Integrand size = 20, antiderivative size = 157 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {a^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 a^2 f \text {Chi}\left (\frac {c f}{d}+f x\right ) \sinh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {2 a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}+\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \] Output:
-4*a^2*cosh(1/2*f*x+1/2*e)^4/d/(d*x+c)-a^2*f*Chi(2*c*f/d+2*f*x)*sinh(-2*e+ 2*c*f/d)/d^2-2*a^2*f*Chi(c*f/d+f*x)*sinh(-e+c*f/d)/d^2+2*a^2*f*cosh(-e+c*f /d)*Shi(c*f/d+f*x)/d^2+a^2*f*cosh(-2*e+2*c*f/d)*Shi(2*c*f/d+2*f*x)/d^2
Time = 0.61 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.32 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 \left (-3 d-4 d \cosh (e+f x)-d \cosh (2 (e+f x))+2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )+4 f (c+d x) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right ) \sinh \left (e-\frac {c f}{d}\right )+4 c f \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 d f x \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+2 c f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )+2 d f x \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d^2 (c+d x)} \] Input:
Integrate[(a + a*Cosh[e + f*x])^2/(c + d*x)^2,x]
Output:
(a^2*(-3*d - 4*d*Cosh[e + f*x] - d*Cosh[2*(e + f*x)] + 2*f*(c + d*x)*CoshI ntegral[(2*f*(c + d*x))/d]*Sinh[2*e - (2*c*f)/d] + 4*f*(c + d*x)*CoshInteg ral[f*(c/d + x)]*Sinh[e - (c*f)/d] + 4*c*f*Cosh[e - (c*f)/d]*SinhIntegral[ f*(c/d + x)] + 4*d*f*x*Cosh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)] + 2*c*f *Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d] + 2*d*f*x*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*f*(c + d*x))/d]))/(2*d^2*(c + d*x))
Result contains complex when optimal does not.
Time = 0.60 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 3799, 3042, 3794, 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.
| \(\displaystyle \int \frac {(a \cosh (e+f x)+a)^2}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a+a \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^2}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle 4 a^2 \int \frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 a^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{2}\right )^4}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle 4 a^2 \left (-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {2 i f \int \left (-\frac {i \sinh (e+f x)}{4 (c+d x)}-\frac {i \sinh (2 e+2 f x)}{8 (c+d x)}\right )dx}{d}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 a^2 \left (-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {2 i f \left (-\frac {i \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 d}-\frac {i \text {Chi}\left (x f+\frac {c f}{d}\right ) \sinh \left (e-\frac {c f}{d}\right )}{4 d}-\frac {i \cosh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{4 d}-\frac {i \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{8 d}\right )}{d}\right )\) |
Input:
Int[(a + a*Cosh[e + f*x])^2/(c + d*x)^2,x]
Output:
4*a^2*(-(Cosh[e/2 + (f*x)/2]^4/(d*(c + d*x))) + ((2*I)*f*(((-1/8*I)*CoshIn tegral[(2*c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d - ((I/4)*CoshIntegral[( c*f)/d + f*x]*Sinh[e - (c*f)/d])/d - ((I/4)*Cosh[e - (c*f)/d]*SinhIntegral [(c*f)/d + f*x])/d - ((I/8)*Cosh[2*e - (2*c*f)/d]*SinhIntegral[(2*c*f)/d + 2*f*x])/d))/d)
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_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Simp[(2*a)^n Int[(c + d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + f*(x/2)]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2 - b^ 2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])
Time = 2.28 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.96
| method | result | size |
| risch | \(-\frac {f \,a^{2} {\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}+\frac {f \,a^{2} {\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{d^{2}}-\frac {f \,a^{2} {\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,a^{2} {\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}-\frac {f \,a^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}+\frac {f \,a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}-\frac {f \,a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {f \,a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}\) | \(308\) |
Input:
int((a+a*cosh(f*x+e))^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-f*a^2*exp(-f*x-e)/d/(d*f*x+c*f)+f*a^2/d^2*exp((c*f-d*e)/d)*Ei(1,f*x+e+(c* f-d*e)/d)-f*a^2/d^2*exp(f*x+e)/(c*f/d+f*x)-f*a^2/d^2*exp(-(c*f-d*e)/d)*Ei( 1,-f*x-e-(c*f-d*e)/d)-3/2*a^2/d/(d*x+c)-1/4*f*a^2*exp(-2*f*x-2*e)/d/(d*f*x +c*f)+1/2*f*a^2/d^2*exp(2*(c*f-d*e)/d)*Ei(1,2*f*x+2*e+2*(c*f-d*e)/d)-1/4*f *a^2/d^2*exp(2*f*x+2*e)/(c*f/d+f*x)-1/2*f*a^2/d^2*exp(-2*(c*f-d*e)/d)*Ei(1 ,-2*f*x-2*e-2*(c*f-d*e)/d)
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (156) = 312\).
