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

Optimal. Leaf size=410 \[ -\frac{i a^3 3^{-m-1} e^{3 e-\frac{3 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )}{8 f}-\frac{3 a^3 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{15 i a^3 e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{8 f}+\frac{15 i a^3 e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{8 f}+\frac{3 a^3 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}-\frac{i a^3 3^{-m-1} e^{\frac{3 c f}{d}-3 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{5 a^3 (c+d x)^{m+1}}{2 d (m+1)} \]

[Out]

(5*a^3*(c + d*x)^(1 + m))/(2*d*(1 + m)) - ((I/8)*3^(-1 - m)*a^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (
-3*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) - (3*2^(-3 - m)*a^3*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) + (((15*I)/8)*a^3*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -
((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + (((15*I)/8)*a^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*
(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (3*2^(-3 - m)*a^3*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) - ((I/8)*3^(-1 - m)*a^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (
3*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)

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Rubi [A]  time = 0.602034, antiderivative size = 410, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {3318, 3312, 3307, 2181, 3308} \[ -\frac{i a^3 3^{-m-1} e^{3 e-\frac{3 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )}{8 f}-\frac{3 a^3 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{15 i a^3 e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{8 f}+\frac{15 i a^3 e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{8 f}+\frac{3 a^3 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}-\frac{i a^3 3^{-m-1} e^{\frac{3 c f}{d}-3 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )}{8 f}+\frac{5 a^3 (c+d x)^{m+1}}{2 d (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a^3*(c + d*x)^(1 + m))/(2*d*(1 + m)) - ((I/8)*3^(-1 - m)*a^3*E^(3*e - (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (
-3*f*(c + d*x))/d])/(f*(-((f*(c + d*x))/d))^m) - (3*2^(-3 - m)*a^3*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) + (((15*I)/8)*a^3*E^(e - (c*f)/d)*(c + d*x)^m*Gamma[1 + m, -
((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + (((15*I)/8)*a^3*E^(-e + (c*f)/d)*(c + d*x)^m*Gamma[1 + m, (f*
(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (3*2^(-3 - m)*a^3*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) - ((I/8)*3^(-1 - m)*a^3*E^(-3*e + (3*c*f)/d)*(c + d*x)^m*Gamma[1 + m, (
3*f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m)

Rule 3318

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1*(e + (Pi*a)/(2*b)))/2 + (f*x)/2]^(2*n), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[a^2
- b^2, 0] && IntegerQ[n] && (GtQ[n, 0] || IGtQ[m, 0])

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]

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]

Rubi steps

\begin{align*} \int (c+d x)^m (a+i a \sinh (e+f x))^3 \, dx &=\left (8 a^3\right ) \int (c+d x)^m \sin ^6\left (\frac{1}{2} \left (i e+\frac{\pi }{2}\right )+\frac{i f x}{2}\right ) \, dx\\ &=\left (8 a^3\right ) \int \left (\frac{5}{16} (c+d x)^m-\frac{3}{16} (c+d x)^m \cosh (2 e+2 f x)+\frac{15}{32} i (c+d x)^m \sinh (e+f x)-\frac{1}{32} i (c+d x)^m \sinh (3 e+3 f x)\right ) \, dx\\ &=\frac{5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac{1}{4} \left (i a^3\right ) \int (c+d x)^m \sinh (3 e+3 f x) \, dx+\frac{1}{4} \left (15 i a^3\right ) \int (c+d x)^m \sinh (e+f x) \, dx-\frac{1}{2} \left (3 a^3\right ) \int (c+d x)^m \cosh (2 e+2 f x) \, dx\\ &=\frac{5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac{1}{8} \left (i a^3\right ) \int e^{-i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac{1}{8} \left (i a^3\right ) \int e^{i (3 i e+3 i f x)} (c+d x)^m \, dx+\frac{1}{8} \left (15 i a^3\right ) \int e^{-i (i e+i f x)} (c+d x)^m \, dx-\frac{1}{8} \left (15 i a^3\right ) \int e^{i (i e+i f x)} (c+d x)^m \, dx-\frac{1}{4} \left (3 a^3\right ) \int e^{-i (2 i e+2 i f x)} (c+d x)^m \, dx-\frac{1}{4} \left (3 a^3\right ) \int e^{i (2 i e+2 i f x)} (c+d x)^m \, dx\\ &=\frac{5 a^3 (c+d x)^{1+m}}{2 d (1+m)}-\frac{i 3^{-1-m} a^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^3 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{15 i a^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{15 i a^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^3 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{i 3^{-1-m} a^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 = 1.58512, size = 339, normalized size = 0.83 \[ \frac{1}{24} a^3 (c+d x)^m \left (-\frac{i 3^{-m} e^{3 e-\frac{3 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{3 f (c+d x)}{d}\right )}{f}-\frac{9\ 2^{-m} e^{2 e-\frac{2 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{45 i e^{e-\frac{c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{f}+\frac{45 i e^{\frac{c f}{d}-e} \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{f}+\frac{9\ 2^{-m} e^{\frac{2 c f}{d}-2 e} \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}-\frac{i 3^{-m} e^{\frac{3 c f}{d}-3 e} \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{3 f (c+d x)}{d}\right )}{f}+\frac{60 (c+d x)}{d (m+1)}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^3*(c + d*x)^m*((60*(c + d*x))/(d*(1 + m)) - (I*E^(3*e - (3*c*f)/d)*Gamma[1 + m, (-3*f*(c + d*x))/d])/(3^m*f
*(-((f*(c + d*x))/d))^m) - (9*E^(2*e - (2*c*f)/d)*Gamma[1 + m, (-2*f*(c + d*x))/d])/(2^m*f*(-((f*(c + d*x))/d)
)^m) + ((45*I)*E^(e - (c*f)/d)*Gamma[1 + m, -((f*(c + d*x))/d)])/(f*(-((f*(c + d*x))/d))^m) + ((45*I)*E^(-e +
(c*f)/d)*Gamma[1 + m, (f*(c + d*x))/d])/(f*((f*(c + d*x))/d)^m) + (9*E^(-2*e + (2*c*f)/d)*Gamma[1 + m, (2*f*(c
+ d*x))/d])/(2^m*f*((f*(c + d*x))/d)^m) - (I*E^(-3*e + (3*c*f)/d)*Gamma[1 + m, (3*f*(c + d*x))/d])/(3^m*f*((f
*(c + d*x))/d)^m)))/24

