3.2.82 \(\int (c+d x)^m (a+b \sinh (e+f x))^3 \, dx\) [182]

Optimal. Leaf size=543 \[ \frac {a^3 (c+d x)^{1+m}}{d (1+m)}-\frac {3 a b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {3^{-1-m} b^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a b^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a^2 b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}-\frac {3 b^3 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{8 f}+\frac {3 a^2 b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f}-\frac {3 b^3 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{8 f}-\frac {3\ 2^{-3-m} a b^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3^{-1-m} b^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f} \]

[Out]

a^3*(d*x+c)^(1+m)/d/(1+m)-3/2*a*b^2*(d*x+c)^(1+m)/d/(1+m)+1/8*3^(-1-m)*b^3*exp(3*e-3*c*f/d)*(d*x+c)^m*GAMMA(1+
m,-3*f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m)+3*2^(-3-m)*a*b^2*exp(2*e-2*c*f/d)*(d*x+c)^m*GAMMA(1+m,-2*f*(d*x+c)/d)/f
/((-f*(d*x+c)/d)^m)+3/2*a^2*b*exp(e-c*f/d)*(d*x+c)^m*GAMMA(1+m,-f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m)-3/8*b^3*exp(
e-c*f/d)*(d*x+c)^m*GAMMA(1+m,-f*(d*x+c)/d)/f/((-f*(d*x+c)/d)^m)+3/2*a^2*b*exp(-e+c*f/d)*(d*x+c)^m*GAMMA(1+m,f*
(d*x+c)/d)/f/((f*(d*x+c)/d)^m)-3/8*b^3*exp(-e+c*f/d)*(d*x+c)^m*GAMMA(1+m,f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)-3*2^
(-3-m)*a*b^2*exp(-2*e+2*c*f/d)*(d*x+c)^m*GAMMA(1+m,2*f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)+1/8*3^(-1-m)*b^3*exp(-3*
e+3*c*f/d)*(d*x+c)^m*GAMMA(1+m,3*f*(d*x+c)/d)/f/((f*(d*x+c)/d)^m)

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Rubi [A]
time = 0.56, antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3398, 3389, 2212, 3393, 3388} \begin {gather*} \frac {3 a^2 b 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 {3 a^2 b 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 {3 a b^2 2^{-m-3} e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {2 f (c+d x)}{d}\right )}{f}-\frac {3 a b^2 2^{-m-3} e^{\frac {2 c f}{d}-2 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {b^3 3^{-m-1} e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,-\frac {3 f (c+d x)}{d}\right )}{8 f}-\frac {3 b^3 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 )}{8 f}-\frac {3 b^3 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 )}{8 f}+\frac {b^3 3^{-m-1} e^{\frac {3 c f}{d}-3 e} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \text {Gamma}\left (m+1,\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {a^3 (c+d x)^{m+1}}{d (m+1)}-\frac {3 a b^2 (c+d x)^{m+1}}{2 d (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(c + d*x)^(1 + m))/(d*(1 + m)) - (3*a*b^2*(c + d*x)^(1 + m))/(2*d*(1 + m)) + (3^(-1 - m)*b^3*E^(3*e - (3*
c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-3*f*(c + d*x))/d])/(8*f*(-((f*(c + d*x))/d))^m) + (3*2^(-3 - m)*a*b^2*E^(2*
e - (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (-2*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) + (3*a^2*b*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) - (3*b^3*E^(e - (c*f)/d)*(c +
d*x)^m*Gamma[1 + m, -((f*(c + d*x))/d)])/(8*f*(-((f*(c + d*x))/d))^m) + (3*a^2*b*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) - (3*b^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f
*(c + d*x))/d])/(8*f*((f*(c + d*x))/d)^m) - (3*2^(-3 - m)*a*b^2*E^(-2*e + (2*c*f)/d)*(c + d*x)^m*Gamma[1 + m,
(2*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (3^(-1 - m)*b^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (3
*f*(c + d*x))/d])/(8*f*((f*(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 3388

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 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]

Rule 3393

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 3398

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])

