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

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

[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)

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Rubi [A]  time = 0.807379, antiderivative size = 543, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3317, 3308, 2181, 3312, 3307} \[ \frac{3 a^2 b e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{2 f}+\frac{3 a^2 b e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{2 f}+\frac{3 a b^2 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}-\frac{3 a b^2 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}+\frac{b^3 3^{-m-1} e^{3 e-\frac{3 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )}{8 f}-\frac{3 b^3 e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{8 f}-\frac{3 b^3 e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{8 f}+\frac{b^3 3^{-m-1} e^{\frac{3 c f}{d}-3 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{a^3 (c+d x)^{m+1}}{d (m+1)}-\frac{3 a b^2 (c+d x)^{m+1}}{2 d (m+1)} \]

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 3317

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])

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]

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 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]

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*}

Mathematica [A]  time = 1.74719, size = 448, normalized size = 0.83 \[ \frac{2^{-m-3} 3^{-m-1} e^{-3 \left (\frac{c f}{d}+e\right )} (c+d x)^m \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (2^m e^{\frac{3 c f}{d}} \left (b^3 d (m+1) e^{\frac{3 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )+4 a e^{3 e} f 3^{m+1} \left (2 a^2-3 b^2\right ) (c+d x) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^m\right )-b d 2^m 3^{m+2} (m+1) \left (b^2-4 a^2\right ) e^{\frac{2 c f}{d}+4 e} \left (\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )-b d 2^m 3^{m+2} (m+1) \left (b^2-4 a^2\right ) e^{\frac{4 c f}{d}+2 e} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )+a b^2 d 3^{m+2} (m+1) e^{\frac{c f}{d}+5 e} \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )-a b^2 d 3^{m+2} (m+1) e^{\frac{5 c f}{d}+e} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )+b^3 d e^{6 e} 2^m (m+1) \left (\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )\right )}{d f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2^(-3 - m)*3^(-1 - m)*(c + d*x)^m*(2^m*b^3*d*E^(6*e)*(1 + m)*((f*(c + d*x))/d)^m*Gamma[1 + m, (-3*f*(c + d*x)
)/d] + 3^(2 + m)*a*b^2*d*E^(5*e + (c*f)/d)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, (-2*f*(c + d*x))/d] - 2^m*3^(2
+ m)*b*(-4*a^2 + b^2)*d*E^(4*e + (2*c*f)/d)*(1 + m)*((f*(c + d*x))/d)^m*Gamma[1 + m, -((f*(c + d*x))/d)] - 2^
m*3^(2 + m)*b*(-4*a^2 + b^2)*d*E^(2*e + (4*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (f*(c + d*x))/d
] - 3^(2 + m)*a*b^2*d*E^(e + (5*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (2*f*(c + d*x))/d] + 2^m*E
^((3*c*f)/d)*(4*3^(1 + m)*a*(2*a^2 - 3*b^2)*E^(3*e)*f*(c + d*x)*(-((f^2*(c + d*x)^2)/d^2))^m + b^3*d*E^((3*c*f
)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m, (3*f*(c + d*x))/d])))/(d*E^(3*(e + (c*f)/d))*f*(1 + m)*(-((f^
2*(c + d*x)^2)/d^2))^m)

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Maple [F]  time = 0.108, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+b\sinh \left ( fx+e \right ) \right ) ^{3}\, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 2.96735, size = 1885, normalized size = 3.47 \begin{align*} \text{result too large to display} \end{align*}

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*d*e - 3*c*f)/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*d*e - 2*c*f)/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) + d*e - c*f)/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) - d*e + c*f)/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*d*e + 2*c*f)/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*d*e + 3*c*f)/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*d*e - 3*c*f)/d) + 9*(a*b^2*d*m + a*b^2*d)*gamma(m + 1, 2*(d*f*x + c*f)/d)*sin
h((d*m*log(2*f/d) + 2*d*e - 2*c*f)/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) + d*e - c*f)/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) - d*e + c*f)/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*d*e + 2*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*d*e + 3*c*f)/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., 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*(a+b*sinh(f*x+e))**3,x)

[Out]

Exception raised: TypeError

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

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)