Integrand size = 23, antiderivative size = 170 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=-\frac {4 a^2 \cosh ^4\left (\frac {e}{2}+\frac {i \pi }{4}+\frac {f x}{2}\right )}{d (c+d x)}+\frac {2 i a^2 f \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {a^2 f \text {Chi}\left (\frac {2 c f}{d}+2 f x\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{d^2}+\frac {2 i a^2 f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (\frac {c f}{d}+f x\right )}{d^2}-\frac {a^2 f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 c f}{d}+2 f x\right )}{d^2} \] Output:
-4*a^2*cosh(1/2*e+1/4*I*Pi+1/2*f*x)^4/d/(d*x+c)+2*I*a^2*f*cosh(-e+c*f/d)*C hi(c*f/d+f*x)/d^2+a^2*f*Chi(2*c*f/d+2*f*x)*sinh(-2*e+2*c*f/d)/d^2-2*I*a^2* f*sinh(-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 = 0.73 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.26 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^2 \left (-3 d+d \cosh (2 (e+f x))+4 i f (c+d x) \cosh \left (e-\frac {c f}{d}\right ) \text {Chi}\left (f \left (\frac {c}{d}+x\right )\right )-2 f (c+d x) \text {Chi}\left (\frac {2 f (c+d x)}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )-4 i d \sinh (e+f x)+4 i c f \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )+4 i d f x \sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (f \left (\frac {c}{d}+x\right )\right )-2 c f \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )-2 d f x \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (\frac {2 f (c+d x)}{d}\right )\right )}{2 d^2 (c+d x)} \] Input:
Integrate[(a + I*a*Sinh[e + f*x])^2/(c + d*x)^2,x]
Output:
(a^2*(-3*d + d*Cosh[2*(e + f*x)] + (4*I)*f*(c + d*x)*Cosh[e - (c*f)/d]*Cos hIntegral[f*(c/d + x)] - 2*f*(c + d*x)*CoshIntegral[(2*f*(c + d*x))/d]*Sin h[2*e - (2*c*f)/d] - (4*I)*d*Sinh[e + f*x] + (4*I)*c*f*Sinh[e - (c*f)/d]*S inhIntegral[f*(c/d + x)] + (4*I)*d*f*x*Sinh[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))
Time = 0.65 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.02, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, 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.
| \(\displaystyle \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+a \sin (i e+i f x))^2}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3799 |
| \(\displaystyle 4 a^2 \int \frac {\sinh ^4\left (\frac {e}{2}+\frac {f x}{2}-\frac {i \pi }{4}\right )}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle 4 a^2 \int \frac {\sin \left (\frac {i e}{2}+\frac {i f x}{2}+\frac {\pi }{4}\right )^4}{(c+d x)^2}dx\) |
| \(\Big \downarrow \) 3794 |
| \(\displaystyle 4 a^2 \left (\frac {2 i f \int \left (\frac {\cosh (e+f x)}{4 (c+d x)}+\frac {i \sinh (2 e+2 f x)}{8 (c+d x)}\right )dx}{d}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d (c+d x)}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle 4 a^2 \left (\frac {2 i f \left (\frac {i \text {Chi}\left (2 x f+\frac {2 c f}{d}\right ) \sinh \left (2 e-\frac {2 c f}{d}\right )}{8 d}+\frac {\text {Chi}\left (x f+\frac {c f}{d}\right ) \cosh \left (e-\frac {c f}{d}\right )}{4 d}+\frac {\sinh \left (e-\frac {c f}{d}\right ) \text {Shi}\left (x f+\frac {c f}{d}\right )}{4 d}+\frac {i \cosh \left (2 e-\frac {2 c f}{d}\right ) \text {Shi}\left (2 x f+\frac {2 c f}{d}\right )}{8 d}\right )}{d}-\frac {\cosh ^4\left (\frac {e}{2}+\frac {f x}{2}+\frac {i \pi }{4}\right )}{d (c+d x)}\right )\) |
Input:
Int[(a + I*a*Sinh[e + f*x])^2/(c + d*x)^2,x]