Rubi steps

\begin {align*} \int (c+d x)^m (a+b \sinh (e+f x))^3 \, dx &=\int \left (a^3 (c+d x)^m+3 a^2 b (c+d x)^m \sinh (e+f x)+3 a b^2 (c+d x)^m \sinh ^2(e+f x)+b^3 (c+d x)^m \sinh ^3(e+f x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}+\left (3 a^2 b\right ) \int (c+d x)^m \sinh (e+f x) \, dx+\left (3 a b^2\right ) \int (c+d x)^m \sinh ^2(e+f x) \, dx+b^3 \int (c+d x)^m \sinh ^3(e+f x) \, dx\\ &=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}+\frac {1}{2} \left (3 a^2 b\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx-\frac {1}{2} \left (3 a^2 b\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx-\left (3 a b^2\right ) \int \left (\frac {1}{2} (c+d x)^m-\frac {1}{2} (c+d x)^m \cosh (2 e+2 f x)\right ) \, dx+\left (i b^3\right ) \int \left (\frac {3}{4} i (c+d x)^m \sinh (e+f x)-\frac {1}{4} i (c+d x)^m \sinh (3 e+3 f x)\right ) \, dx\\ &=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}-\frac {3 a b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {3 a^2 b 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 {3 a^2 b 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 {1}{2} \left (3 a b^2\right ) \int (c+d x)^m \cosh (2 e+2 f x) \, dx+\frac {1}{4} b^3 \int (c+d x)^m \sinh (3 e+3 f x) \, dx-\frac {1}{4} \left (3 b^3\right ) \int (c+d x)^m \sinh (e+f x) \, dx\\ &=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}-\frac {3 a b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {3 a^2 b 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 {3 a^2 b 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 {1}{4} \left (3 a b^2\right ) \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac {1}{4} \left (3 a b^2\right ) \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx+\frac {1}{8} b^3 \int e^{-i (3 i e+3 i f x)} (c+d x)^m \, dx-\frac {1}{8} b^3 \int e^{i (3 i e+3 i f x)} (c+d x)^m \, dx-\frac {1}{8} \left (3 b^3\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx+\frac {1}{8} \left (3 b^3\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx\\ &=\frac {a^3 (c+d x)^{1+m}}{d (1+m)}-\frac {3 a b^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {3^{-1-m} b^3 e^{3 e-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {3 f (c+d x)}{d}\right )}{8 f}+\frac {3\ 2^{-3-m} a b^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3 a^2 b 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 {3 b^3 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 )}{8 f}+\frac {3 a^2 b 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 {3 b^3 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 )}{8 f}-\frac {3\ 2^{-3-m} a b^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f}+\frac {3^{-1-m} b^3 e^{-3 e+\frac {3 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {3 f (c+d x)}{d}\right )}{8 f}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2639\) vs. \(2(543)=1086\).
time = 18.52, size = 2639, normalized size = 4.86 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(c + d*x)^m*(c*f + d*f*x))/(d*f*(1 + m)) + (3*a^2*b*((f*Cosh[((-c + (d*e)/f)*f)/d]*(-((c - (d*e)/f + (d*(
e + f*x))/f)^(1 + m)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, -((f*(c - (d*e)/f + (d*(
e + f*x))/f))/d)]) + (c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^(-1 - m)*
Gamma[1 + m, (f*(c - (d*e)/f + (d*(e + f*x))/f))/d]))/(2*d) + (f*(-((c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*(-
((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, -((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)]) - (c
- (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^(-1 - m)*Gamma[1 + m, (f*(c - (d
*e)/f + (d*(e + f*x))/f))/d])*Sinh[((-c + (d*e)/f)*f)/d])/(2*d)))/f + (3*a*b^2*((f*Cosh[((-c + (d*e)/f)*f)/d]^
