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

Optimal. Leaf size=98 \[ \frac{6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac{3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{i a (c+d x)^3 \cosh (e+f x)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 i a d^3 \sinh (e+f x)}{f^4} \]

[Out]

(a*(c + d*x)^4)/(4*d) + ((6*I)*a*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (I*a*(c + d*x)^3*Cosh[e + f*x])/f - ((6*I)
*a*d^3*Sinh[e + f*x])/f^4 - ((3*I)*a*d*(c + d*x)^2*Sinh[e + f*x])/f^2

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Rubi [A]  time = 0.139952, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3317, 3296, 2637} \[ \frac{6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}-\frac{3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{i a (c+d x)^3 \cosh (e+f x)}{f}+\frac{a (c+d x)^4}{4 d}-\frac{6 i a d^3 \sinh (e+f x)}{f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x)^4)/(4*d) + ((6*I)*a*d^2*(c + d*x)*Cosh[e + f*x])/f^3 + (I*a*(c + d*x)^3*Cosh[e + f*x])/f - ((6*I)
*a*d^3*Sinh[e + f*x])/f^4 - ((3*I)*a*d*(c + d*x)^2*Sinh[e + f*x])/f^2

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 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin{align*} \int (c+d x)^3 (a+i a \sinh (e+f x)) \, dx &=\int \left (a (c+d x)^3+i a (c+d x)^3 \sinh (e+f x)\right ) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+(i a) \int (c+d x)^3 \sinh (e+f x) \, dx\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i a (c+d x)^3 \cosh (e+f x)}{f}-\frac{(3 i a d) \int (c+d x)^2 \cosh (e+f x) \, dx}{f}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{i a (c+d x)^3 \cosh (e+f x)}{f}-\frac{3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}+\frac{\left (6 i a d^2\right ) \int (c+d x) \sinh (e+f x) \, dx}{f^2}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac{i a (c+d x)^3 \cosh (e+f x)}{f}-\frac{3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}-\frac{\left (6 i a d^3\right ) \int \cosh (e+f x) \, dx}{f^3}\\ &=\frac{a (c+d x)^4}{4 d}+\frac{6 i a d^2 (c+d x) \cosh (e+f x)}{f^3}+\frac{i a (c+d x)^3 \cosh (e+f x)}{f}-\frac{6 i a d^3 \sinh (e+f x)}{f^4}-\frac{3 i a d (c+d x)^2 \sinh (e+f x)}{f^2}\\ \end{align*}

Mathematica [A]  time = 0.832429, size = 128, normalized size = 1.31 \[ \frac{a \left (-12 i d \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+2\right )\right ) \sinh (e+f x)+4 i f (c+d x) \left (c^2 f^2+2 c d f^2 x+d^2 \left (f^2 x^2+6\right )\right ) \cosh (e+f x)+f^4 x \left (6 c^2 d x+4 c^3+4 c d^2 x^2+d^3 x^3\right )\right )}{4 f^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(f^4*x*(4*c^3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) + (4*I)*f*(c + d*x)*(c^2*f^2 + 2*c*d*f^2*x + d^2*(6 + f^
2*x^2))*Cosh[e + f*x] - (12*I)*d*(c^2*f^2 + 2*c*d*f^2*x + d^2*(2 + f^2*x^2))*Sinh[e + f*x]))/(4*f^4)