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Maple [F]  time = 0.12, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+ia\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+I*a*sinh(f*x+e))^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.8322, size = 857, normalized size = 2.09 \begin{align*} \frac{{\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac{d m \log \left (\frac{3 \, f}{d}\right ) + 3 \, d e - 3 \, c f}{d}\right )} \Gamma \left (m + 1, \frac{3 \,{\left (d f x + c f\right )}}{d}\right ) + 9 \,{\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac{d m \log \left (\frac{2 \, f}{d}\right ) + 2 \, d e - 2 \, c f}{d}\right )} \Gamma \left (m + 1, \frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (45 i \, a^{3} d m + 45 i \, a^{3} d\right )} e^{\left (-\frac{d m \log \left (\frac{f}{d}\right ) + d e - c f}{d}\right )} \Gamma \left (m + 1, \frac{d f x + c f}{d}\right ) +{\left (45 i \, a^{3} d m + 45 i \, a^{3} d\right )} e^{\left (-\frac{d m \log \left (-\frac{f}{d}\right ) - d e + c f}{d}\right )} \Gamma \left (m + 1, -\frac{d f x + c f}{d}\right ) - 9 \,{\left (a^{3} d m + a^{3} d\right )} e^{\left (-\frac{d m \log \left (-\frac{2 \, f}{d}\right ) - 2 \, d e + 2 \, c f}{d}\right )} \Gamma \left (m + 1, -\frac{2 \,{\left (d f x + c f\right )}}{d}\right ) +{\left (-i \, a^{3} d m - i \, a^{3} d\right )} e^{\left (-\frac{d m \log \left (-\frac{3 \, f}{d}\right ) - 3 \, d e + 3 \, c f}{d}\right )} \Gamma \left (m + 1, -\frac{3 \,{\left (d f x + c f\right )}}{d}\right ) + 60 \,{\left (a^{3} d f x + a^{3} c f\right )}{\left (d x + c\right )}^{m}}{24 \,{\left (d f m + d f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*((-I*a^3*d*m - I*a^3*d)*e^(-(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^3*
d*m + a^3*d)*e^(-(d*m*log(2*f/d) + 2*d*e - 2*c*f)/d)*gamma(m + 1, 2*(d*f*x + c*f)/d) + (45*I*a^3*d*m + 45*I*a^
3*d)*e^(-(d*m*log(f/d) + d*e - c*f)/d)*gamma(m + 1, (d*f*x + c*f)/d) + (45*I*a^3*d*m + 45*I*a^3*d)*e^(-(d*m*lo
g(-f/d) - d*e + c*f)/d)*gamma(m + 1, -(d*f*x + c*f)/d) - 9*(a^3*d*m + a^3*d)*e^(-(d*m*log(-2*f/d) - 2*d*e + 2*
c*f)/d)*gamma(m + 1, -2*(d*f*x + c*f)/d) + (-I*a^3*d*m - I*a^3*d)*e^(-(d*m*log(-3*f/d) - 3*d*e + 3*c*f)/d)*gam
ma(m + 1, -3*(d*f*x + c*f)/d) + 60*(a^3*d*f*x + a^3*c*f)*(d*x + c)^m)/(d*f*m + d*f)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (i \, a \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+I*a*sinh(f*x+e))^3,x, algorithm="giac")

[Out]

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