Time = 0.10 (sec) , antiderivative size = 359, normalized size of antiderivative = 2.29 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {a^{2} d \cosh \left (f x + e\right )^{2} + a^{2} d \sinh \left (f x + e\right )^{2} + 4 \, a^{2} d \cosh \left (f x + e\right ) + 3 \, a^{2} d - 2 \, {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \cosh \left (-\frac {d e - c f}{d}\right ) - {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \cosh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right ) + 2 \, {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right )\right )} \sinh \left (-\frac {d e - c f}{d}\right ) + {\left ({\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right )\right )} \sinh \left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}{2 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:
integrate((a+a*cosh(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")
Output:
-1/2*(a^2*d*cosh(f*x + e)^2 + a^2*d*sinh(f*x + e)^2 + 4*a^2*d*cosh(f*x + e ) + 3*a^2*d - 2*((a^2*d*f*x + a^2*c*f)*Ei((d*f*x + c*f)/d) - (a^2*d*f*x + a^2*c*f)*Ei(-(d*f*x + c*f)/d))*cosh(-(d*e - c*f)/d) - ((a^2*d*f*x + a^2*c* f)*Ei(2*(d*f*x + c*f)/d) - (a^2*d*f*x + a^2*c*f)*Ei(-2*(d*f*x + c*f)/d))*c osh(-2*(d*e - c*f)/d) + 2*((a^2*d*f*x + a^2*c*f)*Ei((d*f*x + c*f)/d) + (a^ 2*d*f*x + a^2*c*f)*Ei(-(d*f*x + c*f)/d))*sinh(-(d*e - c*f)/d) + ((a^2*d*f* x + a^2*c*f)*Ei(2*(d*f*x + c*f)/d) + (a^2*d*f*x + a^2*c*f)*Ei(-2*(d*f*x + c*f)/d))*sinh(-2*(d*e - c*f)/d))/(d^3*x + c*d^2)
\[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=a^{2} \left (\int \frac {2 \cosh {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {\cosh ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx\right ) \] Input:
integrate((a+a*cosh(f*x+e))**2/(d*x+c)**2,x)
Output:
a**2*(Integral(2*cosh(e + f*x)/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral (cosh(e + f*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integral(1/(c**2 + 2* c*d*x + d**2*x**2), x))
Time = 0.14 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.16 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {1}{4} \, a^{2} {\left (\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {2}{d^{2} x + c d}\right )} - a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a^{2}}{d^{2} x + c d} \] Input:
integrate((a+a*cosh(f*x+e))^2/(d*x+c)^2,x, algorithm="maxima")
Output:
-1/4*a^2*(e^(-2*e + 2*c*f/d)*exp_integral_e(2, 2*(d*x + c)*f/d)/((d*x + c) *d) + e^(2*e - 2*c*f/d)*exp_integral_e(2, -2*(d*x + c)*f/d)/((d*x + c)*d) + 2/(d^2*x + c*d)) - a^2*(e^(-e + c*f/d)*exp_integral_e(2, (d*x + c)*f/d)/ ((d*x + c)*d) + e^(e - c*f/d)*exp_integral_e(2, -(d*x + c)*f/d)/((d*x + c) *d)) - a^2/(d^2*x + c*d)
Leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (156) = 312\).
