Integrand size = 23, antiderivative size = 79 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {1}{2} (a+b) (a+5 b) x+\frac {(a+b) (a+5 b) \tanh (c+d x)}{2 d}+\frac {(a+b)^2 \sinh ^2(c+d x) \tanh (c+d x)}{2 d}+\frac {b^2 \tanh ^3(c+d x)}{3 d} \]
-1/2*(a+b)*(a+5*b)*x+1/2*(a+b)*(a+5*b)*tanh(d*x+c)/d+1/2*(a+b)^2*sinh(d*x+ c)^2*tanh(d*x+c)/d+1/3*b^2*tanh(d*x+c)^3/d
Time = 1.55 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.89 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {-6 \left (a^2+6 a b+5 b^2\right ) (c+d x)+3 (a+b)^2 \sinh (2 (c+d x))+4 b \left (6 a+7 b-b \text {sech}^2(c+d x)\right ) \tanh (c+d x)}{12 d} \]
(-6*(a^2 + 6*a*b + 5*b^2)*(c + d*x) + 3*(a + b)^2*Sinh[2*(c + d*x)] + 4*b* (6*a + 7*b - b*Sech[c + d*x]^2)*Tanh[c + d*x])/(12*d)
Time = 0.30 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 25, 4146, 366, 363, 262, 219}
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.
| \(\displaystyle \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\sin (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^2dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \sin (i c+i d x)^2 \left (a-b \tan (i c+i d x)^2\right )^2dx\) |
| \(\Big \downarrow \) 4146 |
| \(\displaystyle \frac {\int \frac {\tanh ^2(c+d x) \left (b \tanh ^2(c+d x)+a\right )^2}{\left (1-\tanh ^2(c+d x)\right )^2}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 366 |
| \(\displaystyle \frac {\frac {(a+b)^2 \tanh ^3(c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}-\frac {1}{2} \int \frac {\tanh ^2(c+d x) \left (a^2+6 b a+3 b^2+2 b^2 \tanh ^2(c+d x)\right )}{1-\tanh ^2(c+d x)}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2}{3} b^2 \tanh ^3(c+d x)-(a+b) (a+5 b) \int \frac {\tanh ^2(c+d x)}{1-\tanh ^2(c+d x)}d\tanh (c+d x)\right )+\frac {(a+b)^2 \tanh ^3(c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2}{3} b^2 \tanh ^3(c+d x)-(a+b) (a+5 b) \left (\int \frac {1}{1-\tanh ^2(c+d x)}d\tanh (c+d x)-\tanh (c+d x)\right )\right )+\frac {(a+b)^2 \tanh ^3(c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {2}{3} b^2 \tanh ^3(c+d x)-(a+b) (a+5 b) (\text {arctanh}(\tanh (c+d x))-\tanh (c+d x))\right )+\frac {(a+b)^2 \tanh ^3(c+d x)}{2 \left (1-\tanh ^2(c+d x)\right )}}{d}\) |
(((a + b)^2*Tanh[c + d*x]^3)/(2*(1 - Tanh[c + d*x]^2)) + (-((a + b)*(a + 5 *b)*(ArcTanh[Tanh[c + d*x]] - Tanh[c + d*x])) + (2*b^2*Tanh[c + d*x]^3)/3) /2)/d
3.1.11.3.1 Defintions of rubi rules used
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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 0]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^2, x_Symbol] :> Simp[(-(b*c - a*d)^2)*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(2*a* b^2*e*(p + 1))), x] + Simp[1/(2*a*b^2*(p + 1)) Int[(e*x)^m*(a + b*x^2)^(p + 1)*Simp[(b*c - a*d)^2*(m + 1) + 2*b^2*c^2*(p + 1) + 2*a*b*d^2*(p + 1)*x^ 2, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && LtQ[p , -1]
Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_ )])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Sim p[c*(ff^(m + 1)/f) Subst[Int[x^m*((a + b*(ff*x)^n)^p/(c^2 + ff^2*x^2)^(m/ 2 + 1)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x ] && IntegerQ[m/2]
