3.73 \(\int (c+d x)^m \sinh ^3(a+b x) \, dx\)

Optimal. Leaf size=237 \[ \frac{3^{-m-1} e^{3 a-\frac{3 b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 b (c+d x)}{d}\right )}{8 b}-\frac{3 e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )}{8 b}-\frac{3 e^{\frac{b c}{d}-a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )}{8 b}+\frac{3^{-m-1} e^{\frac{3 b c}{d}-3 a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 b (c+d x)}{d}\right )}{8 b} \]

[Out]

(3^(-1 - m)*E^(3*a - (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-3*b*(c + d*x))/d])/(8*b*(-((b*(c + d*x))/d))^m) - (
3*E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(8*b*(-((b*(c + d*x))/d))^m) - (3*E^(-a + (b*c
)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m) + (3^(-1 - m)*E^(-3*a + (3*b*c)/d)*(
c + d*x)^m*Gamma[1 + m, (3*b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m)

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Rubi [A]  time = 0.317832, antiderivative size = 237, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {3312, 3308, 2181} \[ \frac{3^{-m-1} e^{3 a-\frac{3 b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 b (c+d x)}{d}\right )}{8 b}-\frac{3 e^{a-\frac{b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )}{8 b}-\frac{3 e^{\frac{b c}{d}-a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )}{8 b}+\frac{3^{-m-1} e^{\frac{3 b c}{d}-3 a} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 b (c+d x)}{d}\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^m*Sinh[a + b*x]^3,x]

[Out]

(3^(-1 - m)*E^(3*a - (3*b*c)/d)*(c + d*x)^m*Gamma[1 + m, (-3*b*(c + d*x))/d])/(8*b*(-((b*(c + d*x))/d))^m) - (
3*E^(a - (b*c)/d)*(c + d*x)^m*Gamma[1 + m, -((b*(c + d*x))/d)])/(8*b*(-((b*(c + d*x))/d))^m) - (3*E^(-a + (b*c
)/d)*(c + d*x)^m*Gamma[1 + m, (b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m) + (3^(-1 - m)*E^(-3*a + (3*b*c)/d)*(
c + d*x)^m*Gamma[1 + m, (3*b*(c + d*x))/d])/(8*b*((b*(c + d*x))/d)^m)

Rule 3312

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 3308

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]

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 \sinh ^3(a+b x) \, dx &=i \int \left (\frac{3}{4} i (c+d x)^m \sinh (a+b x)-\frac{1}{4} i (c+d x)^m \sinh (3 a+3 b x)\right ) \, dx\\ &=\frac{1}{4} \int (c+d x)^m \sinh (3 a+3 b x) \, dx-\frac{3}{4} \int (c+d x)^m \sinh (a+b x) \, dx\\ &=\frac{1}{8} \int e^{-i (3 i a+3 i b x)} (c+d x)^m \, dx-\frac{1}{8} \int e^{i (3 i a+3 i b x)} (c+d x)^m \, dx-\frac{3}{8} \int e^{-i (i a+i b x)} (c+d x)^m \, dx+\frac{3}{8} \int e^{i (i a+i b x)} (c+d x)^m \, dx\\ &=\frac{3^{-1-m} e^{3 a-\frac{3 b c}{d}} (c+d x)^m \left (-\frac{b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac{3 b (c+d x)}{d}\right )}{8 b}-\frac{3 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 )}{8 b}-\frac{3 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 )}{8 b}+\frac{3^{-1-m} e^{-3 a+\frac{3 b c}{d}} (c+d x)^m \left (\frac{b (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac{3 b (c+d x)}{d}\right )}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.198479, size = 206, normalized size = 0.87 \[ \frac{3^{-m-1} e^{-3 \left (a+\frac{b c}{d}\right )} (c+d x)^m \left (-\frac{b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{6 a} \left (b \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{3 b (c+d x)}{d}\right )-3^{m+2} e^{4 a+\frac{2 b c}{d}} \left (b \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{b (c+d x)}{d}\right )+e^{\frac{4 b c}{d}} \left (-\frac{b (c+d x)}{d}\right )^m \left (e^{\frac{2 b c}{d}} \text{Gamma}\left (m+1,\frac{3 b (c+d x)}{d}\right )-e^{2 a} 3^{m+2} \text{Gamma}\left (m+1,\frac{b (c+d x)}{d}\right )\right )\right )}{8 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^m*Sinh[a + b*x]^3,x]

[Out]

