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

Optimal. Leaf size=131 \[ \frac{a e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{2 f}-\frac{a e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.147174, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3317, 3307, 2181} \[ \frac{a e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{2 f}-\frac{a e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{2 f}+\frac{a (c+d x)^{m+1}}{d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x)^(1 + m))/(d*(1 + m)) + (a*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) - (a*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 3317

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])

Rule 3307

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 2181

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

Rubi steps

\begin{align*} \int (c+d x)^m (a+a \cosh (e+f x)) \, dx &=\int \left (a (c+d x)^m+a (c+d x)^m \cosh (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+a \int (c+d x)^m \cosh (e+f x) \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+\frac{1}{2} a \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac{1}{2} a \int e^{i (i e+i f x)} (c+d x)^m \, dx\\ &=\frac{a (c+d x)^{1+m}}{d (1+m)}+\frac{a 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{a 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]  time = 0.31201, size = 189, normalized size = 1.44 \[ -\frac{a e^{-\frac{c f}{d}-e} (c+d x)^m (\cosh (e+f x)+1) \text{sech}^2\left (\frac{1}{2} (e+f x)\right ) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-d e^{2 e} (m+1) \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )+d (m+1) e^{\frac{2 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )-2 f (c+d x) e^{\frac{c f}{d}+e} \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^m\right )}{4 d f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a*E^(-e - (c*f)/d)*(c + d*x)^m*(1 + Cosh[e + f*x])*(-2*E^(e + (c*f)/d)*f*(c + d*x)*(-((f^2*(c + d*x)^2)/d^2)
)^m - d*E^(2*e)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, -((f*(c + d*x))/d)] + d*E^((2*c*f)/d)*(1 + m)*(-((f*(c +
d*x))/d))^m*Gamma[1 + m, (f*(c + d*x))/d])*Sech[(e + f*x)/2]^2)/(4*d*f*(1 + m)*(-((f^2*(c + d*x)^2)/d^2))^m)

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Maple [F]  time = 0.069, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+a\cosh \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.21396, size = 586, normalized size = 4.47 \begin{align*} -\frac{{\left (a d m + a 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 (a d m + a 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 (a d m + a 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 (a d m + a 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 )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cosh \left (f x + e\right ) + a\right )}{\left (d x + c\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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