\(\int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx\) [180]

Optimal result

Integrand size = 20, antiderivative size = 282 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\frac {a^2 (c+d x)^{1+m}}{d (1+m)}+\frac {b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {2^{-3-m} b^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}-\frac {a b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}-\frac {2^{-3-m} b^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f} \] Output:

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

Mathematica [A] (verified)

Time = 0.48 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.90 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\frac {(c+d x)^m \left (8 a^2 f (c+d x)+4 b^2 f (c+d x)+2^{-m} b^2 d e^{2 e-\frac {2 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )+8 a b d e^{e-\frac {c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )-8 a b d e^{-e+\frac {c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )-2^{-m} b^2 d e^{-2 e+\frac {2 c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )\right )}{8 d f (1+m)} \] Input:

Integrate[(c + d*x)^m*(a + b*Cosh[e + f*x])^2,x]
 

Output:

((c + d*x)^m*(8*a^2*f*(c + d*x) + 4*b^2*f*(c + d*x) + (b^2*d*E^(2*e - (2*c 
*f)/d)*(1 + m)*Gamma[1 + m, (-2*f*(c + d*x))/d])/(2^m*(-((f*(c + d*x))/d)) 
^m) + (8*a*b*d*E^(e - (c*f)/d)*(1 + m)*Gamma[1 + m, -((f*(c + d*x))/d)])/( 
-((f*(c + d*x))/d))^m - (8*a*b*d*E^(-e + (c*f)/d)*(1 + m)*Gamma[1 + m, (f* 
(c + d*x))/d])/((f*(c + d*x))/d)^m - (b^2*d*E^(-2*e + (2*c*f)/d)*(1 + m)*G 
amma[1 + m, (2*f*(c + d*x))/d])/(2^m*((f*(c + d*x))/d)^m)))/(8*d*f*(1 + m) 
)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 282, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3798

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

\(\Big \downarrow \) 2009

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

Input:

Int[(c + d*x)^m*(a + b*Cosh[e + f*x])^2,x]
 

Output:

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

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])
 

Maple [F]

\[\int \left (d x +c \right )^{m} \left (a +b \cosh \left (f x +e \right )\right )^{2}d x\]

Input:

int((d*x+c)^m*(a+b*cosh(f*x+e))^2,x)
 

Output:

int((d*x+c)^m*(a+b*cosh(f*x+e))^2,x)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.80 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=-\frac {{\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 8 \, {\left (a b d m + a b d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - 8 \, {\left (a b d m + a b d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - {\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 8 \, {\left (a b d m + a b d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) + 8 \, {\left (a b d m + a b d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) + {\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) - 4 \, {\left ({\left (2 \, a^{2} + b^{2}\right )} d f x + {\left (2 \, a^{2} + b^{2}\right )} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 4 \, {\left ({\left (2 \, a^{2} + b^{2}\right )} d f x + {\left (2 \, a^{2} + b^{2}\right )} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{8 \, {\left (d f m + d f\right )}} \] Input:

integrate((d*x+c)^m*(a+b*cosh(f*x+e))^2,x, algorithm="fricas")
 

Output:

-1/8*((b^2*d*m + 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) + 8*(a*b*d*m + a*b*d)*cosh((d*m*log(f/d) + d*e - c* 
f)/d)*gamma(m + 1, (d*f*x + c*f)/d) - 8*(a*b*d*m + a*b*d)*cosh((d*m*log(-f 
/d) - d*e + c*f)/d)*gamma(m + 1, -(d*f*x + c*f)/d) - (b^2*d*m + b^2*d)*cos 
h((d*m*log(-2*f/d) - 2*d*e + 2*c*f)/d)*gamma(m + 1, -2*(d*f*x + c*f)/d) - 
(b^2*d*m + 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) - 8*(a*b*d*m + a*b*d)*gamma(m + 1, (d*f*x + c*f)/d)*sinh( 
(d*m*log(f/d) + d*e - c*f)/d) + 8*(a*b*d*m + a*b*d)*gamma(m + 1, -(d*f*x + 
 c*f)/d)*sinh((d*m*log(-f/d) - d*e + c*f)/d) + (b^2*d*m + 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) - 4*((2* 
a^2 + b^2)*d*f*x + (2*a^2 + b^2)*c*f)*cosh(m*log(d*x + c)) - 4*((2*a^2 + b 
^2)*d*f*x + (2*a^2 + b^2)*c*f)*sinh(m*log(d*x + c)))/(d*f*m + d*f)
 

