Integrand size = 21, antiderivative size = 95 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=-\frac {a}{d (c+d x)}+\frac {i a f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {i a \sinh (e+f x)}{d (c+d x)}+\frac {i a f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2} \] Output:
-a/d/(d*x+c)+I*a*f*cosh(-e+c*f/d)*Chi(c*f/d+f*x)/d^2-I*a*sinh(f*x+e)/d/(d* x+c)-I*a*f*sinh(-e+c*f/d)*Shi(c*f/d+f*x)/d^2
Time = 0.43 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {i a \left (f (c+d x) \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-d (-i+\sinh (e+f x))+f (c+d x) \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )\right )}{d^2 (c+d x)} \] Input:
Integrate[(a + I*a*Sinh[e + f*x])/(c + d*x)^2,x]
Output:
(I*a*(f*(c + d*x)*Cosh[e - (c*f)/d]*CoshIntegral[f*(c/d + x)] - d*(-I + Si nh[e + f*x]) + f*(c + d*x)*Sinh[e - (c*f)/d]*SinhIntegral[f*(c/d + x)]))/( d^2*(c + d*x))
Time = 0.41 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3042, 3798, 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.
\(\displaystyle \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {a+a \sin (i e+i f x)}{(c+d x)^2}dx\) |
\(\Big \downarrow \) 3798 |
\(\displaystyle \int \left (\frac {a}{(c+d x)^2}+\frac {i a \sinh (e+f x)}{(c+d x)^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i a f \text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{d^2}+\frac {i a f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{d^2}-\frac {i a \sinh (e+f x)}{d (c+d x)}-\frac {a}{d (c+d x)}\) |
Input:
Int[(a + I*a*Sinh[e + f*x])/(c + d*x)^2,x]
Output:
-(a/(d*(c + d*x))) + (I*a*f*Cosh[e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x]) /d^2 - (I*a*Sinh[e + f*x])/(d*(c + d*x)) + (I*a*f*Sinh[e - (c*f)/d]*SinhIn tegral[(c*f)/d + f*x])/d^2
Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.) , x_Symbol] :> Int[ExpandIntegrand[(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] || IGtQ[ m, 0] || NeQ[a^2 - b^2, 0])
Time = 0.43 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.61
method | result | size |
risch | \(-\frac {a}{d \left (d x +c \right )}+\frac {i a f \,{\mathrm e}^{-f x -e}}{2 d \left (d x f +c f \right )}-\frac {i a f \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{2 d^{2}}-\frac {i a f \,{\mathrm e}^{f x +e}}{2 d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {i a f \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{2 d^{2}}\) | \(153\) |
Input:
int((a+I*a*sinh(f*x+e))/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-a/d/(d*x+c)+1/2*I*a*f*exp(-f*x-e)/d/(d*f*x+c*f)-1/2*I*a*f/d^2*exp((c*f-d* e)/d)*Ei(1,f*x+e+(c*f-d*e)/d)-1/2*I*a*f/d^2*exp(f*x+e)/(c*f/d+f*x)-1/2*I*a *f/d^2*exp(-(c*f-d*e)/d)*Ei(1,-f*x-e-(c*f-d*e)/d)
Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.41 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {{\left (-i \, a d e^{\left (2 \, f x + 2 \, e\right )} + i \, a d + {\left ({\left (i \, a d f x + i \, a c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + {\left (i \, a d f x + i \, a c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - 2 \, a d\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{2 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)^2,x, algorithm="fricas")
Output:
1/2*(-I*a*d*e^(2*f*x + 2*e) + I*a*d + ((I*a*d*f*x + I*a*c*f)*Ei((d*f*x + c *f)/d)*e^((d*e - c*f)/d) + (I*a*d*f*x + I*a*c*f)*Ei(-(d*f*x + c*f)/d)*e^(- (d*e - c*f)/d) - 2*a*d)*e^(f*x + e))*e^(-f*x - e)/(d^3*x + c*d^2)
Timed out. \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\text {Timed out} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)**2,x)
Output:
Timed out
Time = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} i \, a {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a}{d^{2} x + c d} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)^2,x, algorithm="maxima")
Output:
1/2*I*a*(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/(d^2*x + c*d)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 630 vs. \(2 (91) = 182\).
Time = 0.16 (sec) , antiderivative size = 630, normalized size of antiderivative = 6.63 \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {1}{2} i \, a {\left (\frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} - d f^{2} e^{\left (\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f} + \frac {{\left ({\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - d e f^{2} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + c f^{3} {\rm Ei}\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d e + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} + d f^{2} e^{\left (-\frac {{\left (d x + c\right )} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )}}{d}\right )}\right )} d^{2}}{{\left ({\left (d x + c\right )} d^{4} {\left (\frac {d e}{d x + c} - \frac {c f}{d x + c} + f\right )} - d^{5} e + c d^{4} f\right )} f}\right )} - \frac {a}{{\left (d x + c\right )} d} \] Input:
integrate((a+I*a*sinh(f*x+e))/(d*x+c)^2,x, algorithm="giac")
Output:
1/2*I*a*(((d*x + c)*(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) - 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) + 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) - d*f^2*e^((d*x + c)*(d*e/(d*x + c) - c* f/(d*x + c) + f)/d))*d^2/(((d*x + c)*d^4*(d*e/(d*x + c) - c*f/(d*x + c) + f) - d^5*e + c*d^4*f)*f) + ((d*x + c)*(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) - 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) + 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) + d*f^2*e^(-(d*x + c)*(d*e/(d*x + c) - c*f/(d*x + c) + f)/d))*d^2/(((d*x + c)*d^4*(d*e/(d* x + c) - c*f/(d*x + c) + f) - d^5*e + c*d^4*f)*f)) - a/((d*x + c)*d)
Timed out. \[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\int \frac {a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + a*sinh(e + f*x)*1i)/(c + d*x)^2,x)
Output:
int((a + a*sinh(e + f*x)*1i)/(c + d*x)^2, x)
\[ \int \frac {a+i a \sinh (e+f x)}{(c+d x)^2} \, dx=\frac {a \left (e^{2 e} \left (\int \frac {e^{f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2} i +e^{2 e} \left (\int \frac {e^{f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d i x +2 e^{e} x -\left (\int \frac {1}{e^{f x} c^{2}+2 e^{f x} c d x +e^{f x} d^{2} x^{2}}d x \right ) c^{2} i -\left (\int \frac {1}{e^{f x} c^{2}+2 e^{f x} c d x +e^{f x} d^{2} x^{2}}d x \right ) c d i x \right )}{2 e^{e} c \left (d x +c \right )} \] Input:
int((a+I*a*sinh(f*x+e))/(d*x+c)^2,x)
Output:
(a*(e**(2*e)*int(e**(f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c**2*i + e**(2*e )*int(e**(f*x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c*d*i*x + 2*e**e*x - int(1/ (e**(f*x)*c**2 + 2*e**(f*x)*c*d*x + e**(f*x)*d**2*x**2),x)*c**2*i - int(1/ (e**(f*x)*c**2 + 2*e**(f*x)*c*d*x + e**(f*x)*d**2*x**2),x)*c*d*i*x))/(2*e* *e*c*(c + d*x))