Integrand size = 16, antiderivative size = 112 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=-\frac {\cosh ^2(a+b x)}{2 d (c+d x)^2}+\frac {b^2 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3}-\frac {b \cosh (a+b x) \sinh (a+b x)}{d^2 (c+d x)}+\frac {b^2 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{d^3} \] Output:
-1/2*cosh(b*x+a)^2/d/(d*x+c)^2+b^2*cosh(2*a-2*b*c/d)*Chi(2*b*c/d+2*b*x)/d^ 3-b*cosh(b*x+a)*sinh(b*x+a)/d^2/(d*x+c)+b^2*sinh(2*a-2*b*c/d)*Shi(2*b*c/d+ 2*b*x)/d^3
Time = 0.61 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.91 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=\frac {2 b^2 \cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b (c+d x)}{d}\right )-\frac {d \left (d \cosh ^2(a+b x)+b (c+d x) \sinh (2 (a+b x))\right )}{(c+d x)^2}+2 b^2 \sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b (c+d x)}{d}\right )}{2 d^3} \] Input:
Integrate[Cosh[a + b*x]^2/(c + d*x)^3,x]
Output:
(2*b^2*Cosh[2*a - (2*b*c)/d]*CoshIntegral[(2*b*(c + d*x))/d] - (d*(d*Cosh[ a + b*x]^2 + b*(c + d*x)*Sinh[2*(a + b*x)]))/(c + d*x)^2 + 2*b^2*Sinh[2*a - (2*b*c)/d]*SinhIntegral[(2*b*(c + d*x))/d])/(2*d^3)
Time = 0.50 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.32, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3042, 3795, 16, 3042, 3793, 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.
| \(\displaystyle \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^2}{(c+d x)^3}dx\) |
| \(\Big \downarrow \) 3795 |
| \(\displaystyle \frac {2 b^2 \int \frac {\cosh ^2(a+b x)}{c+d x}dx}{d^2}-\frac {b^2 \int \frac {1}{c+d x}dx}{d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\cosh ^2(a+b x)}{2 d (c+d x)^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 b^2 \int \frac {\cosh ^2(a+b x)}{c+d x}dx}{d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac {b^2 \log (c+d x)}{d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 b^2 \int \frac {\sin \left (i a+i b x+\frac {\pi }{2}\right )^2}{c+d x}dx}{d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac {b^2 \log (c+d x)}{d^3}\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {2 b^2 \int \left (\frac {\cosh (2 a+2 b x)}{2 (c+d x)}+\frac {1}{2 (c+d x)}\right )dx}{d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac {b^2 \log (c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2 b^2 \left (\frac {\cosh \left (2 a-\frac {2 b c}{d}\right ) \text {Chi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\sinh \left (2 a-\frac {2 b c}{d}\right ) \text {Shi}\left (\frac {2 b c}{d}+2 b x\right )}{2 d}+\frac {\log (c+d x)}{2 d}\right )}{d^2}-\frac {b \sinh (a+b x) \cosh (a+b x)}{d^2 (c+d x)}-\frac {\cosh ^2(a+b x)}{2 d (c+d x)^2}-\frac {b^2 \log (c+d x)}{d^3}\) |
Input:
Int[Cosh[a + b*x]^2/(c + d*x)^3,x]
Output:
-1/2*Cosh[a + b*x]^2/(d*(c + d*x)^2) - (b^2*Log[c + d*x])/d^3 - (b*Cosh[a + b*x]*Sinh[a + b*x])/(d^2*(c + d*x)) + (2*b^2*((Cosh[2*a - (2*b*c)/d]*Cos hIntegral[(2*b*c)/d + 2*b*x])/(2*d) + Log[c + d*x]/(2*d) + (Sinh[2*a - (2* b*c)/d]*SinhIntegral[(2*b*c)/d + 2*b*x])/(2*d)))/d^2
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && 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. \(298\) vs. \(2(110)=220\).