Time = 0.19 (sec) , antiderivative size = 1134, normalized size of antiderivative = 7.22 \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((a+a*cosh(f*x+e))^2/(d*x+c)^2,x, algorithm="giac")
Output:
1/4*(2*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(2*(d*e - c*f)/d) - 2*a^2*d*e*f^2*Ei(2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(2*(d*e - c*f)/d) + 2*a^2*c*f^3*Ei(2*((d*x + c)*(d*e/(d*x + c ) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(2*(d*e - c*f)/d) + 4*(d*x + c)*a ^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^((d*e - c*f)/d) - 4*a^2*d*e*f^2*Ei((( d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^((d*e - c*f )/d) + 4*a^2*c*f^3*Ei(((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^((d*e - c*f)/d) - 4*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c* f)/d)*e^(-(d*e - c*f)/d) + 4*a^2*d*e*f^2*Ei(-((d*x + c)*(d*e/(d*x + c) - c *f/(d*x + c) + f) - d*e + c*f)/d)*e^(-(d*e - c*f)/d) - 4*a^2*c*f^3*Ei(-((d *x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-(d*e - c*f )/d) - 2*(d*x + c)*a^2*(d*e/(d*x + c) - c*f/(d*x + c) + f)*f^2*Ei(-2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f )/d) + 2*a^2*d*e*f^2*Ei(-2*((d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) - 2*a^2*c*f^3*Ei(-2*((d*x + c)*(d*e/( d*x + c) - c*f/(d*x + c) + f) - d*e + c*f)/d)*e^(-2*(d*e - c*f)/d) - a^2*d *f^2*e^(2*(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d) - 4*a^2*d*f^...
Timed out. \[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + a*cosh(e + f*x))^2/(c + d*x)^2,x)
Output:
int((a + a*cosh(e + f*x))^2/(c + d*x)^2, x)
\[ \int \frac {(a+a \cosh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^{2} \left (e^{3 e} \left (\int \frac {e^{2 f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2}+e^{3 e} \left (\int \frac {e^{2 f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d x +4 e^{2 e} \left (\int \frac {e^{f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2}+4 e^{2 e} \left (\int \frac {e^{f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d x +e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{2}+2 e^{2 f x +2 e} c d x +e^{2 f x +2 e} d^{2} x^{2}}d x \right ) c^{2}+e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{2}+2 e^{2 f x +2 e} c d x +e^{2 f x +2 e} d^{2} x^{2}}d x \right ) c d x +6 e^{e} x +4 \left (\int \frac {1}{e^{f x} c^{2}+2 e^{f x} c d x +e^{f x} d^{2} x^{2}}d x \right ) c^{2}+4 \left (\int \frac {1}{e^{f x} c^{2}+2 e^{f x} c d x +e^{f x} d^{2} x^{2}}d x \right ) c d x \right )}{4 e^{e} c \left (d x +c \right )} \] Input:
int((a+a*cosh(f*x+e))^2/(d*x+c)^2,x)
Output:
(a**2*(e**(3*e)*int(e**(2*f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c**2 + e**( 3*e)*int(e**(2*f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c*d*x + 4*e**(2*e)*int (e**(f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c**2 + 4*e**(2*e)*int(e**(f*x)/( c**2 + 2*c*d*x + d**2*x**2),x)*c*d*x + e**e*int(1/(e**(2*e + 2*f*x)*c**2 + 2*e**(2*e + 2*f*x)*c*d*x + e**(2*e + 2*f*x)*d**2*x**2),x)*c**2 + e**e*int (1/(e**(2*e + 2*f*x)*c**2 + 2*e**(2*e + 2*f*x)*c*d*x + e**(2*e + 2*f*x)*d* *2*x**2),x)*c*d*x + 6*e**e*x + 4*int(1/(e**(f*x)*c**2 + 2*e**(f*x)*c*d*x + e**(f*x)*d**2*x**2),x)*c**2 + 4*int(1/(e**(f*x)*c**2 + 2*e**(f*x)*c*d*x + e**(f*x)*d**2*x**2),x)*c*d*x))/(4*e**e*c*(c + d*x))