3.2.84 \(\int (c+d x)^m (a+b \sinh (e+f x)) \, dx\) [184]

Optimal. Leaf size=131 \[ \frac {a (c+d x)^{1+m}}{d (1+m)}+\frac {b e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{2 f}+\frac {b e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{2 f} \]

[Out]

a*(d*x+c)^(1+m)/d/(1+m)+1/2*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)+1/2*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)

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Rubi [A]
time = 0.10, antiderivative size = 131, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3398, 3389, 2212} \begin {gather*} \frac {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 {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 {a (c+d x)^{m+1}}{d (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(c + d*x)^(1 + m))/(d*(1 + m)) + (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) + (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)

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

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Mathematica [A]
time = 17.29, size = 201, normalized size = 1.53 \begin {gather*} \frac {e^{-\frac {3 c f}{d}} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} \left (\cosh \left (\frac {3 c f}{d}\right )+\sinh \left (\frac {3 c f}{d}\right )\right ) \left (2 a f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m+b d (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,f \left (\frac {c}{d}+x\right )\right ) (\cosh (e)-\sinh (e)) \left (\cosh \left (\frac {c f}{d}\right )+\sinh \left (\frac {c f}{d}\right )\right )+b d (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right ) \left (\cosh \left (e-\frac {c f}{d}\right )+\sinh \left (e-\frac {c f}{d}\right )\right )\right )}{2 d f (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.06, size = 103, normalized size = 0.79 \begin {gather*} \frac {1}{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 )} b + \frac {{\left (d x + c\right )}^{m + 1} a}{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)),x, algorithm="maxima")

[Out]

1/2*((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)*b + (d*x + c)^(m + 1)*a/(d*(m + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 271 vs. \(2 (127) = 254\).
time = 0.10, size = 271, normalized size = 2.07 \begin {gather*} \frac {{\left (b d m + b 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 ) + {\left (b d m + b 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 ) - {\left (b d m + b 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 ) - {\left (b d m + b 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 ) + 2 \, {\left (a d f x + a c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) + 2 \, {\left (a d f x + a c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{2 \, {\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)),x, algorithm="fricas")

[Out]

1/2*((b*d*m + b*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) + (b*d*m
+ b*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) - (b*d*m + b*d)*g
amma(m + 1, (d*f*x + c*f)/d)*sinh((d*m*log(f/d) - c*f + d*cosh(1) + d*sinh(1))/d) - (b*d*m + b*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) + 2*(a*d*f*x + a*c*f)*cosh(m*log(d*x
+ c)) + 2*(a*d*f*x + a*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)),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)),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {sinh}\left (e+f\,x\right )\right )\,{\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))*(c + d*x)^m,x)

[Out]

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

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