Time = 1.64 (sec) , antiderivative size = 299, normalized size of antiderivative = 2.67
| method | result | size |
| risch | \(-\frac {1}{4 \left (d x +c \right )^{2} d}+\frac {b^{3} {\mathrm e}^{-2 b x -2 a} x}{4 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}+\frac {b^{3} {\mathrm e}^{-2 b x -2 a} c}{4 d^{2} \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{2} {\mathrm e}^{-2 b x -2 a}}{8 d \left (d^{2} x^{2} b^{2}+2 b^{2} c d x +b^{2} c^{2}\right )}-\frac {b^{2} {\mathrm e}^{-\frac {2 \left (a d -c b \right )}{d}} \operatorname {expIntegral}_{1}\left (2 b x +2 a -\frac {2 \left (a d -c b \right )}{d}\right )}{2 d^{3}}-\frac {b^{2} {\mathrm e}^{2 b x +2 a}}{8 d^{3} \left (\frac {b c}{d}+b x \right )^{2}}-\frac {b^{2} {\mathrm e}^{2 b x +2 a}}{4 d^{3} \left (\frac {b c}{d}+b x \right )}-\frac {b^{2} {\mathrm e}^{\frac {2 a d -2 c b}{d}} \operatorname {expIntegral}_{1}\left (-2 b x -2 a -\frac {2 \left (-a d +c b \right )}{d}\right )}{2 d^{3}}\) | \(299\) |
Input:
int(cosh(b*x+a)^2/(d*x+c)^3,x,method=_RETURNVERBOSE)
Output:
-1/4/(d*x+c)^2/d+1/4*b^3*exp(-2*b*x-2*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^ 2)*x+1/4*b^3*exp(-2*b*x-2*a)/d^2/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)*c-1/8*b ^2*exp(-2*b*x-2*a)/d/(b^2*d^2*x^2+2*b^2*c*d*x+b^2*c^2)-1/2*b^2/d^3*exp(-2* (a*d-b*c)/d)*Ei(1,2*b*x+2*a-2*(a*d-b*c)/d)-1/8*b^2/d^3*exp(2*b*x+2*a)/(b*c /d+b*x)^2-1/4*b^2/d^3*exp(2*b*x+2*a)/(b*c/d+b*x)-1/2*b^2/d^3*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. 278 vs. \(2 (110) = 220\).
Time = 0.10 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.48 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=-\frac {d^{2} \cosh \left (b x + a\right )^{2} + d^{2} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b d^{2} x + b c d\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + d^{2} - 2 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right ) - 2 \, {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) - {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (b c - a d\right )}}{d}\right )}{4 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate(cosh(b*x+a)^2/(d*x+c)^3,x, algorithm="fricas")
Output:
-1/4*(d^2*cosh(b*x + a)^2 + d^2*sinh(b*x + a)^2 + 4*(b*d^2*x + b*c*d)*cosh (b*x + a)*sinh(b*x + a) + d^2 - 2*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*E i(2*(b*d*x + b*c)/d) + (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-2*(b*d*x + b*c)/d))*cosh(-2*(b*c - a*d)/d) - 2*((b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^ 2)*Ei(2*(b*d*x + b*c)/d) - (b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*Ei(-2*(b* d*x + b*c)/d))*sinh(-2*(b*c - a*d)/d))/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
\[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=\int \frac {\cosh ^{2}{\left (a + b x \right )}}{\left (c + d x\right )^{3}}\, dx \] Input:
integrate(cosh(b*x+a)**2/(d*x+c)**3,x)
Output:
Integral(cosh(a + b*x)**2/(c + d*x)**3, x)
Time = 0.08 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.88 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=-\frac {1}{4 \, {\left (d^{3} x^{2} + 2 \, c d^{2} x + c^{2} d\right )}} - \frac {e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} E_{3}\left (\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{2} d} - \frac {e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} E_{3}\left (-\frac {2 \, {\left (d x + c\right )} b}{d}\right )}{4 \, {\left (d x + c\right )}^{2} d} \] Input:
integrate(cosh(b*x+a)^2/(d*x+c)^3,x, algorithm="maxima")
Output:
-1/4/(d^3*x^2 + 2*c*d^2*x + c^2*d) - 1/4*e^(-2*a + 2*b*c/d)*exp_integral_e (3, 2*(d*x + c)*b/d)/((d*x + c)^2*d) - 1/4*e^(2*a - 2*b*c/d)*exp_integral_ e(3, -2*(d*x + c)*b/d)/((d*x + c)^2*d)
Leaf count of result is larger than twice the leaf count of optimal. 330 vs. \(2 (110) = 220\).