Output:
4*a^2*(-(Cosh[e/2 + (I/4)*Pi + (f*x)/2]^4/(d*(c + d*x))) + ((2*I)*f*((Cosh [e - (c*f)/d]*CoshIntegral[(c*f)/d + f*x])/(4*d) + ((I/8)*CoshIntegral[(2* c*f)/d + 2*f*x]*Sinh[2*e - (2*c*f)/d])/d + (Sinh[e - (c*f)/d]*SinhIntegral [(c*f)/d + f*x])/(4*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 = 1.10 (sec) , antiderivative size = 313, normalized size of antiderivative = 1.84
| method | result | size |
| risch | \(-\frac {i a^{2} f \,{\mathrm e}^{f x +e}}{d^{2} \left (\frac {c f}{d}+f x \right )}-\frac {i a^{2} f \,{\mathrm e}^{-\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (-f x -e -\frac {c f -d e}{d}\right )}{d^{2}}-\frac {3 a^{2}}{2 d \left (d x +c \right )}+\frac {f \,a^{2} {\mathrm e}^{-2 f x -2 e}}{4 d \left (d x f +c f \right )}-\frac {f \,a^{2} {\mathrm e}^{\frac {2 c f -2 d e}{d}} \operatorname {expIntegral}_{1}\left (2 f x +2 e +\frac {2 c f -2 d e}{d}\right )}{2 d^{2}}+\frac {f \,a^{2} {\mathrm e}^{2 f x +2 e}}{4 d^{2} \left (\frac {c f}{d}+f x \right )}+\frac {f \,a^{2} {\mathrm e}^{-\frac {2 \left (c f -d e \right )}{d}} \operatorname {expIntegral}_{1}\left (-2 f x -2 e -\frac {2 \left (c f -d e \right )}{d}\right )}{2 d^{2}}+\frac {i a^{2} f \,{\mathrm e}^{-f x -e}}{d \left (d x f +c f \right )}-\frac {i a^{2} f \,{\mathrm e}^{\frac {c f -d e}{d}} \operatorname {expIntegral}_{1}\left (f x +e +\frac {c f -d e}{d}\right )}{d^{2}}\) | \(313\) |
Input:
int((a+I*a*sinh(f*x+e))^2/(d*x+c)^2,x,method=_RETURNVERBOSE)
Output:
-I*a^2*f/d^2*exp(f*x+e)/(c*f/d+f*x)-I*a^2*f/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)+I*a^2*f*exp(-f*x-e)/d/(d*f*x+c*f)-I*a^2*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 = 266, normalized size of antiderivative = 1.56 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=\frac {{\left (a^{2} d e^{\left (4 \, f x + 4 \, e\right )} - 4 i \, a^{2} d e^{\left (3 \, f x + 3 \, e\right )} + 4 i \, a^{2} d e^{\left (f x + e\right )} + a^{2} d - 2 \, {\left (3 \, a^{2} d + {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (\frac {2 \, {\left (d e - c f\right )}}{d}\right )} + 2 \, {\left (-i \, a^{2} d f x - i \, a^{2} c f\right )} {\rm Ei}\left (\frac {d f x + c f}{d}\right ) e^{\left (\frac {d e - c f}{d}\right )} + 2 \, {\left (-i \, a^{2} d f x - i \, a^{2} c f\right )} {\rm Ei}\left (-\frac {d f x + c f}{d}\right ) e^{\left (-\frac {d e - c f}{d}\right )} - {\left (a^{2} d f x + a^{2} c f\right )} {\rm Ei}\left (-\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) e^{\left (-\frac {2 \, {\left (d e - c f\right )}}{d}\right )}\right )} e^{\left (2 \, f x + 2 \, e\right )}\right )} e^{\left (-2 \, f x - 2 \, e\right )}}{4 \, {\left (d^{3} x + c d^{2}\right )}} \] Input:
integrate((a+I*a*sinh(f*x+e))^2/(d*x+c)^2,x, algorithm="fricas")
Output:
1/4*(a^2*d*e^(4*f*x + 4*e) - 4*I*a^2*d*e^(3*f*x + 3*e) + 4*I*a^2*d*e^(f*x + e) + a^2*d - 2*(3*a^2*d + (a^2*d*f*x + a^2*c*f)*Ei(2*(d*f*x + c*f)/d)*e^ (2*(d*e - c*f)/d) + 2*(-I*a^2*d*f*x - I*a^2*c*f)*Ei((d*f*x + c*f)/d)*e^((d *e - c*f)/d) + 2*(-I*a^2*d*f*x - I*a^2*c*f)*Ei(-(d*f*x + c*f)/d)*e^(-(d*e - c*f)/d) - (a^2*d*f*x + a^2*c*f)*Ei(-2*(d*f*x + c*f)/d)*e^(-2*(d*e - c*f) /d))*e^(2*f*x + 2*e))*e^(-2*f*x - 2*e)/(d^3*x + c*d^2)
\[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=- a^{2} \left (\int \frac {\sinh ^{2}{\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\, dx + \int \left (- \frac {2 i \sinh {\left (e + f x \right )}}{c^{2} + 2 c d x + d^{2} x^{2}}\right )\, dx + \int \left (- \frac {1}{c^{2} + 2 c d x + d^{2} x^{2}}\right )\, dx\right ) \] Input:
