-10*d*x - 10*c) + (a^10 + 3* a^9*b + 3*a^8*b^2 + a^7*b^3)*e^(-12*d*x - 12*c))*d) + (d*x + c)/(a^4*d)
\[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\int { \frac {1}{{\left (b \operatorname {sech}\left (d x + c\right )^{2} + a\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {1}{\left (a+b \text {sech}^2(c+d x)\right )^4} \, dx=\int \frac {1}{{\left (a+\frac {b}{{\mathrm {cosh}\left (c+d\,x\right )}^2}\right )}^4} \,d x \]