Time = 0.13 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.95 \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=\frac {4 \, b^{2} d^{2} x^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + 4 \, b^{2} d^{2} x^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 8 \, b^{2} c d x {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + 8 \, b^{2} c d x {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} + 4 \, b^{2} c^{2} {\rm Ei}\left (\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (2 \, a - \frac {2 \, b c}{d}\right )} + 4 \, b^{2} c^{2} {\rm Ei}\left (-\frac {2 \, {\left (b d x + b c\right )}}{d}\right ) e^{\left (-2 \, a + \frac {2 \, b c}{d}\right )} - 2 \, b d^{2} x e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b d^{2} x e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, b c d e^{\left (2 \, b x + 2 \, a\right )} + 2 \, b c d e^{\left (-2 \, b x - 2 \, a\right )} - d^{2} e^{\left (2 \, b x + 2 \, a\right )} - d^{2} e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, d^{2}}{8 \, {\left (d^{5} x^{2} + 2 \, c d^{4} x + c^{2} d^{3}\right )}} \] Input:
integrate(cosh(b*x+a)^2/(d*x+c)^3,x, algorithm="giac")
Output:
1/8*(4*b^2*d^2*x^2*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) + 4*b^2*d^2*x^2 *Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2*b*c/d) + 8*b^2*c*d*x*Ei(2*(b*d*x + b*c )/d)*e^(2*a - 2*b*c/d) + 8*b^2*c*d*x*Ei(-2*(b*d*x + b*c)/d)*e^(-2*a + 2*b* c/d) + 4*b^2*c^2*Ei(2*(b*d*x + b*c)/d)*e^(2*a - 2*b*c/d) + 4*b^2*c^2*Ei(-2 *(b*d*x + b*c)/d)*e^(-2*a + 2*b*c/d) - 2*b*d^2*x*e^(2*b*x + 2*a) + 2*b*d^2 *x*e^(-2*b*x - 2*a) - 2*b*c*d*e^(2*b*x + 2*a) + 2*b*c*d*e^(-2*b*x - 2*a) - d^2*e^(2*b*x + 2*a) - d^2*e^(-2*b*x - 2*a) - 2*d^2)/(d^5*x^2 + 2*c*d^4*x + c^2*d^3)
Timed out. \[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=\int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^3} \,d x \] Input:
int(cosh(a + b*x)^2/(c + d*x)^3,x)
Output:
int(cosh(a + b*x)^2/(c + d*x)^3, x)
\[ \int \frac {\cosh ^2(a+b x)}{(c+d x)^3} \, dx=\frac {e^{2 a} \left (\int \frac {e^{2 b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) c^{2} d +2 e^{2 a} \left (\int \frac {e^{2 b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) c \,d^{2} x +e^{2 a} \left (\int \frac {e^{2 b x}}{d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}}d x \right ) d^{3} x^{2}+\left (\int \frac {1}{e^{2 b x +2 a} c^{3}+3 e^{2 b x +2 a} c^{2} d x +3 e^{2 b x +2 a} c \,d^{2} x^{2}+e^{2 b x +2 a} d^{3} x^{3}}d x \right ) c^{2} d +2 \left (\int \frac {1}{e^{2 b x +2 a} c^{3}+3 e^{2 b x +2 a} c^{2} d x +3 e^{2 b x +2 a} c \,d^{2} x^{2}+e^{2 b x +2 a} d^{3} x^{3}}d x \right ) c \,d^{2} x +\left (\int \frac {1}{e^{2 b x +2 a} c^{3}+3 e^{2 b x +2 a} c^{2} d x +3 e^{2 b x +2 a} c \,d^{2} x^{2}+e^{2 b x +2 a} d^{3} x^{3}}d x \right ) d^{3} x^{2}-1}{4 d \left (d^{2} x^{2}+2 c d x +c^{2}\right )} \] Input:
int(cosh(b*x+a)^2/(d*x+c)^3,x)
Output:
(e**(2*a)*int(e**(2*b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x )*c**2*d + 2*e**(2*a)*int(e**(2*b*x)/(c**3 + 3*c**2*d*x + 3*c*d**2*x**2 + d**3*x**3),x)*c*d**2*x + e**(2*a)*int(e**(2*b*x)/(c**3 + 3*c**2*d*x + 3*c* d**2*x**2 + d**3*x**3),x)*d**3*x**2 + int(1/(e**(2*a + 2*b*x)*c**3 + 3*e** (2*a + 2*b*x)*c**2*d*x + 3*e**(2*a + 2*b*x)*c*d**2*x**2 + e**(2*a + 2*b*x) *d**3*x**3),x)*c**2*d + 2*int(1/(e**(2*a + 2*b*x)*c**3 + 3*e**(2*a + 2*b*x )*c**2*d*x + 3*e**(2*a + 2*b*x)*c*d**2*x**2 + e**(2*a + 2*b*x)*d**3*x**3), x)*c*d**2*x + int(1/(e**(2*a + 2*b*x)*c**3 + 3*e**(2*a + 2*b*x)*c**2*d*x + 3*e**(2*a + 2*b*x)*c*d**2*x**2 + e**(2*a + 2*b*x)*d**3*x**3),x)*d**3*x**2 - 1)/(4*d*(c**2 + 2*c*d*x + d**2*x**2))