Integrand size = 16, antiderivative size = 162 \[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=-\frac {b^2}{3 d^3 (c+d x)}+\frac {2 b^3 \text {Chi}\left (\frac {2 b c}{d}+2 b x\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )}{3 d^4}-\frac {b \cosh (a+b x) \sinh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {2 b^2 \sinh ^2(a+b x)}{3 d^3 (c+d x)}+\frac {2 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{3 d^4} \]
-1/3*b^2/d^3/(d*x+c)+2/3*b^3*cosh(2*a-2*b*c/d)*Shi(2*b*c/d+2*b*x)/d^4+2/3* b^3*Chi(2*b*c/d+2*b*x)*sinh(2*a-2*b*c/d)/d^4-1/3*b*cosh(b*x+a)*sinh(b*x+a) /d^2/(d*x+c)^2-1/3*sinh(b*x+a)^2/d/(d*x+c)^3-2/3*b^2*sinh(b*x+a)^2/d^3/(d* x+c)
Time = 0.58 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.76 \[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {4 b^3 \text {Chi}\left (\frac {2 b (c+d x)}{d}\right ) \sinh \left (2 a-\frac {2 b c}{d}\right )-\frac {d \left (\left (d^2+2 b^2 (c+d x)^2\right ) \cosh (2 (a+b x))+d (-d+b (c+d x) \sinh (2 (a+b x)))\right )}{(c+d x)^3}+4 b^3 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )}{6 d^4} \]
(4*b^3*CoshIntegral[(2*b*(c + d*x))/d]*Sinh[2*a - (2*b*c)/d] - (d*((d^2 + 2*b^2*(c + d*x)^2)*Cosh[2*(a + b*x)] + d*(-d + b*(c + d*x)*Sinh[2*(a + b*x )])))/(c + d*x)^3 + 4*b^3*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*(c + d*x ))/d])/(6*d^4)
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.05, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.062, Rules used = {3042, 25, 3795, 17, 25, 3042, 25, 3794, 27, 3042, 26, 3784, 26, 3042, 26, 3779, 3782}
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.
| \(\displaystyle \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin (i a+i b x)^2}{(c+d x)^4}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin (i a+i b x)^2}{(c+d x)^4}dx\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle -\frac {2 b^2 \int -\frac {\sinh ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}+\frac {b^2 \int \frac {1}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}\) |
| \(\Big \downarrow \) 17 |
| \(\displaystyle -\frac {2 b^2 \int -\frac {\sinh ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 b^2 \int \frac {\sinh ^2(a+b x)}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b^2 \int -\frac {\sin (i a+i b x)^2}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 b^2 \int \frac {\sin (i a+i b x)^2}{(c+d x)^2}dx}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {2 i b \int \frac {i \sinh (2 a+2 b x)}{2 (c+d x)}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}-\frac {b \int \frac {\sinh (2 a+2 b x)}{c+d x}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}-\frac {b \int -\frac {i \sin (2 i a+2 i b x)}{c+d x}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \int \frac {\sin (2 i a+2 i b x)}{c+d x}dx}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3784 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx+\cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {i \sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\cosh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx+i \cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sinh \left (\frac {2 b c}{d}+2 b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x+\frac {\pi }{2}\right )}{c+d x}dx+i \cosh \left (2 a-\frac {2 b c}{d}\right ) \int -\frac {i \sin \left (\frac {2 i b c}{d}+2 i b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x+\frac {\pi }{2}\right )}{c+d x}dx+\cosh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x\right )}{c+d x}dx\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3779 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \left (i \sinh \left (2 a-\frac {2 b c}{d}\right ) \int \frac {\sin \left (\frac {2 i b c}{d}+2 i b x+\frac {\pi }{2}\right )}{c+d x}dx+\frac {i \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
| \(\Big \downarrow \) 3782 |
| \(\displaystyle -\frac {2 b^2 \left (\frac {\sinh ^2(a+b x)}{d (c+d x)}+\frac {i b \left (\frac {i \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d}+\frac {i \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d}\right )}{d}\right )}{3 d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{3 d^2 (c+d x)^2}-\frac {\sinh ^2(a+b x)}{3 d (c+d x)^3}-\frac {b^2}{3 d^3 (c+d x)}\) |