integrate((a+I*a*sinh(f*x+e))**2/(d*x+c)**2,x)
Output:
-a**2*(Integral(sinh(e + f*x)**2/(c**2 + 2*c*d*x + d**2*x**2), x) + Integr al(-2*I*sinh(e + f*x)/(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 = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.08 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=\frac {1}{4} \, a^{2} {\left (\frac {e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{2}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} + \frac {e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{2}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {2}{d^{2} x + c d}\right )} + i \, a^{2} {\left (\frac {e^{\left (-e + \frac {c f}{d}\right )} E_{2}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d} - \frac {e^{\left (e - \frac {c f}{d}\right )} E_{2}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{{\left (d x + c\right )} d}\right )} - \frac {a^{2}}{d^{2} x + c d} \] Input:
integrate((a+I*a*sinh(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)) + I*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)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1134 vs. \(2 (159) = 318\).
Time = 0.18 (sec) , antiderivative size = 1134, normalized size of antiderivative = 6.67 \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=\text {Too large to display} \] Input:
integrate((a+I*a*sinh(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*I*(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*I*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*I*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*I*(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*I*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*I*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)...
Timed out. \[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=\int \frac {{\left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^2}{{\left (c+d\,x\right )}^2} \,d x \] Input:
int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^2,x)
Output:
int((a + a*sinh(e + f*x)*1i)^2/(c + d*x)^2, x)
\[ \int \frac {(a+i a \sinh (e+f x))^2}{(c+d x)^2} \, dx=\frac {a^{2} \left (-e^{3 e} \left (\int \frac {e^{2 f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2}-e^{3 e} \left (\int \frac {e^{2 f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d x +4 e^{2 e} \left (\int \frac {e^{f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c^{2} i +4 e^{2 e} \left (\int \frac {e^{f x}}{d^{2} x^{2}+2 c d x +c^{2}}d x \right ) c d i x -e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{2}+2 e^{2 f x +2 e} c d x +e^{2 f x +2 e} d^{2} x^{2}}d x \right ) c^{2}-e^{e} \left (\int \frac {1}{e^{2 f x +2 e} c^{2}+2 e^{2 f x +2 e} c d x +e^{2 f x +2 e} d^{2} x^{2}}d x \right ) c d x +6 e^{e} x -4 \left (\int \frac {1}{e^{f x} c^{2}+2 e^{f x} c d x +e^{f x} d^{2} x^{2}}d x \right ) c^{2} i -4 \left (\int \frac {1}{e^{f x} c^{2}+2 e^{f x} c d x +e^{f x} d^{2} x^{2}}d x \right ) c d i x \right )}{4 e^{e} c \left (d x +c \right )} \] Input:
int((a+I*a*sinh(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*i + 4*e**(2*e)*int(e**(f *x)/(c**2 + 2*c*d*x + d**2*x**2),x)*c*d*i*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*i - 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*i*x))/(4*e**e*c*(c + d*x))