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Maple [B]  time = 0.01, size = 494, normalized size = 5. \begin{align*}{\frac{1}{f} \left ({\frac{{d}^{3}a \left ( fx+e \right ) ^{4}}{4\,{f}^{3}}}-{\frac{3\,ide{c}^{2}a\cosh \left ( fx+e \right ) }{f}}-{\frac{{d}^{3}ea \left ( fx+e \right ) ^{3}}{{f}^{3}}}-{\frac{3\,i{d}^{3}ea \left ( \left ( fx+e \right ) ^{2}\cosh \left ( fx+e \right ) -2\, \left ( fx+e \right ) \sinh \left ( fx+e \right ) +2\,\cosh \left ( fx+e \right ) \right ) }{{f}^{3}}}+{\frac{c{d}^{2}a \left ( fx+e \right ) ^{3}}{{f}^{2}}}-{\frac{i{d}^{3}{e}^{3}a\cosh \left ( fx+e \right ) }{{f}^{3}}}+{\frac{3\,{d}^{3}{e}^{2}a \left ( fx+e \right ) ^{2}}{2\,{f}^{3}}}-{\frac{6\,i{d}^{2}eca \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{{f}^{2}}}-3\,{\frac{{d}^{2}eca \left ( fx+e \right ) ^{2}}{{f}^{2}}}+{\frac{3\,i{d}^{2}{e}^{2}ca\cosh \left ( fx+e \right ) }{{f}^{2}}}+{\frac{3\,{c}^{2}da \left ( fx+e \right ) ^{2}}{2\,f}}+{\frac{3\,i{d}^{2}ca \left ( \left ( fx+e \right ) ^{2}\cosh \left ( fx+e \right ) -2\, \left ( fx+e \right ) \sinh \left ( fx+e \right ) +2\,\cosh \left ( fx+e \right ) \right ) }{{f}^{2}}}-{\frac{{d}^{3}{e}^{3}a \left ( fx+e \right ) }{{f}^{3}}}+i{c}^{3}a\cosh \left ( fx+e \right ) +3\,{\frac{{d}^{2}{e}^{2}ca \left ( fx+e \right ) }{{f}^{2}}}+{\frac{3\,i{d}^{3}{e}^{2}a \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{{f}^{3}}}-3\,{\frac{de{c}^{2}a \left ( fx+e \right ) }{f}}+{\frac{3\,id{c}^{2}a \left ( \left ( fx+e \right ) \cosh \left ( fx+e \right ) -\sinh \left ( fx+e \right ) \right ) }{f}}+{c}^{3}a \left ( fx+e \right ) +{\frac{i{d}^{3}a \left ( \left ( fx+e \right ) ^{3}\cosh \left ( fx+e \right ) -3\, \left ( fx+e \right ) ^{2}\sinh \left ( fx+e \right ) +6\, \left ( fx+e \right ) \cosh \left ( fx+e \right ) -6\,\sinh \left ( fx+e \right ) \right ) }{{f}^{3}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(1/4/f^3*d^3*a*(f*x+e)^4-3*I*d*e/f*c^2*a*cosh(f*x+e)-1/f^3*d^3*e*a*(f*x+e)^3-3*I/f^3*d^3*e*a*((f*x+e)^2*co
sh(f*x+e)-2*(f*x+e)*sinh(f*x+e)+2*cosh(f*x+e))+1/f^2*d^2*c*a*(f*x+e)^3-I*d^3*e^3/f^3*a*cosh(f*x+e)+3/2/f^3*d^3
*e^2*a*(f*x+e)^2-6*I/f^2*d^2*e*c*a*((f*x+e)*cosh(f*x+e)-sinh(f*x+e))-3/f^2*d^2*e*c*a*(f*x+e)^2+3*I*d^2*e^2/f^2
*c*a*cosh(f*x+e)+3/2/f*d*c^2*a*(f*x+e)^2+3*I/f^2*d^2*c*a*((f*x+e)^2*cosh(f*x+e)-2*(f*x+e)*sinh(f*x+e)+2*cosh(f
*x+e))-d^3*e^3/f^3*a*(f*x+e)+I*c^3*a*cosh(f*x+e)+3*d^2*e^2/f^2*c*a*(f*x+e)+3*I/f^3*d^3*e^2*a*((f*x+e)*cosh(f*x
+e)-sinh(f*x+e))-3*d*e/f*c^2*a*(f*x+e)+3*I/f*d*c^2*a*((f*x+e)*cosh(f*x+e)-sinh(f*x+e))+c^3*a*(f*x+e)+I/f^3*d^3
*a*((f*x+e)^3*cosh(f*x+e)-3*(f*x+e)^2*sinh(f*x+e)+6*(f*x+e)*cosh(f*x+e)-6*sinh(f*x+e)))