-1/3*b^2/(d^3*(c + d*x)) - (b*Cosh[a + b*x]*Sinh[a + b*x])/(3*d^2*(c + d*x )^2) - Sinh[a + b*x]^2/(3*d*(c + d*x)^3) - (2*b^2*(Sinh[a + b*x]^2/(d*(c + d*x)) + (I*b*((I*CoshIntegral[(2*b*c)/d + 2*b*x]*Sinh[2*a - (2*b*c)/d])/d + (I*Cosh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/d))/d))/(3*d^ 2)
3.1.15.3.1 Defintions of rubi rules used
Int[(c_.)*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[c*((a + b*x)^(m + 1 )/(b*(m + 1))), x] /; FreeQ[{a, b, c, m}, x] && NeQ[m, -1]
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_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[I*(SinhIntegral[c*f*(fz/d) + f*fz*x]/d), x] /; FreeQ[{c, d, e, f , fz}, x] && EqQ[d*e - c*f*fz*I, 0]
Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbo l] :> Simp[CoshIntegral[c*f*(fz/d) + f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz }, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]
Int[sin[(e_.) + (f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[Cos[(d* e - c*f)/d] Int[Sin[c*(f/d) + f*x]/(c + d*x), x], x] + Simp[Sin[(d*e - c* f)/d] Int[Cos[c*(f/d) + f*x]/(c + d*x), x], x] /; FreeQ[{c, d, e, f}, x] && NeQ[d*e - c*f, 0]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Si mp[(c + d*x)^(m + 1)*(Sin[e + f*x]^n/(d*(m + 1))), x] - Simp[f*(n/(d*(m + 1 ))) Int[ExpandTrigReduce[(c + d*x)^(m + 1), Cos[e + f*x]*Sin[e + f*x]^(n - 1), x], x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && GeQ[m, -2] & & LtQ[m, -1]
Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbo l] :> Simp[(c + d*x)^(m + 1)*((b*Sin[e + f*x])^n/(d*(m + 1))), x] + (-Simp[ b*f*n*(c + d*x)^(m + 2)*Cos[e + f*x]*((b*Sin[e + f*x])^(n - 1)/(d^2*(m + 1) *(m + 2))), x] + Simp[b^2*f^2*n*((n - 1)/(d^2*(m + 1)*(m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[f^2*(n^2/(d^2*(m + 1)* (m + 2))) Int[(c + d*x)^(m + 2)*(b*Sin[e + f*x])^n, x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && LtQ[m, -2]
Leaf count of result is larger than twice the leaf count of optimal. \(554\) vs. \(2(150)=300\).
Time = 3.11 (sec) , antiderivative size = 555, normalized size of antiderivative = 3.43
| method | result | size |
| risch | \(\frac {1}{6 \left (d x +c \right )^{3} d}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} x^{2}}{6 d \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} c x}{3 d^{2} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{5} {\mathrm e}^{-2 b x -2 a} c^{2}}{6 d^{3} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{4} {\mathrm e}^{-2 b x -2 a} x}{12 d \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{4} {\mathrm e}^{-2 b x -2 a} c}{12 d^{2} \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}-\frac {b^{3} {\mathrm e}^{-2 b x -2 a}}{12 d \left (d^{3} x^{3} b^{3}+3 b^{3} c \,d^{2} x^{2}+3 b^{3} c^{2} d x +b^{3} c^{3}\right )}+\frac {b^{3} {\mathrm e}^{-\frac {2 \left (a d -b c \right )}{d}} \operatorname {Ei}_{1}\left (2 b x +2 a -\frac {2 \left (a d -b c \right )}{d}\right )}{3 d^{4}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{12 d^{4} \left (\frac {b c}{d}+b x \right )^{3}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{12 d^{4} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{3} {\mathrm e}^{2 b x +2 a}}{6 d^{4} \left (\frac {b c}{d}+b x \right )}-\frac {b^{3} {\mathrm e}^{\frac {2 a d -2 b c}{d}} \operatorname {Ei}_{1}\left (-2 b x -2 a -\frac {2 \left (-a d +b c \right )}{d}\right )}{3 d^{4}}\) | \(555\) |
1/6/(d*x+c)^3/d-1/6*b^5*exp(-2*b*x-2*a)/d/(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b ^3*c^2*d*x+b^3*c^3)*x^2-1/3*b^5*exp(-2*b*x-2*a)/d^2/(b^3*d^3*x^3+3*b^3*c*d ^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*c*x-1/6*b^5*exp(-2*b*x-2*a)/d^3/(b^3*d^3*x^3 +3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*c^2+1/12*b^4*exp(-2*b*x-2*a)/d/(b^ 3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*x+1/12*b^4*exp(-2*b*x-2*a )/d^2/(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)*c-1/12*b^3*exp(- 2*b*x-2*a)/d/(b^3*d^3*x^3+3*b^3*c*d^2*x^2+3*b^3*c^2*d*x+b^3*c^3)+1/3*b^3/d ^4*exp(-2*(a*d-b*c)/d)*Ei(1,2*b*x+2*a-2*(a*d-b*c)/d)-1/12*b^3/d^4*exp(2*b* x+2*a)/(b*c/d+b*x)^3-1/12*b^3/d^4*exp(2*b*x+2*a)/(b*c/d+b*x)^2-1/6*b^3/d^4 *exp(2*b*x+2*a)/(b*c/d+b*x)-1/3*b^3/d^4*exp(2*(a*d-b*c)/d)*Ei(1,-2*b*x-2*a -2*(-a*d+b*c)/d)
Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (150) = 300\).
