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

Optimal. Leaf size=110 \[ \frac{e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )}{2 b}-\frac{e^{\frac{b c}{d}-a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0913823, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3307, 2181} \[ \frac{e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )}{2 b}-\frac{e^{\frac{b c}{d}-a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^m*Cosh[a + b*x],x]

[Out]

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

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

Mathematica [A]  time = 0.0558259, size = 102, normalized size = 0.93 \[ \frac{e^{-a-\frac{b c}{d}} (c+d x)^m \left (e^{2 a} \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )-e^{\frac{2 b c}{d}} \left (b \left (\frac{c}{d}+x\right )\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )\right )}{2 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m}\cosh \left ( bx+a \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.24449, size = 107, normalized size = 0.97 \begin{align*} -\frac{{\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 )}{2 \, d} - \frac{{\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 )}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.90513, size = 378, normalized size = 3.44 \begin{align*} -\frac{\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{b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac{b d x + b c}{d}\right ) - \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{b d x + b c}{d}\right ) \sinh \left (\frac{d m \log \left (-\frac{b}{d}\right ) + b c - a d}{d}\right )}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*cosh(b*x+a),x)

[Out]

Exception raised: TypeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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