2*(-1/2*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)/(1 + m) + (-(2^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m
)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, (-2*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])
- 2^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^(-1 - m)*Gamma[1
+ m, (2*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])/4))/d + (f*Cosh[((-c + (d*e)/f)*f)/d]*(-(2^(-1 - m)*(c - (d*e)/
f + (d*(e + f*x))/f)^(1 + m)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, (-2*f*(c - (d*e)
/f + (d*(e + f*x))/f))/d]) + 2^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x
))/f))/d)^(-1 - m)*Gamma[1 + m, (2*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])*Sinh[((-c + (d*e)/f)*f)/d])/(2*d) +
(f*((c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)/(2*(1 + m)) + (-(2^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m
)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, (-2*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])
- 2^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^(-1 - m)*Gamma[1
+ m, (2*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])/4)*Sinh[((-c + (d*e)/f)*f)/d]^2)/d))/f + (b^3*((f*Cosh[((-c + (
d*e)/f)*f)/d]^3*((-3*(-((c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1
- m)*Gamma[1 + m, -((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)]) + (c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c
- (d*e)/f + (d*(e + f*x))/f))/d)^(-1 - m)*Gamma[1 + m, (f*(c - (d*e)/f + (d*(e + f*x))/f))/d]))/8 + (-(3^(-1
- m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m,
(-3*f*(c - (d*e)/f + (d*(e + f*x))/f))/d]) + 3^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)
/f + (d*(e + f*x))/f))/d)^(-1 - m)*Gamma[1 + m, (3*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])/8))/d + (f*((3*(-((c
- (d*e)/f + (d*(e + f*x))/f)^(1 + m)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, -((f*(c
- (d*e)/f + (d*(e + f*x))/f))/d)]) - (c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))
/f))/d)^(-1 - m)*Gamma[1 + m, (f*(c - (d*e)/f + (d*(e + f*x))/f))/d]))/8 + (-(3^(-1 - m)*(c - (d*e)/f + (d*(e
+ f*x))/f)^(1 + m)*(-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^(-1 - m)*Gamma[1 + m, (-3*f*(c - (d*e)/f + (d*(e
+ f*x))/f))/d]) - 3^(-1 - m)*(c - (d*e)/f + (d*(e + f*x))/f)^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^
(-1 - m)*Gamma[1 + m, (3*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])/8)*Sinh[((-c + (d*e)/f)*f)/d]^3)/d + ((c - (d*
e)/f + (d*(e + f*x))/f)^m*Cosh[e - (c*f)/d]^2*((-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^m*((f*(c - (d*e)/f +
(d*(e + f*x))/f))/d)^(2*m)*Gamma[1 + m, (-3*f*(c - (d*e)/f + (d*(e + f*x))/f))/d] - (-((f^2*(c - (d*e)/f + (d
*(e + f*x))/f)^2)/d^2))^m*(3^(1 + m)*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^m*Gamma[1 + m, -((f*(c - (d*e)/f
+ (d*(e + f*x))/f))/d)] + (-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^m*(-(3^(1 + m)*Gamma[1 + m, (f*(c - (d*e)
/f + (d*(e + f*x))/f))/d]) + Gamma[1 + m, (3*f*(c - (d*e)/f + (d*(e + f*x))/f))/d])))*Sinh[e - (c*f)/d])/(8*3^
m*(-((f^2*(c - (d*e)/f + (d*(e + f*x))/f)^2)/d^2))^(2*m)) + ((c - (d*e)/f + (d*(e + f*x))/f)^m*Cosh[e - (c*f)/
d]*((-((f*(c - (d*e)/f + (d*(e + f*x))/f))/d))^m*((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)^(2*m)*Gamma[1 + m, (-
3*f*(c - (d*e)/f + (d*(e + f*x))/f))/d] + (-((f^2*(c - (d*e)/f + (d*(e + f*x))/f)^2)/d^2))^m*(3^(1 + m)*((f*(c
- (d*e)/f + (d*(e + f*x))/f))/d)^m*Gamma[1 + m, -((f*(c - (d*e)/f + (d*(e + f*x))/f))/d)] + (-((f*(c - (d*e)/
f + (d*(e + f*x))/f))/d))^m*(3^(1 + m)*Gamma[1 + m, (f*(c - (d*e)/f + (d*(e + f*x))/f))/d] + Gamma[1 + m, (3*f
*(c - (d*e)/f + (d*(e + f*x))/f))/d])))*Sinh[e - (c*f)/d]^2)/(8*3^m*(-((f^2*(c - (d*e)/f + (d*(e + f*x))/f)^2)
/d^2))^(2*m))))/f