Sympy [F(-2)]

Exception generated. \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*x+c)**m*(a+b*cosh(f*x+e))**2,x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.74 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=-{\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a b - \frac {1}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} b^{2} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \] Input:

integrate((d*x+c)^m*(a+b*cosh(f*x+e))^2,x, algorithm="maxima")
 

Output:

-((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*b - 
 1/4*((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)))*b^2 + (d*x + c)^(m + 1)*a^2/( 
d*(m + 1))
 

Giac [F]

\[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\int { {\left (b \cosh \left (f x + e\right ) + a\right )}^{2} {\left (d x + c\right )}^{m} \,d x } \] Input:

integrate((d*x+c)^m*(a+b*cosh(f*x+e))^2,x, algorithm="giac")
 

Output:

integrate((b*cosh(f*x + e) + a)^2*(d*x + c)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\int {\left (a+b\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^m \,d x \] Input:

int((a + b*cosh(e + f*x))^2*(c + d*x)^m,x)
 

Output:

int((a + b*cosh(e + f*x))^2*(c + d*x)^m, x)
 

Reduce [F]

\[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\frac {e^{4 f x +4 e} \left (d x +c \right )^{m} b^{2} d m +e^{4 f x +4 e} \left (d x +c \right )^{m} b^{2} d +8 e^{3 f x +3 e} \left (d x +c \right )^{m} a b d m +8 e^{3 f x +3 e} \left (d x +c \right )^{m} a b d +8 e^{2 f x +2 e} \left (d x +c \right )^{m} a^{2} c f +8 e^{2 f x +2 e} \left (d x +c \right )^{m} a^{2} d f x +4 e^{2 f x +2 e} \left (d x +c \right )^{m} b^{2} c f +4 e^{2 f x +2 e} \left (d x +c \right )^{m} b^{2} d f x -8 e^{f x +e} \left (d x +c \right )^{m} a b d m -8 e^{f x +e} \left (d x +c \right )^{m} a b d -\left (d x +c \right )^{m} b^{2} d m -\left (d x +c \right )^{m} b^{2} d -e^{2 f x +4 e} \left (\int \frac {e^{2 f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) b^{2} d^{2} m^{2}-e^{2 f x +4 e} \left (\int \frac {e^{2 f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) b^{2} d^{2} m -8 e^{2 f x +3 e} \left (\int \frac {e^{f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) a b \,d^{2} m^{2}-8 e^{2 f x +3 e} \left (\int \frac {e^{f x} \left (d x +c \right )^{m}}{d x +c}d x \right ) a b \,d^{2} m +e^{2 f x +2 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x +2 e} c +e^{2 f x +2 e} d x}d x \right ) b^{2} d^{2} m^{2}+e^{2 f x +2 e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{2 f x +2 e} c +e^{2 f x +2 e} d x}d x \right ) b^{2} d^{2} m +8 e^{2 f x +e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{f x} c +e^{f x} d x}d x \right ) a b \,d^{2} m^{2}+8 e^{2 f x +e} \left (\int \frac {\left (d x +c \right )^{m}}{e^{f x} c +e^{f x} d x}d x \right ) a b \,d^{2} m}{8 e^{2 f x +2 e} d f \left (m +1\right )} \] Input:

int((d*x+c)^m*(a+b*cosh(f*x+e))^2,x)
 

Output:

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