Integrand size = 16, antiderivative size = 237 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {3^{-1-m} e^{3 a-\frac {3 b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )}{8 b}+\frac {3^{-1-m} e^{-3 a+\frac {3 b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )}{8 b} \]
1/8*3^(-1-m)*exp(3*a-3*b*c/d)*(d*x+c)^m*GAMMA(1+m,-3*b*(d*x+c)/d)/b/((-b*( d*x+c)/d)^m)-3/8*exp(a-b*c/d)*(d*x+c)^m*GAMMA(1+m,-b*(d*x+c)/d)/b/((-b*(d* x+c)/d)^m)-3/8*exp(-a+b*c/d)*(d*x+c)^m*GAMMA(1+m,b*(d*x+c)/d)/b/((b*(d*x+c )/d)^m)+1/8*3^(-1-m)*exp(-3*a+3*b*c/d)*(d*x+c)^m*GAMMA(1+m,3*b*(d*x+c)/d)/ b/((b*(d*x+c)/d)^m)
Time = 0.15 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.87 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {3^{-1-m} e^{-3 \left (a+\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{6 a} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )-3^{2+m} e^{4 a+\frac {2 b c}{d}} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \left (-\frac {b (c+d x)}{d}\right )^m \left (-3^{2+m} e^{2 a} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )\right )\right )}{8 b} \]
(3^(-1 - m)*(c + d*x)^m*(E^(6*a)*(b*(c/d + x))^m*Gamma[1 + m, (-3*b*(c + d *x))/d] - 3^(2 + m)*E^(4*a + (2*b*c)/d)*(b*(c/d + x))^m*Gamma[1 + m, -((b* (c + d*x))/d)] + E^((4*b*c)/d)*(-((b*(c + d*x))/d))^m*(-(3^(2 + m)*E^(2*a) *Gamma[1 + m, (b*(c + d*x))/d]) + E^((2*b*c)/d)*Gamma[1 + m, (3*b*(c + d*x ))/d])))/(8*b*E^(3*(a + (b*c)/d))*(-((b^2*(c + d*x)^2)/d^2))^m)
Result contains complex when optimal does not.
Time = 0.54 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3042, 26, 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.
\(\displaystyle \int \sinh ^3(a+b x) (c+d x)^m \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int i \sin (i a+i b x)^3 (c+d x)^mdx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle i \int (c+d x)^m \sin (i a+i b x)^3dx\) |
\(\Big \downarrow \) 3793 |
\(\displaystyle i \int \left (\frac {3}{4} i (c+d x)^m \sinh (a+b x)-\frac {1}{4} i (c+d x)^m \sinh (3 a+3 b x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle i \left (-\frac {i 3^{-m-1} e^{3 a-\frac {3 b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 b (c+d x)}{d}\right )}{8 b}+\frac {3 i e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {b (c+d x)}{d}\right )}{8 b}+\frac {3 i e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {b (c+d x)}{d}\right )}{8 b}-\frac {i 3^{-m-1} e^{\frac {3 b c}{d}-3 a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 b (c+d x)}{d}\right )}{8 b}\right )\) |
I*(((-1/8*I)*3^(-1 - m)*E^(3*a - (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-3*b *(c + d*x))/d])/(b*(-((b*(c + d*x))/d))^m) + (((3*I)/8)*E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(b*(-((b*(c + d*x))/d))^m) + ( ((3*I)/8)*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(b*( (b*(c + d*x))/d)^m) - ((I/8)*3^(-1 - m)*E^(-3*a + (3*b*c)/d)*(c + d*x)^m*G amma[1 + m, (3*b*(c + d*x))/d])/(b*((b*(c + d*x))/d)^m))
3.1.73.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]))
\[\int \left (d x +c \right )^{m} \sinh \left (b x +a \right )^{3}d x\]
Time = 0.09 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.43 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {\cosh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) + \cosh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) + 9 \, \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) + 9 \, \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) - \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right )}{24 \, b} \]
1/24*(cosh((d*m*log(3*b/d) - 3*b*c + 3*a*d)/d)*gamma(m + 1, 3*(b*d*x + b*c )/d) - 9*cosh((d*m*log(b/d) - b*c + a*d)/d)*gamma(m + 1, (b*d*x + b*c)/d) - 9*cosh((d*m*log(-b/d) + b*c - a*d)/d)*gamma(m + 1, -(b*d*x + b*c)/d) + c osh((d*m*log(-3*b/d) + 3*b*c - 3*a*d)/d)*gamma(m + 1, -3*(b*d*x + b*c)/d) - gamma(m + 1, 3*(b*d*x + b*c)/d)*sinh((d*m*log(3*b/d) - 3*b*c + 3*a*d)/d) + 9*gamma(m + 1, (b*d*x + b*c)/d)*sinh((d*m*log(b/d) - b*c + a*d)/d) + 9* gamma(m + 1, -(b*d*x + b*c)/d)*sinh((d*m*log(-b/d) + b*c - a*d)/d) - gamma (m + 1, -3*(b*d*x + b*c)/d)*sinh((d*m*log(-3*b/d) + 3*b*c - 3*a*d)/d))/b
\[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\int \left (c + d x\right )^{m} \sinh ^{3}{\left (a + b x \right )}\, dx \]
Time = 0.10 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-a + \frac {b c}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (a - \frac {b c}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \]
1/8*(d*x + c)^(m + 1)*e^(-3*a + 3*b*c/d)*exp_integral_e(-m, 3*(d*x + c)*b/ d)/d - 3/8*(d*x + c)^(m + 1)*e^(-a + b*c/d)*exp_integral_e(-m, (d*x + c)*b /d)/d + 3/8*(d*x + c)^(m + 1)*e^(a - b*c/d)*exp_integral_e(-m, -(d*x + c)* b/d)/d - 1/8*(d*x + c)^(m + 1)*e^(3*a - 3*b*c/d)*exp_integral_e(-m, -3*(d* x + c)*b/d)/d
\[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \sinh \left (b x + a\right )^{3} \,d x } \]
Timed out. \[ \int (c+d x)^m \sinh ^3(a+b x) \, dx=\int {\mathrm {sinh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]