Integrand size = 21, antiderivative size = 135 \[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {i a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}+\frac {i a e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f} \] Output:
a*(d*x+c)^(1+m)/d/(1+m)+1/2*I*a*exp(e-c*f/d)*(d*x+c)^m*GAMMA(1+m,-f*(d*x+c )/d)/f/((-f*(d*x+c)/d)^m)+1/2*I*a*exp(-e+c*f/d)*(d*x+c)^m*GAMMA(1+m,f*(d*x +c)/d)/f/((f*(d*x+c)/d)^m)
Time = 0.37 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.53 \[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=-\frac {a e^{-e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-2 i e^{e+\frac {c f}{d}} f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m+d e^{2 e} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )+d e^{\frac {2 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )\right ) (-i+\sinh (e+f x))}{2 d f (1+m) \left (\cosh \left (\frac {1}{2} (e+f x)\right )+i \sinh \left (\frac {1}{2} (e+f x)\right )\right )^2} \] Input:
Integrate[(c + d*x)^m*(a + I*a*Sinh[e + f*x]),x]
Output:
-1/2*(a*E^(-e - (c*f)/d)*(c + d*x)^m*((-2*I)*E^(e + (c*f)/d)*f*(c + d*x)*( -((f^2*(c + d*x)^2)/d^2))^m + d*E^(2*e)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, -((f*(c + d*x))/d)] + d*E^((2*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Ga mma[1 + m, (f*(c + d*x))/d])*(-I + Sinh[e + f*x]))/(d*f*(1 + m)*(-((f^2*(c + d*x)^2)/d^2))^m*(Cosh[(e + f*x)/2] + I*Sinh[(e + f*x)/2])^2)
Time = 0.40 (sec) , antiderivative size = 135, 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.
\(\displaystyle \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int (c+d x)^m (a+a \sin (i e+i f x))dx\) |
\(\Big \downarrow \) 3798 |
\(\displaystyle \int \left (a (c+d x)^m+i a (c+d x)^m \sinh (e+f x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i a e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}+\frac {i a e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{2 f}+\frac {a (c+d x)^{m+1}}{d (m+1)}\) |
Input:
Int[(c + d*x)^m*(a + I*a*Sinh[e + f*x]),x]
Output:
(a*(c + d*x)^(1 + m))/(d*(1 + m)) + ((I/2)*a*E^(e - (c*f)/d)*(c + d*x)^m*G amma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + ((I/2)*a*E^( -e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x)) /d)^m)
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])
\[\int \left (d x +c \right )^{m} \left (a +i a \sinh \left (f x +e \right )\right )d x\]
Input:
int((d*x+c)^m*(a+I*a*sinh(f*x+e)),x)
Output:
int((d*x+c)^m*(a+I*a*sinh(f*x+e)),x)
Time = 0.10 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.99 \[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\frac {{\left (i \, a d m + i \, a d\right )} e^{\left (-\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) + {\left (i \, a d m + i \, a d\right )} e^{\left (-\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) + 2 \, {\left (a d f x + a c f\right )} {\left (d x + c\right )}^{m}}{2 \, {\left (d f m + d f\right )}} \] Input:
integrate((d*x+c)^m*(a+I*a*sinh(f*x+e)),x, algorithm="fricas")
Output:
1/2*((I*a*d*m + I*a*d)*e^(-(d*m*log(f/d) + d*e - c*f)/d)*gamma(m + 1, (d*f *x + c*f)/d) + (I*a*d*m + I*a*d)*e^(-(d*m*log(-f/d) - d*e + c*f)/d)*gamma( m + 1, -(d*f*x + c*f)/d) + 2*(a*d*f*x + a*c*f)*(d*x + c)^m)/(d*f*m + d*f)
Exception generated. \[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x+c)**m*(a+I*a*sinh(f*x+e)),x)
Output:
Exception raised: TypeError >> cannot determine truth value of Relational
Time = 0.09 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.75 \[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\frac {1}{2} i \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \] Input:
integrate((d*x+c)^m*(a+I*a*sinh(f*x+e)),x, algorithm="maxima")
Output:
1/2*I*((d*x + c)^(m + 1)*e^(-e + c*f/d)*exp_integral_e(-m, (d*x + c)*f/d)/ d - (d*x + c)^(m + 1)*e^(e - c*f/d)*exp_integral_e(-m, -(d*x + c)*f/d)/d)* a + (d*x + c)^(m + 1)*a/(d*(m + 1))
\[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\int { {\left (i \, a \sinh \left (f x + e\right ) + a\right )} {\left (d x + c\right )}^{m} \,d x } \] Input:
integrate((d*x+c)^m*(a+I*a*sinh(f*x+e)),x, algorithm="giac")
Output:
integrate((I*a*sinh(f*x + e) + a)*(d*x + c)^m, x)
Timed out. \[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\int \left (a+a\,\mathrm {sinh}\left (e+f\,x\right )\,1{}\mathrm {i}\right )\,{\left (c+d\,x\right )}^m \,d x \] Input:
int((a + a*sinh(e + f*x)*1i)*(c + d*x)^m,x)
Output:
int((a + a*sinh(e + f*x)*1i)*(c + d*x)^m, x)
\[ \int (c+d x)^m (a+i a \sinh (e+f x)) \, dx=\frac {a \left (e^{2 f x +2 e} \left (d x +c \right )^{m} d i m +e^{2 f x +2 e} \left (d x +c \right )^{m} d i +2 e^{f x +e} \left (d x +c \right )^{m} c f +2 e^{f x +e} \left (d x +c \right )^{m} d f x +\left (d x +c \right )^{m} d i m +\left (d x +c \right )^{m} d i -e^{f x +2 e} \left (\int \frac {e^{f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} i \,m^{2}-e^{f x +2 e} \left (\int \frac {e^{f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d^{2} i m -e^{f x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{f x} c +e^{f x} d x}d x \right ) d^{2} i \,m^{2}-e^{f x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{f x} c +e^{f x} d x}d x \right ) d^{2} i m \right )}{2 e^{f x +e} d f \left (m +1\right )} \] Input:
int((d*x+c)^m*(a+I*a*sinh(f*x+e)),x)
Output:
(a*(e**(2*e + 2*f*x)*(c + d*x)**m*d*i*m + e**(2*e + 2*f*x)*(c + d*x)**m*d* i + 2*e**(e + f*x)*(c + d*x)**m*c*f + 2*e**(e + f*x)*(c + d*x)**m*d*f*x + (c + d*x)**m*d*i*m + (c + d*x)**m*d*i - e**(2*e + f*x)*int((e**(f*x)*(c + d*x)**m)/(c + d*x),x)*d**2*i*m**2 - e**(2*e + f*x)*int((e**(f*x)*(c + d*x) **m)/(c + d*x),x)*d**2*i*m - e**(f*x)*int((c + d*x)**m/(e**(f*x)*c + e**(f *x)*d*x),x)*d**2*i*m**2 - e**(f*x)*int((c + d*x)**m/(e**(f*x)*c + e**(f*x) *d*x),x)*d**2*i*m))/(2*e**(e + f*x)*d*f*(m + 1))