3.1.75 \(\int (c+d x)^m \sinh (a+b x) \, dx\) [75]

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

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 110, normalized size of antiderivative = 1.00, 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 = {3389, 2212} \begin {gather*} \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} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^m*Sinh[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 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 3389

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

Rubi steps

\begin {align*} \int (c+d x)^m \sinh (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*}

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Mathematica [A]
time = 0.04, size = 101, normalized size = 0.92 \begin {gather*} \frac {e^{-a-\frac {b c}{d}} (c+d x)^m \left (e^{2 a} \left (-\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {b (c+d x)}{d}\right )+e^{\frac {2 b c}{d}} \left (b \left (\frac {c}{d}+x\right )\right )^{-m} \Gamma \left (1+m,\frac {b (c+d x)}{d}\right )\right )}{2 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^m*Sinh[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 = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \sinh \left (b x +a \right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.06, size = 79, normalized size = 0.72 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*sinh(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 = 0.09, size = 168, normalized size = 1.53 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*sinh(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)*ga
mma(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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \mathrm {sinh}\left (a+b\,x\right )\,{\left (c+d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)*(c + d*x)^m,x)

[Out]

int(sinh(a + b*x)*(c + d*x)^m, x)

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