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

Optimal result

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

[Out]

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*exp(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*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)-2^(-3-m)*b^2*exp(-2*e+2*c*f/d)*(d*x+c)^m*GAM
MA(1+m,2*f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3398, 3388, 2212, 3393} \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\frac {a^2 (c+d x)^{m+1}}{d (m+1)}+\frac {a b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {f (c+d x)}{d}\right )}{f}-\frac {a b e^{\frac {c f}{d}-e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {f (c+d x)}{d}\right )}{f}+\frac {b^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}-\frac {b^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {b^2 (c+d x)^{m+1}}{2 d (m+1)} \]

[In]

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

[Out]

(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)

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

Mathematica [A] (verified)

Time = 0.52 (sec) , antiderivative size = 254, normalized size of antiderivative = 0.90 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=\frac {(c+d x)^m \left (8 a^2 f (c+d x)+4 b^2 f (c+d x)+2^{-m} b^2 d e^{2 e-\frac {2 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )+8 a b d e^{e-\frac {c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )-8 a b d e^{-e+\frac {c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )-2^{-m} b^2 d e^{-2 e+\frac {2 c f}{d}} (1+m) \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )\right )}{8 d f (1+m)} \]

[In]

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

[Out]

((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)*Gamma[1 + m, (2*f*(c + d*x))/d])/(2^m*((f*(c + d*x))/d)^m)))/(8*d*f*(1
+ m))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.09 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.80 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=-\frac {{\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 8 \, {\left (a b d m + a b d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) - 8 \, {\left (a b d m + a b d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) - {\left (b^{2} d m + b^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right ) - 8 \, {\left (a b d m + a b d\right )} \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {f}{d}\right ) + d e - c f}{d}\right ) + 8 \, {\left (a b d m + a b d\right )} \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {f}{d}\right ) - d e + c f}{d}\right ) + {\left (b^{2} d m + b^{2} d\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right ) - 4 \, {\left ({\left (2 \, a^{2} + b^{2}\right )} d f x + {\left (2 \, a^{2} + b^{2}\right )} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 4 \, {\left ({\left (2 \, a^{2} + b^{2}\right )} d f x + {\left (2 \, a^{2} + b^{2}\right )} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{8 \, {\left (d f m + d f\right )}} \]

[In]

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

[Out]

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

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.10 (sec) , antiderivative size = 208, normalized size of antiderivative = 0.74 \[ \int (c+d x)^m (a+b \cosh (e+f x))^2 \, dx=-{\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-e + \frac {c f}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (e - \frac {c f}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a b - \frac {1}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (-2 \, e + \frac {2 \, c f}{d}\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (2 \, e - \frac {2 \, c f}{d}\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} b^{2} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \]

[In]

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

[Out]

-((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_i
ntegral_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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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