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Maxima [B]  time = 1.0986, size = 317, normalized size = 3.23 \begin{align*} \frac{1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac{3}{2} \, a c^{2} d x^{2} + a c^{3} x + \frac{3}{2} i \, a c^{2} d{\left (\frac{{\left (f x e^{e} - e^{e}\right )} e^{\left (f x\right )}}{f^{2}} + \frac{{\left (f x + 1\right )} e^{\left (-f x - e\right )}}{f^{2}}\right )} + \frac{3}{2} i \, a c d^{2}{\left (\frac{{\left (f^{2} x^{2} e^{e} - 2 \, f x e^{e} + 2 \, e^{e}\right )} e^{\left (f x\right )}}{f^{3}} + \frac{{\left (f^{2} x^{2} + 2 \, f x + 2\right )} e^{\left (-f x - e\right )}}{f^{3}}\right )} + \frac{1}{2} i \, a d^{3}{\left (\frac{{\left (f^{3} x^{3} e^{e} - 3 \, f^{2} x^{2} e^{e} + 6 \, f x e^{e} - 6 \, e^{e}\right )} e^{\left (f x\right )}}{f^{4}} + \frac{{\left (f^{3} x^{3} + 3 \, f^{2} x^{2} + 6 \, f x + 6\right )} e^{\left (-f x - e\right )}}{f^{4}}\right )} + \frac{i \, a c^{3} \cosh \left (f x + e\right )}{f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x + 3/2*I*a*c^2*d*((f*x*e^e - e^e)*e^(f*x)/f^2 + (f*x +
1)*e^(-f*x - e)/f^2) + 3/2*I*a*c*d^2*((f^2*x^2*e^e - 2*f*x*e^e + 2*e^e)*e^(f*x)/f^3 + (f^2*x^2 + 2*f*x + 2)*e^
(-f*x - e)/f^3) + 1/2*I*a*d^3*((f^3*x^3*e^e - 3*f^2*x^2*e^e + 6*f*x*e^e - 6*e^e)*e^(f*x)/f^4 + (f^3*x^3 + 3*f^
2*x^2 + 6*f*x + 6)*e^(-f*x - e)/f^4) + I*a*c^3*cosh(f*x + e)/f

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Fricas [B]  time = 2.78733, size = 656, normalized size = 6.69 \begin{align*} \frac{{\left (2 i \, a d^{3} f^{3} x^{3} + 2 i \, a c^{3} f^{3} + 6 i \, a c^{2} d f^{2} + 12 i \, a c d^{2} f + 12 i \, a d^{3} +{\left (6 i \, a c d^{2} f^{3} + 6 i \, a d^{3} f^{2}\right )} x^{2} +{\left (6 i \, a c^{2} d f^{3} + 12 i \, a c d^{2} f^{2} + 12 i \, a d^{3} f\right )} x +{\left (2 i \, a d^{3} f^{3} x^{3} + 2 i \, a c^{3} f^{3} - 6 i \, a c^{2} d f^{2} + 12 i \, a c d^{2} f - 12 i \, a d^{3} +{\left (6 i \, a c d^{2} f^{3} - 6 i \, a d^{3} f^{2}\right )} x^{2} +{\left (6 i \, a c^{2} d f^{3} - 12 i \, a c d^{2} f^{2} + 12 i \, a d^{3} f\right )} x\right )} e^{\left (2 \, f x + 2 \, e\right )} +{\left (a d^{3} f^{4} x^{4} + 4 \, a c d^{2} f^{4} x^{3} + 6 \, a c^{2} d f^{4} x^{2} + 4 \, a c^{3} f^{4} x\right )} e^{\left (f x + e\right )}\right )} e^{\left (-f x - e\right )}}{4 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(2*I*a*d^3*f^3*x^3 + 2*I*a*c^3*f^3 + 6*I*a*c^2*d*f^2 + 12*I*a*c*d^2*f + 12*I*a*d^3 + (6*I*a*c*d^2*f^3 + 6*
I*a*d^3*f^2)*x^2 + (6*I*a*c^2*d*f^3 + 12*I*a*c*d^2*f^2 + 12*I*a*d^3*f)*x + (2*I*a*d^3*f^3*x^3 + 2*I*a*c^3*f^3
- 6*I*a*c^2*d*f^2 + 12*I*a*c*d^2*f - 12*I*a*d^3 + (6*I*a*c*d^2*f^3 - 6*I*a*d^3*f^2)*x^2 + (6*I*a*c^2*d*f^3 - 1
2*I*a*c*d^2*f^2 + 12*I*a*d^3*f)*x)*e^(2*f*x + 2*e) + (a*d^3*f^4*x^4 + 4*a*c*d^2*f^4*x^3 + 6*a*c^2*d*f^4*x^2 +
4*a*c^3*f^4*x)*e^(f*x + e))*e^(-f*x - e)/f^4

