\(\int (c+d x)^m \cosh ^3(a+b x) \, dx\) [76]

Optimal result

Integrand size = 16, antiderivative size = 237 \[ \int (c+d x)^m \cosh ^3(a+b x) \, dx=\frac {3^{-1-m} e^{3 a-\frac {3 b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )}{8 b}+\frac {3 e^{a-\frac {b c}{d}} (c+d x)^m \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )}{8 b}-\frac {3 e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )}{8 b}-\frac {3^{-1-m} e^{-3 a+\frac {3 b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )}{8 b} \] Output:

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)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.86 \[ \int (c+d x)^m \cosh ^3(a+b x) \, dx=\frac {3^{-1-m} e^{-3 \left (a+\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{6 a} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {3 b (c+d x)}{d}\right )+3^{2+m} e^{4 a+\frac {2 b c}{d}} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )-e^{\frac {4 b c}{d}} \left (-\frac {b (c+d x)}{d}\right )^m \left (3^{2+m} e^{2 a} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \Gamma \left (1+m,\frac {3 b (c+d x)}{d}\right )\right )\right )}{8 b} \] Input:

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

Output:

(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)*G 
amma[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)
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {3042, 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.

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 3793

\(\displaystyle \int \left (\frac {3}{4} \cosh (a+b x) (c+d x)^m+\frac {1}{4} \cosh (3 a+3 b x) (c+d x)^m\right )dx\)

\(\Big \downarrow \) 2009

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

Input:

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

Output:

(3^(-1 - m)*E^(3*a - (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-3*b*(c + d*x))/ 
d])/(8*b*(-((b*(c + d*x))/d))^m) + (3*E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 
+ m, -((b*(c + d*x))/d)])/(8*b*(-((b*(c + d*x))/d))^m) - (3*E^(-a + (b*c)/ 
d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m) - 
(3^(-1 - m)*E^(-3*a + (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (3*b*(c + d*x))/ 
d])/(8*b*((b*(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 3793

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]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 340, normalized size of antiderivative = 1.43 \[ \int (c+d x)^m \cosh ^3(a+b x) \, dx=-\frac {\cosh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) + 9 \, \cosh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) - \cosh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) - 9 \, \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) + 9 \, \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) + \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right )}{24 \, b} \] Input:

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

Output:

-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) - 
cosh((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) + gamm 
a(m + 1, -3*(b*d*x + b*c)/d)*sinh((d*m*log(-3*b/d) + 3*b*c - 3*a*d)/d))/b
 

Sympy [F]

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

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

Output:

Integral((c + d*x)**m*cosh(a + b*x)**3, x)
 

Maxima [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.68 \[ \int (c+d x)^m \cosh ^3(a+b x) \, dx=-\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-a + \frac {b c}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (a - \frac {b c}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \] Input:

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

Output:

-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
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (c+d x)^m \cosh ^3(a+b x) \, dx=\frac {e^{6 b x +6 a} \left (d x +c \right )^{m}+9 e^{4 b x +4 a} \left (d x +c \right )^{m}-9 e^{2 b x +2 a} \left (d x +c \right )^{m}-\left (d x +c \right )^{m}-e^{3 b x +6 a} \left (\int \frac {e^{3 b x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d m -9 e^{3 b x +4 a} \left (\int \frac {e^{b x} \left (d x +c \right )^{m}}{d x +c}d x \right ) d m +e^{3 b x +3 a} \left (\int \frac {\left (d x +c \right )^{m}}{e^{3 b x +3 a} c +e^{3 b x +3 a} d x}d x \right ) d m +9 e^{3 b x +2 a} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) d m}{24 e^{3 b x +3 a} b} \] Input:

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

Output:

(e**(6*a + 6*b*x)*(c + d*x)**m + 9*e**(4*a + 4*b*x)*(c + d*x)**m - 9*e**(2 
*a + 2*b*x)*(c + d*x)**m - (c + d*x)**m - e**(6*a + 3*b*x)*int((e**(3*b*x) 
*(c + d*x)**m)/(c + d*x),x)*d*m - 9*e**(4*a + 3*b*x)*int((e**(b*x)*(c + d* 
x)**m)/(c + d*x),x)*d*m + e**(3*a + 3*b*x)*int((c + d*x)**m/(e**(3*a + 3*b 
*x)*c + e**(3*a + 3*b*x)*d*x),x)*d*m + 9*e**(2*a + 3*b*x)*int((c + d*x)**m 
/(e**(b*x)*c + e**(b*x)*d*x),x)*d*m)/(24*e**(3*a + 3*b*x)*b)