Time = 0.88 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.49
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )}{d}\) | \(118\) |
| default | \(\frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+2 a b \left (\frac {\sinh \left (d x +c \right )^{3}}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )+b^{2} \left (\frac {\sinh \left (d x +c \right )^{5}}{2 \cosh \left (d x +c \right )^{3}}-\frac {5 d x}{2}-\frac {5 c}{2}+\frac {5 \tanh \left (d x +c \right )}{2}+\frac {5 \tanh \left (d x +c \right )^{3}}{6}\right )}{d}\) | \(118\) |
| risch | \(-\frac {a^{2} x}{2}-3 a b x -\frac {5 b^{2} x}{2}+\frac {{\mathrm e}^{2 d x +2 c} a^{2}}{8 d}+\frac {{\mathrm e}^{2 d x +2 c} a b}{4 d}+\frac {{\mathrm e}^{2 d x +2 c} b^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{2}}{8 d}-\frac {{\mathrm e}^{-2 d x -2 c} a b}{4 d}-\frac {{\mathrm e}^{-2 d x -2 c} b^{2}}{8 d}-\frac {2 b \left (6 a \,{\mathrm e}^{4 d x +4 c}+9 b \,{\mathrm e}^{4 d x +4 c}+12 \,{\mathrm e}^{2 d x +2 c} a +12 b \,{\mathrm e}^{2 d x +2 c}+6 a +7 b \right )}{3 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{3}}\) | \(193\) |
1/d*(a^2*(1/2*cosh(d*x+c)*sinh(d*x+c)-1/2*d*x-1/2*c)+2*a*b*(1/2*sinh(d*x+c )^3/cosh(d*x+c)-3/2*d*x-3/2*c+3/2*tanh(d*x+c))+b^2*(1/2*sinh(d*x+c)^5/cosh (d*x+c)^3-5/2*d*x-5/2*c+5/2*tanh(d*x+c)+5/6*tanh(d*x+c)^3))
Leaf count of result is larger than twice the leaf count of optimal. 291 vs. \(2 (71) = 142\).
Time = 0.26 (sec) , antiderivative size = 291, normalized size of antiderivative = 3.68 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{5} - 4 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} d x + 12 \, a b + 14 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} - 12 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} d x + 12 \, a b + 14 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + {\left (30 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + 9 \, a^{2} + 66 \, a b + 65 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} - 12 \, {\left (3 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} d x + 12 \, a b + 14 \, b^{2}\right )} \cosh \left (d x + c\right ) + 3 \, {\left (5 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (9 \, a^{2} + 66 \, a b + 65 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, a^{2} + 20 \, a b + 10 \, b^{2}\right )} \sinh \left (d x + c\right )}{24 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
1/24*(3*(a^2 + 2*a*b + b^2)*sinh(d*x + c)^5 - 4*(3*(a^2 + 6*a*b + 5*b^2)*d *x + 12*a*b + 14*b^2)*cosh(d*x + c)^3 - 12*(3*(a^2 + 6*a*b + 5*b^2)*d*x + 12*a*b + 14*b^2)*cosh(d*x + c)*sinh(d*x + c)^2 + (30*(a^2 + 2*a*b + b^2)*c osh(d*x + c)^2 + 9*a^2 + 66*a*b + 65*b^2)*sinh(d*x + c)^3 - 12*(3*(a^2 + 6 *a*b + 5*b^2)*d*x + 12*a*b + 14*b^2)*cosh(d*x + c) + 3*(5*(a^2 + 2*a*b + b ^2)*cosh(d*x + c)^4 + (9*a^2 + 66*a*b + 65*b^2)*cosh(d*x + c)^2 + 2*a^2 + 20*a*b + 10*b^2)*sinh(d*x + c))/(d*cosh(d*x + c)^3 + 3*d*cosh(d*x + c)*sin h(d*x + c)^2 + 3*d*cosh(d*x + c))
\[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \sinh ^{2}{\left (c + d x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (71) = 142\).