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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +b \sinh \left (f x +e \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.12, size = 385, normalized size = 0.71 \begin {gather*} \frac {3}{2} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (\frac {c f}{d} - e\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {c f}{d} + e\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d}\right )} a^{2} b - \frac {3}{4} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (\frac {2 \, c f}{d} - 2 \, e\right )} E_{-m}\left (\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {2 \, c f}{d} + 2 \, e\right )} E_{-m}\left (-\frac {2 \, {\left (d x + c\right )} f}{d}\right )}{d} + \frac {2 \, {\left (d x + c\right )}^{m + 1}}{d {\left (m + 1\right )}}\right )} a b^{2} + \frac {1}{8} \, {\left (\frac {{\left (d x + c\right )}^{m + 1} e^{\left (\frac {3 \, c f}{d} - 3 \, e\right )} E_{-m}\left (\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d} - \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (\frac {c f}{d} - e\right )} E_{-m}\left (\frac {{\left (d x + c\right )} f}{d}\right )}{d} + \frac {3 \, {\left (d x + c\right )}^{m + 1} e^{\left (-\frac {c f}{d} + e\right )} E_{-m}\left (-\frac {{\left (d x + c\right )} f}{d}\right )}{d} - \frac {{\left (d x + c\right )}^{m + 1} e^{\left (-\frac {3 \, c f}{d} + 3 \, e\right )} E_{-m}\left (-\frac {3 \, {\left (d x + c\right )} f}{d}\right )}{d}\right )} b^{3} + \frac {{\left (d x + c\right )}^{m + 1} a^{3}}{d {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2*((d*x + c)^(m + 1)*e^(c*f/d - e)*exp_integral_e(-m, (d*x + c)*f/d)/d - (d*x + c)^(m + 1)*e^(-c*f/d + e)*ex
p_integral_e(-m, -(d*x + c)*f/d)/d)*a^2*b - 3/4*((d*x + c)^(m + 1)*e^(2*c*f/d - 2*e)*exp_integral_e(-m, 2*(d*x
+ c)*f/d)/d + (d*x + c)^(m + 1)*e^(-2*c*f/d + 2*e)*exp_integral_e(-m, -2*(d*x + c)*f/d)/d + 2*(d*x + c)^(m +
1)/(d*(m + 1)))*a*b^2 + 1/8*((d*x + c)^(m + 1)*e^(3*c*f/d - 3*e)*exp_integral_e(-m, 3*(d*x + c)*f/d)/d - 3*(d*
x + c)^(m + 1)*e^(c*f/d - e)*exp_integral_e(-m, (d*x + c)*f/d)/d + 3*(d*x + c)^(m + 1)*e^(-c*f/d + e)*exp_inte
gral_e(-m, -(d*x + c)*f/d)/d - (d*x + c)^(m + 1)*e^(-3*c*f/d + 3*e)*exp_integral_e(-m, -3*(d*x + c)*f/d)/d)*b^
3 + (d*x + c)^(m + 1)*a^3/(d*(m + 1))