Time = 0.27 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.54 \[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {d^{3} - {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (b d^{3} x + b c d^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) - {\left (2 \, b^{2} d^{3} x^{2} + 4 \, b^{2} c d^{2} x + 2 \, b^{2} c^{2} d + d^{3}\right )} \sinh \left (b x + a\right )^{2} + 2 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) + 2 \, {\left ({\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{6 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
1/6*(d^3 - (2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*cosh(b*x + a)^2 - 2*(b*d^3*x + b*c*d^2)*cosh(b*x + a)*sinh(b*x + a) - (2*b^2*d^3*x^2 + 4*b^2*c*d^2*x + 2*b^2*c^2*d + d^3)*sinh(b*x + a)^2 + 2*((b^3*d^3*x^3 + 3 *b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*Ei(2*(b*d*x + b*c)/d) - (b^3*d^3 *x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*Ei(-2*(b*d*x + b*c)/d))* cosh(-2*(b*c - a*d)/d) + 2*((b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*Ei(2*(b*d*x + b*c)/d) + (b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3 *c^2*d*x + b^3*c^3)*Ei(-2*(b*d*x + b*c)/d))*sinh(-2*(b*c - a*d)/d))/(d^7*x ^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
\[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {\sinh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{4}}\, dx \]
Time = 0.25 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.68 \[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {1}{6 \, {\left (d^{4} x^{3} + 3 \, c d^{3} x^{2} + 3 \, c^{2} d^{2} x + c^{3} d\right )}} - \frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{4}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{3} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{4}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{3} d} \]
1/6/(d^4*x^3 + 3*c*d^3*x^2 + 3*c^2*d^2*x + c^3*d) - 1/4*e^(-2*a + 2*b*c/d) *exp_integral_e(4, 2*(d*x + c)*b/d)/((d*x + c)^3*d) - 1/4*e^(2*a - 2*b*c/d )*exp_integral_e(4, -2*(d*x + c)*b/d)/((d*x + c)^3*d)
Leaf count of result is larger than twice the leaf count of optimal. 537 vs. \(2 (150) = 300\).
Time = 0.27 (sec) , antiderivative size = 537, normalized size of antiderivative = 3.31 \[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=\frac {4 \, b^{3} d^{3} x^{3} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 4 \, b^{3} d^{3} x^{3} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 12 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 12 \, b^{3} c d^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 12 \, b^{3} c^{2} d x {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 12 \, b^{3} c^{2} d x {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 2 \, b^{2} d^{3} x^{2} e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b^{2} d^{3} x^{2} e^{\left (-2 \, b x - 2 \, a\right )} + 4 \, b^{3} c^{3} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} - 4 \, b^{3} c^{3} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 4 \, b^{2} c d^{2} x e^{\left (2 \, b x + 2 \, a\right )} - 4 \, b^{2} c d^{2} x e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, b^{2} c^{2} d e^{\left (2 \, b x + 2 \, a\right )} - b d^{3} x e^{\left (2 \, b x + 2 \, a\right )} - 2 \, b^{2} c^{2} d e^{\left (-2 \, b x - 2 \, a\right )} + b d^{3} x e^{\left (-2 \, b x - 2 \, a\right )} - b c d^{2} e^{\left (2 \, b x + 2 \, a\right )} + b c d^{2} e^{\left (-2 \, b x - 2 \, a\right )} - d^{3} e^{\left (2 \, b x + 2 \, a\right )} - d^{3} e^{\left (-2 \, b x - 2 \, a\right )} + 2 \, d^{3}}{12 \, {\left (d^{7} x^{3} + 3 \, c d^{6} x^{2} + 3 \, c^{2} d^{5} x + c^{3} d^{4}\right )}} \]
1/12*(4*b^3*d^3*x^3*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) - 4*b^3*d^3*x^ 3*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2*b*c/d) + 12*b^3*c*d^2*x^2*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) - 12*b^3*c*d^2*x^2*Ei(-2*(b*d*x + b*c)/d)*e^( -2*a + 2*b*c/d) + 12*b^3*c^2*d*x*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) - 12*b^3*c^2*d*x*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2*b*c/d) - 2*b^2*d^3*x^2* e^(2*b*x + 2*a) - 2*b^2*d^3*x^2*e^(-2*b*x - 2*a) + 4*b^3*c^3*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) - 4*b^3*c^3*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2* b*c/d) - 4*b^2*c*d^2*x*e^(2*b*x + 2*a) - 4*b^2*c*d^2*x*e^(-2*b*x - 2*a) - 2*b^2*c^2*d*e^(2*b*x + 2*a) - b*d^3*x*e^(2*b*x + 2*a) - 2*b^2*c^2*d*e^(-2* b*x - 2*a) + b*d^3*x*e^(-2*b*x - 2*a) - b*c*d^2*e^(2*b*x + 2*a) + b*c*d^2* e^(-2*b*x - 2*a) - d^3*e^(2*b*x + 2*a) - d^3*e^(-2*b*x - 2*a) + 2*d^3)/(d^ 7*x^3 + 3*c*d^6*x^2 + 3*c^2*d^5*x + c^3*d^4)
Timed out. \[ \int \frac {\sinh ^2(a+b x)}{(c+d x)^4} \, dx=\int \frac {{\mathrm {sinh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^4} \,d x \]