Time = 0.22 (sec) , antiderivative size = 217, normalized size of antiderivative = 2.75 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=-\frac {1}{8} \, a^{2} {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{24} \, b^{2} {\left (\frac {60 \, {\left (d x + c\right )}}{d} + \frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {121 \, e^{\left (-2 \, d x - 2 \, c\right )} + 201 \, e^{\left (-4 \, d x - 4 \, c\right )} + 147 \, e^{\left (-6 \, d x - 6 \, c\right )} + 3}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3 \, e^{\left (-6 \, d x - 6 \, c\right )} + e^{\left (-8 \, d x - 8 \, c\right )}\right )}}\right )} - \frac {1}{4} \, a b {\left (\frac {12 \, {\left (d x + c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \]
-1/8*a^2*(4*x - e^(2*d*x + 2*c)/d + e^(-2*d*x - 2*c)/d) - 1/24*b^2*(60*(d* x + c)/d + 3*e^(-2*d*x - 2*c)/d - (121*e^(-2*d*x - 2*c) + 201*e^(-4*d*x - 4*c) + 147*e^(-6*d*x - 6*c) + 3)/(d*(e^(-2*d*x - 2*c) + 3*e^(-4*d*x - 4*c) + 3*e^(-6*d*x - 6*c) + e^(-8*d*x - 8*c)))) - 1/4*a*b*(12*(d*x + c)/d + e^ (-2*d*x - 2*c)/d - (17*e^(-2*d*x - 2*c) + 1)/(d*(e^(-2*d*x - 2*c) + e^(-4* d*x - 4*c))))
Leaf count of result is larger than twice the leaf count of optimal. 213 vs. \(2 (71) = 142\).
Time = 0.34 (sec) , antiderivative size = 213, normalized size of antiderivative = 2.70 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 12 \, {\left (a^{2} + 6 \, a b + 5 \, b^{2}\right )} {\left (d x + c\right )} + 3 \, {\left (2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 10 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a^{2} - 2 \, a b - b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - \frac {16 \, {\left (6 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 12 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 12 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 6 \, a b + 7 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{24 \, d} \]
1/24*(3*a^2*e^(2*d*x + 2*c) + 6*a*b*e^(2*d*x + 2*c) + 3*b^2*e^(2*d*x + 2*c ) - 12*(a^2 + 6*a*b + 5*b^2)*(d*x + c) + 3*(2*a^2*e^(2*d*x + 2*c) + 12*a*b *e^(2*d*x + 2*c) + 10*b^2*e^(2*d*x + 2*c) - a^2 - 2*a*b - b^2)*e^(-2*d*x - 2*c) - 16*(6*a*b*e^(4*d*x + 4*c) + 9*b^2*e^(4*d*x + 4*c) + 12*a*b*e^(2*d* x + 2*c) + 12*b^2*e^(2*d*x + 2*c) + 6*a*b + 7*b^2)/(e^(2*d*x + 2*c) + 1)^3 )/d
Time = 1.87 (sec) , antiderivative size = 248, normalized size of antiderivative = 3.14 \[ \int \sinh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx=\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-x\,\left (\frac {a^2}{2}+3\,a\,b+\frac {5\,b^2}{2}\right )-\frac {\frac {2\,\left (3\,b^2+2\,a\,b\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (3\,b^2+2\,a\,b\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {2\,\left (3\,b^2+2\,a\,b\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-\frac {\frac {2\,\left (b^2+2\,a\,b\right )}{3\,d}+\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (3\,b^2+2\,a\,b\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1} \]
(exp(2*c + 2*d*x)*(a + b)^2)/(8*d) - x*(3*a*b + a^2/2 + (5*b^2)/2) - ((2*( 2*a*b + 3*b^2))/(3*d) + (4*exp(2*c + 2*d*x)*(2*a*b + b^2))/(3*d) + (2*exp( 4*c + 4*d*x)*(2*a*b + 3*b^2))/(3*d))/(3*exp(2*c + 2*d*x) + 3*exp(4*c + 4*d *x) + exp(6*c + 6*d*x) + 1) - (2*(2*a*b + 3*b^2))/(3*d*(exp(2*c + 2*d*x) + 1)) - (exp(- 2*c - 2*d*x)*(a + b)^2)/(8*d) - ((2*(2*a*b + b^2))/(3*d) + ( 2*exp(2*c + 2*d*x)*(2*a*b + 3*b^2))/(3*d))/(2*exp(2*c + 2*d*x) + exp(4*c + 4*d*x) + 1)