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Fricas [A]
time = 0.11, size = 899, normalized size = 1.66 \begin {gather*} \frac {{\left (b^{3} d m + b^{3} d\right )} \cosh \left (\frac {d m \log \left (\frac {3 \, f}{d}\right ) - 3 \, c f + 3 \, d \cosh \left (1\right ) + 3 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) - 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \cosh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) - 2 \, c f + 2 \, d \cosh \left (1\right ) + 2 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + 9 \, {\left ({\left (4 \, a^{2} b - b^{3}\right )} d m + {\left (4 \, a^{2} b - b^{3}\right )} d\right )} \cosh \left (\frac {d m \log \left (\frac {f}{d}\right ) - c f + d \cosh \left (1\right ) + d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, \frac {d f x + c f}{d}\right ) + 9 \, {\left ({\left (4 \, a^{2} b - b^{3}\right )} d m + {\left (4 \, a^{2} b - b^{3}\right )} d\right )} \cosh \left (\frac {d m \log \left (-\frac {f}{d}\right ) + c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, -\frac {d f x + c f}{d}\right ) + 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \cosh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) + 2 \, c f - 2 \, d \cosh \left (1\right ) - 2 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) + {\left (b^{3} d m + b^{3} d\right )} \cosh \left (\frac {d m \log \left (-\frac {3 \, f}{d}\right ) + 3 \, c f - 3 \, d \cosh \left (1\right ) - 3 \, d \sinh \left (1\right )}{d}\right ) \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) - {\left (b^{3} d m + b^{3} d\right )} \Gamma \left (m + 1, \frac {3 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {3 \, f}{d}\right ) - 3 \, c f + 3 \, d \cosh \left (1\right ) + 3 \, d \sinh \left (1\right )}{d}\right ) + 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \Gamma \left (m + 1, \frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (\frac {2 \, f}{d}\right ) - 2 \, c f + 2 \, d \cosh \left (1\right ) + 2 \, d \sinh \left (1\right )}{d}\right ) - 9 \, {\left ({\left (4 \, a^{2} b - b^{3}\right )} d m + {\left (4 \, a^{2} b - b^{3}\right )} 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 ) - c f + d \cosh \left (1\right ) + d \sinh \left (1\right )}{d}\right ) - 9 \, {\left ({\left (4 \, a^{2} b - b^{3}\right )} d m + {\left (4 \, a^{2} b - b^{3}\right )} 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 ) + c f - d \cosh \left (1\right ) - d \sinh \left (1\right )}{d}\right ) - 9 \, {\left (a b^{2} d m + a b^{2} d\right )} \Gamma \left (m + 1, -\frac {2 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {2 \, f}{d}\right ) + 2 \, c f - 2 \, d \cosh \left (1\right ) - 2 \, d \sinh \left (1\right )}{d}\right ) - {\left (b^{3} d m + b^{3} d\right )} \Gamma \left (m + 1, -\frac {3 \, {\left (d f x + c f\right )}}{d}\right ) \sinh \left (\frac {d m \log \left (-\frac {3 \, f}{d}\right ) + 3 \, c f - 3 \, d \cosh \left (1\right ) - 3 \, d \sinh \left (1\right )}{d}\right ) + 12 \, {\left ({\left (2 \, a^{3} - 3 \, a b^{2}\right )} d f x + {\left (2 \, a^{3} - 3 \, a b^{2}\right )} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 12 \, {\left ({\left (2 \, a^{3} - 3 \, a b^{2}\right )} d f x + {\left (2 \, a^{3} - 3 \, a b^{2}\right )} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{24 \, {\left (d f m + d f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((b^3*d*m + b^3*d)*cosh((d*m*log(3*f/d) - 3*c*f + 3*d*cosh(1) + 3*d*sinh(1))/d)*gamma(m + 1, 3*(d*f*x + c
*f)/d) - 9*(a*b^2*d*m + a*b^2*d)*cosh((d*m*log(2*f/d) - 2*c*f + 2*d*cosh(1) + 2*d*sinh(1))/d)*gamma(m + 1, 2*(
d*f*x + c*f)/d) + 9*((4*a^2*b - b^3)*d*m + (4*a^2*b - b^3)*d)*cosh((d*m*log(f/d) - c*f + d*cosh(1) + d*sinh(1)
)/d)*gamma(m + 1, (d*f*x + c*f)/d) + 9*((4*a^2*b - b^3)*d*m + (4*a^2*b - b^3)*d)*cosh((d*m*log(-f/d) + c*f - d
*cosh(1) - d*sinh(1))/d)*gamma(m + 1, -(d*f*x + c*f)/d) + 9*(a*b^2*d*m + a*b^2*d)*cosh((d*m*log(-2*f/d) + 2*c*
f - 2*d*cosh(1) - 2*d*sinh(1))/d)*gamma(m + 1, -2*(d*f*x + c*f)/d) + (b^3*d*m + b^3*d)*cosh((d*m*log(-3*f/d) +
3*c*f - 3*d*cosh(1) - 3*d*sinh(1))/d)*gamma(m + 1, -3*(d*f*x + c*f)/d) - (b^3*d*m + b^3*d)*gamma(m + 1, 3*(d*
f*x + c*f)/d)*sinh((d*m*log(3*f/d) - 3*c*f + 3*d*cosh(1) + 3*d*sinh(1))/d) + 9*(a*b^2*d*m + a*b^2*d)*gamma(m +
1, 2*(d*f*x + c*f)/d)*sinh((d*m*log(2*f/d) - 2*c*f + 2*d*cosh(1) + 2*d*sinh(1))/d) - 9*((4*a^2*b - b^3)*d*m +
(4*a^2*b - b^3)*d)*gamma(m + 1, (d*f*x + c*f)/d)*sinh((d*m*log(f/d) - c*f + d*cosh(1) + d*sinh(1))/d) - 9*((4
*a^2*b - b^3)*d*m + (4*a^2*b - b^3)*d)*gamma(m + 1, -(d*f*x + c*f)/d)*sinh((d*m*log(-f/d) + c*f - d*cosh(1) -
d*sinh(1))/d) - 9*(a*b^2*d*m + a*b^2*d)*gamma(m + 1, -2*(d*f*x + c*f)/d)*sinh((d*m*log(-2*f/d) + 2*c*f - 2*d*c
osh(1) - 2*d*sinh(1))/d) - (b^3*d*m + b^3*d)*gamma(m + 1, -3*(d*f*x + c*f)/d)*sinh((d*m*log(-3*f/d) + 3*c*f -
3*d*cosh(1) - 3*d*sinh(1))/d) + 12*((2*a^3 - 3*a*b^2)*d*f*x + (2*a^3 - 3*a*b^2)*c*f)*cosh(m*log(d*x + c)) + 12
*((2*a^3 - 3*a*b^2)*d*f*x + (2*a^3 - 3*a*b^2)*c*f)*sinh(m*log(d*x + c)))/(d*f*m + d*f)

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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*(a+b*sinh(f*x+e))**3,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*(a+b*sinh(f*x+e))^3,x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )}^3\,{\left (c+d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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