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

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

[Out]

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

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Rubi [A]  time = 0.32, antiderivative size = 237, normalized size of antiderivative = 1.00, 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 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]

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 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])
)

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*}

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Mathematica [A]  time = 0.19, size = 206, normalized size = 0.87 \[ \frac {3^{-m-1} e^{-3 \left (a+\frac {b c}{d}\right )} (c+d x)^m \left (-\frac {b^2 (c+d x)^2}{d^2}\right )^{-m} \left (e^{6 a} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m+1,-\frac {3 b (c+d x)}{d}\right )-3^{m+2} e^{4 a+\frac {2 b c}{d}} \left (b \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (m+1,-\frac {b (c+d x)}{d}\right )+e^{\frac {4 b c}{d}} \left (-\frac {b (c+d x)}{d}\right )^m \left (e^{\frac {2 b c}{d}} \Gamma \left (m+1,\frac {3 b (c+d x)}{d}\right )-e^{2 a} 3^{m+2} \Gamma \left (m+1,\frac {b (c+d x)}{d}\right )\right )\right )}{8 b} \]

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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fricas [A]  time = 0.51, size = 340, normalized size = 1.43 \[ \frac {\cosh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) - 9 \, \cosh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) + \cosh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) - \Gamma \left (m + 1, \frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, b}{d}\right ) - 3 \, b c + 3 \, a d}{d}\right ) + 9 \, \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right ) + 9 \, \Gamma \left (m + 1, -\frac {b d x + b c}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {b}{d}\right ) + b c - a d}{d}\right ) - \Gamma \left (m + 1, -\frac {3 \, {\left (b d x + b c\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, b}{d}\right ) + 3 \, b c - 3 \, a d}{d}\right )}{24 \, b} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d x + c\right )}^{m} \sinh \left (b x + a\right )^{3}\,{d x} \]

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)

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maple [F]  time = 0.15, size = 0, normalized size = 0.00 \[ \int \left (d x +c \right )^{m} \left (\sinh ^{3}\left (b x +a \right )\right )\, dx \]

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 = 0.48, size = 161, normalized size = 0.68 \[ \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-3 \, a + \frac {3 \, b c}{d}\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-a + \frac {b c}{d}\right )} E_{-m}\left (\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (a - \frac {b c}{d}\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} b}{d}\right )}{8 \, d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (3 \, a - \frac {3 \, b c}{d}\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} b}{d}\right )}{8 \, d} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {sinh}\left (a+b\,x\right )}^3\,{\left (c+d\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sinh(a + b*x)^3*(c + d*x)^m,x)

[Out]

int(sinh(a + b*x)^3*(c + d*x)^m, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (c + d x\right )^{m} \sinh ^{3}{\left (a + b x \right )}\, dx \]

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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