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Sympy [A]  time = 2.22986, size = 573, normalized size = 5.85 \begin{align*} a c^{3} x + \frac{3 a c^{2} d x^{2}}{2} + a c d^{2} x^{3} + \frac{a d^{3} x^{4}}{4} + \begin{cases} \frac{\left (\left (2 i a c^{3} f^{19} e^{3 e} + 6 i a c^{2} d f^{19} x e^{3 e} + 6 i a c^{2} d f^{18} e^{3 e} + 6 i a c d^{2} f^{19} x^{2} e^{3 e} + 12 i a c d^{2} f^{18} x e^{3 e} + 12 i a c d^{2} f^{17} e^{3 e} + 2 i a d^{3} f^{19} x^{3} e^{3 e} + 6 i a d^{3} f^{18} x^{2} e^{3 e} + 12 i a d^{3} f^{17} x e^{3 e} + 12 i a d^{3} f^{16} e^{3 e}\right ) e^{- f x} + \left (2 i a c^{3} f^{19} e^{5 e} + 6 i a c^{2} d f^{19} x e^{5 e} - 6 i a c^{2} d f^{18} e^{5 e} + 6 i a c d^{2} f^{19} x^{2} e^{5 e} - 12 i a c d^{2} f^{18} x e^{5 e} + 12 i a c d^{2} f^{17} e^{5 e} + 2 i a d^{3} f^{19} x^{3} e^{5 e} - 6 i a d^{3} f^{18} x^{2} e^{5 e} + 12 i a d^{3} f^{17} x e^{5 e} - 12 i a d^{3} f^{16} e^{5 e}\right ) e^{f x}\right ) e^{- 4 e}}{4 f^{20}} & \text{for}\: 4 f^{20} e^{4 e} \neq 0 \\\frac{x^{4} \left (i a d^{3} e^{2 e} - i a d^{3}\right ) e^{- e}}{8} + \frac{x^{3} \left (i a c d^{2} e^{2 e} - i a c d^{2}\right ) e^{- e}}{2} + \frac{x^{2} \left (3 i a c^{2} d e^{2 e} - 3 i a c^{2} d\right ) e^{- e}}{4} + \frac{x \left (i a c^{3} e^{2 e} - i a c^{3}\right ) e^{- e}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*c**3*x + 3*a*c**2*d*x**2/2 + a*c*d**2*x**3 + a*d**3*x**4/4 + Piecewise((((2*I*a*c**3*f**19*exp(3*e) + 6*I*a*
c**2*d*f**19*x*exp(3*e) + 6*I*a*c**2*d*f**18*exp(3*e) + 6*I*a*c*d**2*f**19*x**2*exp(3*e) + 12*I*a*c*d**2*f**18
*x*exp(3*e) + 12*I*a*c*d**2*f**17*exp(3*e) + 2*I*a*d**3*f**19*x**3*exp(3*e) + 6*I*a*d**3*f**18*x**2*exp(3*e) +
12*I*a*d**3*f**17*x*exp(3*e) + 12*I*a*d**3*f**16*exp(3*e))*exp(-f*x) + (2*I*a*c**3*f**19*exp(5*e) + 6*I*a*c**
2*d*f**19*x*exp(5*e) - 6*I*a*c**2*d*f**18*exp(5*e) + 6*I*a*c*d**2*f**19*x**2*exp(5*e) - 12*I*a*c*d**2*f**18*x*
exp(5*e) + 12*I*a*c*d**2*f**17*exp(5*e) + 2*I*a*d**3*f**19*x**3*exp(5*e) - 6*I*a*d**3*f**18*x**2*exp(5*e) + 12
*I*a*d**3*f**17*x*exp(5*e) - 12*I*a*d**3*f**16*exp(5*e))*exp(f*x))*exp(-4*e)/(4*f**20), Ne(4*f**20*exp(4*e), 0
)), (x**4*(I*a*d**3*exp(2*e) - I*a*d**3)*exp(-e)/8 + x**3*(I*a*c*d**2*exp(2*e) - I*a*c*d**2)*exp(-e)/2 + x**2*
(3*I*a*c**2*d*exp(2*e) - 3*I*a*c**2*d)*exp(-e)/4 + x*(I*a*c**3*exp(2*e) - I*a*c**3)*exp(-e)/2, True))

