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

Optimal result

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

[Out]

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*GAMMA(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)

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3398, 3388, 2212, 3393} \[ \int (c+d x)^m (a+b \cosh (e+f x))^3 \, dx=\frac {a^3 (c+d x)^{m+1}}{d (m+1)}+\frac {3 a^2 b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {3 a^2 b e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{2 f}+\frac {3 a b^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}-\frac {3 a b^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a b^2 (c+d x)^{m+1}}{2 d (m+1)}+\frac {b^3 3^{-m-1} e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3 b^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3 b^3 e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{8 f}-\frac {b^3 3^{-m-1} e^{\frac {3 c f}{d}-3 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {3 f (c+d x)}{d}\right )}{8 f} \]

[In]

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

[Out]

(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)

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3388

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(
I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d
, e, f, m}, x] && IntegerQ[2*k]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[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])
)

Rule 3398

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])

Rubi steps \begin{align*} \text {integral}& = \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 \\ & = \frac {a^3 (c+d x)^{1+m}}{d (1+m)}+\left (3 a^2 b\right ) \int (c+d x)^m \cosh (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^m \cosh ^2(e+f x) \, dx+b^3 \int (c+d x)^m \cosh ^3(e+f x) \, dx \\ & = \frac {a^3 (c+d x)^{1+m}}{d (1+m)}+\frac {1}{2} \left (3 a^2 b\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac {1}{2} \left (3 a^2 b\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx+\left (3 a b^2\right ) \int \left (\frac {1}{2} (c+d x)^m+\frac {1}{2} (c+d x)^m \cosh (2 e+2 f x)\right ) \, dx+b^3 \int \left (\frac {3}{4} (c+d x)^m \cosh (e+f x)+\frac {1}{4} (c+d x)^m \cosh (3 e+3 f x)\right ) \, 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 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 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 {1}{2} \left (3 a b^2\right ) \int (c+d x)^m \cosh (2 e+2 f x) \, dx+\frac {1}{4} b^3 \int (c+d x)^m \cosh (3 e+3 f x) \, dx+\frac {1}{4} \left (3 b^3\right ) \int (c+d x)^m \cosh (e+f x) \, 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 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 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 {1}{4} \left (3 a b^2\right ) \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac {1}{4} \left (3 a b^2\right ) \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac {1}{8} b^3 \int e^{-i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac {1}{8} b^3 \int e^{i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac {1}{8} \left (3 b^3\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac {1}{8} \left (3 b^3\right ) \int e^{i (i e+i f x)} (c+d x)^m \, 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} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

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

[Out]

(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*Gamma[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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

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 )}} \]

[In]

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

[Out]

-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)*gamma(m + 1, 2*(d*f*x + c*f)/d)*si
nh((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)*sinh((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)

Sympy [F(-2)]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

-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)*e
xp_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_in
tegral_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))

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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