3.2.79 \(\int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx\) [179]

3.2.79.1 Optimal result

Integrand size = 20, antiderivative size = 543 \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}+\frac {3 a b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {3^{-1-m} b^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a b^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a^2 b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}+\frac {3 b^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3 a^2 b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f}-\frac {3 b^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a b^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}-\frac {3^{-1-m} b^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \]

output

a^3*(d*x+c)^(1+m)/d/(1+m)+3/2*a*b^2*(d*x+c)^(1+m)/d/(1+m)+1/8*3^(-1-m)*b^3 
*exp(3*e-3*c*f/d)*(d*x+c)^m*GAMMA(1+m,-3*f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m) 
+3*2^(-3-m)*a*b^2*exp(2*e-2*c*f/d)*(d*x+c)^m*GAMMA(1+m,-2*f*(d*x+c)/d)/f/( 
(-f*(d*x+c)/d)^m)+3/2*a^2*b*exp(e-c*f/d)*(d*x+c)^m*GAMMA(1+m,-f*(d*x+c)/d) 
/f/((-f*(d*x+c)/d)^m)+3/8*b^3*exp(e-c*f/d)*(d*x+c)^m*GAMMA(1+m,-f*(d*x+c)/ 
d)/f/((-f*(d*x+c)/d)^m)-3/2*a^2*b*exp(-e+c*f/d)*(d*x+c)^m*GAMMA(1+m,f*(d*x 
+c)/d)/f/((f*(d*x+c)/d)^m)-3/8*b^3*exp(-e+c*f/d)*(d*x+c)^m*GAMMA(1+m,f*(d* 
x+c)/d)/f/((f*(d*x+c)/d)^m)-3*2^(-3-m)*a*b^2*exp(-2*e+2*c*f/d)*(d*x+c)^m*G 
AMMA(1+m,2*f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)-1/8*3^(-1-m)*b^3*exp(-3*e+3*c* 
f/d)*(d*x+c)^m*GAMMA(1+m,3*f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)
 

3.2.79.2 Mathematica [A] (verified)

Time = 1.21 (sec) , antiderivative size = 447, normalized size of antiderivative = 0.82 \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=\frac {2^{-3-m} 3^{-1-m} e^{-3 \left (e+\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (2^m b^3 d e^{6 e} (1+m) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )+3^{2+m} a b^2 d e^{5 e+\frac {c f}{d}} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )+2^m 3^{2+m} b \left (4 a^2+b^2\right ) d e^{4 e+\frac {2 c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )-e^{\frac {3 c f}{d}} \left (2^m 3^{2+m} b \left (4 a^2+b^2\right ) d e^{2 e+\frac {c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )+3^{2+m} a b^2 d e^{e+\frac {2 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )+2^m \left (-4 3^{1+m} a \left (2 a^2+3 b^2\right ) e^{3 e} f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m+b^3 d e^{\frac {3 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )\right )\right )\right )}{d f (1+m)} \]

input

Integrate[(c + d*x)^m*(a + b*Cosh[e + f*x])^3,x]
 

output

(2^(-3 - m)*3^(-1 - m)*(c + d*x)^m*(2^m*b^3*d*E^(6*e)*(1 + m)*((f*(c + d*x 
))/d)^m*Gamma[1 + m, (-3*f*(c + d*x))/d] + 3^(2 + m)*a*b^2*d*E^(5*e + (c*f 
)/d)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, (-2*f*(c + d*x))/d] + 2^m*3^(2 + 
 m)*b*(4*a^2 + b^2)*d*E^(4*e + (2*c*f)/d)*(1 + m)*((f*(c + d*x))/d)^m*Gamm 
a[1 + m, -((f*(c + d*x))/d)] - E^((3*c*f)/d)*(2^m*3^(2 + m)*b*(4*a^2 + b^2 
)*d*E^(2*e + (c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (f*(c + 
d*x))/d] + 3^(2 + m)*a*b^2*d*E^(e + (2*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d) 
)^m*Gamma[1 + m, (2*f*(c + d*x))/d] + 2^m*(-4*3^(1 + m)*a*(2*a^2 + 3*b^2)* 
E^(3*e)*f*(c + d*x)*(-((f^2*(c + d*x)^2)/d^2))^m + b^3*d*E^((3*c*f)/d)*(1 
+ m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (3*f*(c + d*x))/d]))))/(d*E^(3*(e 
 + (c*f)/d))*f*(1 + m)*(-((f^2*(c + d*x)^2)/d^2))^m)
 

3.2.79.3 Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, 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.

