\(\int (c+d x)^m (a+b \cosh (e+f x)) \, dx\) [181]

Optimal result

Integrand size = 18, antiderivative size = 131 \[ \int (c+d x)^m (a+b \cosh (e+f x)) \, dx=\frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3398, 3388, 2212} \[ \int (c+d x)^m (a+b \cosh (e+f x)) \, dx=\frac {a (c+d x)^{m+1}}{d (m+1)}+\frac {b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {b e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{2 f} \]

[In]

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

[Out]

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

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 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 (c+d x)^m+b (c+d x)^m \cosh (e+f x)\right ) \, dx \\ & = \frac {a (c+d x)^{1+m}}{d (1+m)}+b \int (c+d x)^m \cosh (e+f x) \, dx \\ & = \frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {1}{2} b \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac {1}{2} b \int e^{i (i e+i f x)} (c+d x)^m \, dx \\ & = \frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {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 {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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int (c+d x)^m (a+b \cosh (e+f x)) \, dx=\frac {1}{2} (c+d x)^m \left (\frac {2 a (c+d x)}{d (1+m)}+\frac {b e^{e-\frac {c f}{d}} \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}-\frac {b e^{-e+\frac {c f}{d}} \left (f \left (\frac {c}{d}+x\right )\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}\right ) \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.90 \[ \int (c+d x)^m (a+b \cosh (e+f x)) \, dx=-\frac {{\left (b d m + b d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - {\left (b d m + b d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - {\left (b d m + b d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) + {\left (b d m + b d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) - 2 \, {\left (a d f x + a c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 2 \, {\left (a d f x + a c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{2 \, {\left (d f m + d f\right )}} \]

[In]

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

[Out]

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

Sympy [F(-2)]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.07 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.76 \[ \int (c+d x)^m (a+b \cosh (e+f x)) \, dx=-\frac {1}{2} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} b + \frac {{\left (d x + c\right )}^{m + 1} a}{d {\left (m + 1\right )}} \]

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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