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Giac [B]  time = 1.29727, size = 356, normalized size = 3.63 \begin{align*} \frac{1}{4} \, a d^{3} x^{4} + a c d^{2} x^{3} + \frac{3}{2} \, a c^{2} d x^{2} + a c^{3} x - \frac{{\left (-i \, a d^{3} f^{3} x^{3} - 3 i \, a c d^{2} f^{3} x^{2} - 3 i \, a c^{2} d f^{3} x + 3 i \, a d^{3} f^{2} x^{2} - i \, a c^{3} f^{3} + 6 i \, a c d^{2} f^{2} x + 3 i \, a c^{2} d f^{2} - 6 i \, a d^{3} f x - 6 i \, a c d^{2} f + 6 i \, a d^{3}\right )} e^{\left (f x + e\right )}}{2 \, f^{4}} - \frac{{\left (-i \, a d^{3} f^{3} x^{3} - 3 i \, a c d^{2} f^{3} x^{2} - 3 i \, a c^{2} d f^{3} x - 3 i \, a d^{3} f^{2} x^{2} - i \, a c^{3} f^{3} - 6 i \, a c d^{2} f^{2} x - 3 i \, a c^{2} d f^{2} - 6 i \, a d^{3} f x - 6 i \, a c d^{2} f - 6 i \, a d^{3}\right )} e^{\left (-f x - e\right )}}{2 \, f^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*d^3*x^4 + a*c*d^2*x^3 + 3/2*a*c^2*d*x^2 + a*c^3*x - 1/2*(-I*a*d^3*f^3*x^3 - 3*I*a*c*d^2*f^3*x^2 - 3*I*a*
c^2*d*f^3*x + 3*I*a*d^3*f^2*x^2 - I*a*c^3*f^3 + 6*I*a*c*d^2*f^2*x + 3*I*a*c^2*d*f^2 - 6*I*a*d^3*f*x - 6*I*a*c*
d^2*f + 6*I*a*d^3)*e^(f*x + e)/f^4 - 1/2*(-I*a*d^3*f^3*x^3 - 3*I*a*c*d^2*f^3*x^2 - 3*I*a*c^2*d*f^3*x - 3*I*a*d
^3*f^2*x^2 - I*a*c^3*f^3 - 6*I*a*c*d^2*f^2*x - 3*I*a*c^2*d*f^2 - 6*I*a*d^3*f*x - 6*I*a*c*d^2*f - 6*I*a*d^3)*e^
(-f*x - e)/f^4