\(\displaystyle \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (c+d x)^m \left (a+b \sin \left (i e+i f x+\frac {\pi }{2}\right )\right )^3dx\)

\(\Big \downarrow \) 3798

\(\displaystyle \int \left (a^3 (c+d x)^m+3 a^2 b (c+d x)^m \cosh (e+f x)+3 a b^2 (c+d x)^m \cosh ^2(e+f x)+b^3 (c+d x)^m \cosh ^3(e+f x)\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^3 (c+d x)^{m+1}}{d (m+1)}+\frac {3 a^2 b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {3 a^2 b e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{2 f}+\frac {3 a b^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}-\frac {3 a b^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a b^2 (c+d x)^{m+1}}{2 d (m+1)}+\frac {b^3 3^{-m-1} e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3 b^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3 b^3 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{8 f}-\frac {b^3 3^{-m-1} e^{\frac {3 c f}{d}-3 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 f (c+d x)}{d}\right )}{8 f}\)

input

Int[(c + d*x)^m*(a + b*Cosh[e + f*x])^3,x]
 

output

(a^3*(c + d*x)^(1 + m))/(d*(1 + m)) + (3*a*b^2*(c + d*x)^(1 + m))/(2*d*(1 
+ m)) + (3^(-1 - m)*b^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-3*f 
*(c + d*x))/d])/(8*f*(-((f*(c + d*x))/d))^m) + (3*2^(-3 - m)*a*b^2*E^(2*e 
- (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d 
*x))/d))^m) + (3*a^2*b*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -((f*(c + 
d*x))/d)])/(2*f*(-((f*(c + d*x))/d))^m) + (3*b^3*E^(e - (c*f)/d)*(c + d*x) 
^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(8*f*(-((f*(c + d*x))/d))^m) - (3*a^2 
*b*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + d*x))/d])/(2*f*((f*(c 
 + d*x))/d)^m) - (3*b^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*(c + 
d*x))/d])/(8*f*((f*(c + d*x))/d)^m) - (3*2^(-3 - m)*a*b^2*E^(-2*e + (2*c*f 
)/d)*(c + d*x)^m*Gamma[1 + m, (2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) 
- (3^(-1 - m)*b^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (3*f*(c + 
d*x))/d])/(8*f*((f*(c + d*x))/d)^m)
 

3.2.79.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3798

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])
 

3.2.79.4 Maple [F]

\[\int \left (d x +c \right )^{m} \left (a +b \cosh \left (f x +e \right )\right )^{3}d x\]

input

int((d*x+c)^m*(a+b*cosh(f*x+e))^3,x)
 

output

int((d*x+c)^m*(a+b*cosh(f*x+e))^3,x)
 

3.2.79.5 Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 813, normalized size of antiderivative = 1.50 \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=-\frac {{\left (b^{3} d m + b^{3} d\right )} \cosh \left (\frac {d m \log \left (\frac {3 \, f}{d}\right ) + 3 \, d e - 3 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left ({\left (4 \, a^{2} b + b^{3}\right )} d m + {\left (4 \, a^{2} b + b^{3}\right )} d\right )} \cosh \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 ) - 9 \, {\left ({\left (4 \, a^{2} b + b^{3}\right )} d m + {\left (4 \, a^{2} b + b^{3}\right )} d\right )} \cosh \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 ) - 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{3} d m + b^{3} d\right )} \cosh \left (\frac {d m \log \left (-\frac {3 \, f}{d}\right ) - 3 \, d e + 3 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{3} d m + b^{3} d\right )} \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, f}{d}\right ) + 3 \, d e - 3 \, c f}{d}\right ) - 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 9 \, {\left ({\left (4 \, a^{2} b + b^{3}\right )} d m + {\left (4 \, a^{2} b + b^{3}\right )} d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) + 9 \, {\left ({\left (4 \, a^{2} b + b^{3}\right )} d m + {\left (4 \, a^{2} b + b^{3}\right )} d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) + 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) + {\left (b^{3} d m + b^{3} d\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, f}{d}\right ) - 3 \, d e + 3 \, c f}{d}\right ) - 12 \, {\left ({\left (2 \, a^{3} + 3 \, a b^{2}\right )} d f x + {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 12 \, {\left ({\left (2 \, a^{3} + 3 \, a b^{2}\right )} d f x + {\left (2 \, a^{3} + 3 \, a b^{2}\right )} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{24 \, {\left (d f m + d f\right )}} \]

input

integrate((d*x+c)^m*(a+b*cosh(f*x+e))^3,x, algorithm="fricas")
 