(3^(-1 - m)*(c + d*x)^m*(E^(6*a)*(b*(c/d + x))^m*Gamma[1 + m, (-3*b*(c + d*x))/d] - 3^(2 + m)*E^(4*a + (2*b*c)
/d)*(b*(c/d + x))^m*Gamma[1 + m, -((b*(c + d*x))/d)] + E^((4*b*c)/d)*(-((b*(c + d*x))/d))^m*(-(3^(2 + m)*E^(2*
a)*Gamma[1 + m, (b*(c + d*x))/d]) + E^((2*b*c)/d)*Gamma[1 + m, (3*b*(c + d*x))/d])))/(8*b*E^(3*(a + (b*c)/d))*
(-((b^2*(c + d*x)^2)/d^2))^m)

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Maple [F]  time = 0.089, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( \sinh \left ( bx+a \right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m*sinh(b*x+a)^3,x)

[Out]

int((d*x+c)^m*sinh(b*x+a)^3,x)

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Maxima [A]  time = 1.54627, size = 217, normalized size = 0.92 \begin{align*} \frac{{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, a + \frac{3 \, b c}{d}\right )} E_{-m}\left (\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac{3 \,{\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 )}{8 \, d} + \frac{3 \,{\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 )}{8 \, d} - \frac{{\left (d x + c\right )}^{m + 1} e^{\left (3 \, a - \frac{3 \, b c}{d}\right )} E_{-m}\left (-\frac{3 \,{\left (d x + c\right )} b}{d}\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*sinh(b*x+a)^3,x, algorithm="maxima")

[Out]

1/8*(d*x + c)^(m + 1)*e^(-3*a + 3*b*c/d)*exp_integral_e(-m, 3*(d*x + c)*b/d)/d - 3/8*(d*x + c)^(m + 1)*e^(-a +
b*c/d)*exp_integral_e(-m, (d*x + c)*b/d)/d + 3/8*(d*x + c)^(m + 1)*e^(a - b*c/d)*exp_integral_e(-m, -(d*x + c
)*b/d)/d - 1/8*(d*x + c)^(m + 1)*e^(3*a - 3*b*c/d)*exp_integral_e(-m, -3*(d*x + c)*b/d)/d

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Fricas [A]  time = 2.92055, size = 799, normalized size = 3.37 \begin{align*} \frac{\cosh \left (\frac{d m \log \left (\frac{3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) \Gamma \left (m + 1, \frac{3 \,{\left (b d x + b c\right )}}{d}\right ) - 9 \, \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 ) - 9 \, \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{3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right ) \Gamma \left (m + 1, -\frac{3 \,{\left (b d x + b c\right )}}{d}\right ) - \Gamma \left (m + 1, \frac{3 \,{\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (\frac{3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) + 9 \, \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 ) + 9 \, \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{3 \,{\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac{d m \log \left (-\frac{3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right )}{24 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*sinh(b*x+a)^3,x, algorithm="fricas")

[Out]

1/24*(cosh((d*m*log(3*b/d) - 3*b*c + 3*a*d)/d)*gamma(m + 1, 3*(b*d*x + b*c)/d) - 9*cosh((d*m*log(b/d) - b*c +
a*d)/d)*gamma(m + 1, (b*d*x + b*c)/d) - 9*cosh((d*m*log(-b/d) + b*c - a*d)/d)*gamma(m + 1, -(b*d*x + b*c)/d) +
cosh((d*m*log(-3*b/d) + 3*b*c - 3*a*d)/d)*gamma(m + 1, -3*(b*d*x + b*c)/d) - gamma(m + 1, 3*(b*d*x + b*c)/d)*
sinh((d*m*log(3*b/d) - 3*b*c + 3*a*d)/d) + 9*gamma(m + 1, (b*d*x + b*c)/d)*sinh((d*m*log(b/d) - b*c + a*d)/d)
+ 9*gamma(m + 1, -(b*d*x + b*c)/d)*sinh((d*m*log(-b/d) + b*c - a*d)/d) - gamma(m + 1, -3*(b*d*x + b*c)/d)*sinh
((d*m*log(-3*b/d) + 3*b*c - 3*a*d)/d))/b

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (c + d x\right )^{m} \sinh ^{3}{\left (a + b x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**m*sinh(b*x+a)**3,x)

[Out]

Integral((c + d*x)**m*sinh(a + b*x)**3, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{m} \sinh \left (b x + a\right )^{3}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*sinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^m*sinh(b*x + a)^3, x)