output

-1/24*((b^3*d*m + b^3*d)*cosh((d*m*log(3*f/d) + 3*d*e - 3*c*f)/d)*gamma(m 
+ 1, 3*(d*f*x + c*f)/d) + 9*(a*b^2*d*m + a*b^2*d)*cosh((d*m*log(2*f/d) + 2 
*d*e - 2*c*f)/d)*gamma(m + 1, 2*(d*f*x + c*f)/d) + 9*((4*a^2*b + b^3)*d*m 
+ (4*a^2*b + b^3)*d)*cosh((d*m*log(f/d) + d*e - c*f)/d)*gamma(m + 1, (d*f* 
x + c*f)/d) - 9*((4*a^2*b + b^3)*d*m + (4*a^2*b + b^3)*d)*cosh((d*m*log(-f 
/d) - d*e + c*f)/d)*gamma(m + 1, -(d*f*x + c*f)/d) - 9*(a*b^2*d*m + a*b^2* 
d)*cosh((d*m*log(-2*f/d) - 2*d*e + 2*c*f)/d)*gamma(m + 1, -2*(d*f*x + c*f) 
/d) - (b^3*d*m + b^3*d)*cosh((d*m*log(-3*f/d) - 3*d*e + 3*c*f)/d)*gamma(m 
+ 1, -3*(d*f*x + c*f)/d) - (b^3*d*m + b^3*d)*gamma(m + 1, 3*(d*f*x + c*f)/ 
d)*sinh((d*m*log(3*f/d) + 3*d*e - 3*c*f)/d) - 9*(a*b^2*d*m + a*b^2*d)*gamm 
a(m + 1, 2*(d*f*x + c*f)/d)*sinh((d*m*log(2*f/d) + 2*d*e - 2*c*f)/d) - 9*( 
(4*a^2*b + b^3)*d*m + (4*a^2*b + b^3)*d)*gamma(m + 1, (d*f*x + c*f)/d)*sin 
h((d*m*log(f/d) + d*e - c*f)/d) + 9*((4*a^2*b + b^3)*d*m + (4*a^2*b + b^3) 
*d)*gamma(m + 1, -(d*f*x + c*f)/d)*sinh((d*m*log(-f/d) - d*e + c*f)/d) + 9 
*(a*b^2*d*m + a*b^2*d)*gamma(m + 1, -2*(d*f*x + c*f)/d)*sinh((d*m*log(-2*f 
/d) - 2*d*e + 2*c*f)/d) + (b^3*d*m + b^3*d)*gamma(m + 1, -3*(d*f*x + c*f)/ 
d)*sinh((d*m*log(-3*f/d) - 3*d*e + 3*c*f)/d) - 12*((2*a^3 + 3*a*b^2)*d*f*x 
 + (2*a^3 + 3*a*b^2)*c*f)*cosh(m*log(d*x + c)) - 12*((2*a^3 + 3*a*b^2)*d*f 
*x + (2*a^3 + 3*a*b^2)*c*f)*sinh(m*log(d*x + c)))/(d*f*m + d*f)
 

3.2.79.6 Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*x+c)**m*(a+b*cosh(f*x+e))**3,x)
 

output

Exception raised: TypeError >> cannot determine truth value of Relational
 

3.2.79.7 Maxima [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.69 \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=-\frac {3}{2} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{2} b - \frac {3}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} a b^{2} - \frac {1}{8} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, e + \frac {3 \, c f}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, e - \frac {3 \, c f}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d}\right )} b^{3} + \frac {{\left (d x + c\right )}^{m + 1} a^{3}}{d {\left (m + 1\right )}} \]

input

integrate((d*x+c)^m*(a+b*cosh(f*x+e))^3,x, algorithm="maxima")
 

output

-3/2*((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 
^2*b - 3/4*((d*x + c)^(m + 1)*e^(-2*e + 2*c*f/d)*exp_integral_e(-m, 2*(d*x 
 + c)*f/d)/d + (d*x + c)^(m + 1)*e^(2*e - 2*c*f/d)*exp_integral_e(-m, -2*( 
d*x + c)*f/d)/d - 2*(d*x + c)^(m + 1)/(d*(m + 1)))*a*b^2 - 1/8*((d*x + c)^ 
(m + 1)*e^(-3*e + 3*c*f/d)*exp_integral_e(-m, 3*(d*x + c)*f/d)/d + 3*(d*x 
+ c)^(m + 1)*e^(-e + c*f/d)*exp_integral_e(-m, (d*x + c)*f/d)/d + 3*(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^(3*e - 3*c*f/d)*exp_integral_e(-m, -3*(d*x + c)*f/d)/d)*b^3 + ( 
d*x + c)^(m + 1)*a^3/(d*(m + 1))
 

3.2.79.8 Giac [F]

\[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=\int { {\left (b \cosh \left (f x + e\right ) + a\right )}^{3} {\left (d x + c\right )}^{m} \,d x } \]

input

integrate((d*x+c)^m*(a+b*cosh(f*x+e))^3,x, algorithm="giac")
 

output

integrate((b*cosh(f*x + e) + a)^3*(d*x + c)^m, x)
 

3.2.79.9 Mupad [F(-1)]

Timed out. \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=\int {\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]

input

int((a + b*cosh(e + f*x))^3*(c + d*x)^m,x)
 

output

int((a + b*cosh(e + f*x))^3*(c + d*x)^m, x)