3.2.52 \(\int (c+d x)^m (a+a \cosh (e+f x))^2 \, dx\) [152]

Optimal. Leaf size=263 \[ \frac {3 a^2 (c+d x)^{1+m}}{2 d (1+m)}+\frac {2^{-3-m} a^2 e^{2 e-\frac {2 c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )}{f}+\frac {a^2 e^{e-\frac {c f}{d}} (c+d x)^m \left (-\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )}{f}-\frac {a^2 e^{-e+\frac {c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )}{f}-\frac {2^{-3-m} a^2 e^{-2 e+\frac {2 c f}{d}} (c+d x)^m \left (\frac {f (c+d x)}{d}\right )^{-m} \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )}{f} \]

[Out]

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

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Rubi [A]
time = 0.24, antiderivative size = 263, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3399, 3393, 3388, 2212} \begin {gather*} \frac {a^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 {a^2 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 )}{f}-\frac {a^2 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 )}{f}-\frac {a^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 {3 a^2 (c+d x)^{m+1}}{2 d (m+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^m*(a + a*Cosh[e + f*x])^2,x]

[Out]

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

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 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 3399

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[(2*a)^n, Int[(c
+ d*x)^m*Sin[(1/2)*(e + Pi*(a/(2*b))) + 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])

Rubi steps

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

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Mathematica [A]
time = 0.78, size = 302, normalized size = 1.15 \begin {gather*} -\frac {2^{-5-m} a^2 e^{-2 \left (e+\frac {c f}{d}\right )} (c+d x)^m \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^{-m} (1+\cosh (e+f x))^2 \left (-3 2^{2+m} e^{2 \left (e+\frac {c f}{d}\right )} f (c+d x) \left (-\frac {f^2 (c+d x)^2}{d^2}\right )^m-d e^{4 e} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {2 f (c+d x)}{d}\right )-2^{3+m} d e^{3 e+\frac {c f}{d}} (1+m) \left (f \left (\frac {c}{d}+x\right )\right )^m \Gamma \left (1+m,-\frac {f (c+d x)}{d}\right )+2^{3+m} d e^{e+\frac {3 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {f (c+d x)}{d}\right )+d e^{\frac {4 c f}{d}} (1+m) \left (-\frac {f (c+d x)}{d}\right )^m \Gamma \left (1+m,\frac {2 f (c+d x)}{d}\right )\right ) \text {sech}^4\left (\frac {1}{2} (e+f x)\right )}{d f (1+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x)^m*(a + a*Cosh[e + f*x])^2,x]

[Out]

-((2^(-5 - m)*a^2*(c + d*x)^m*(1 + Cosh[e + f*x])^2*(-3*2^(2 + m)*E^(2*(e + (c*f)/d))*f*(c + d*x)*(-((f^2*(c +
d*x)^2)/d^2))^m - d*E^(4*e)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, (-2*f*(c + d*x))/d] - 2^(3 + m)*d*E^(3*e + (
c*f)/d)*(1 + m)*(f*(c/d + x))^m*Gamma[1 + m, -((f*(c + d*x))/d)] + 2^(3 + m)*d*E^(e + (3*c*f)/d)*(1 + m)*(-((f
*(c + d*x))/d))^m*Gamma[1 + m, (f*(c + d*x))/d] + d*E^((4*c*f)/d)*(1 + m)*(-((f*(c + d*x))/d))^m*Gamma[1 + m,
(2*f*(c + d*x))/d])*Sech[(e + f*x)/2]^4)/(d*E^(2*(e + (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 \[\int \left (d x +c \right )^{m} \left (a +a \cosh \left (f x +e \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^m*(a+a*cosh(f*x+e))^2,x)

[Out]

int((d*x+c)^m*(a+a*cosh(f*x+e))^2,x)

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Maxima [A]
time = 0.08, size = 213, normalized size = 0.81 \begin {gather*} -\frac {1}{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^{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} + \frac {{\left (d x + c\right )}^{m + 1} a^{2}}{d {\left (m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^m*(a+a*cosh(f*x+e))^2,x, algorithm="maxima")

[Out]

-1/4*((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^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)*exp_integral_e(-m, -(d*x +
c)*f/d)/d)*a^2 + (d*x + c)^(m + 1)*a^2/(d*(m + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 539 vs. \(2 (261) = 522\).
time = 0.10, size = 539, normalized size = 2.05 \begin {gather*} -\frac {{\left (a^{2} d m + a^{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 ) + 8 \, {\left (a^{2} d m + a^{2} 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 ) - 8 \, {\left (a^{2} d m + a^{2} 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 (a^{2} d m + a^{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 (a^{2} d m + a^{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 ) - 8 \, {\left (a^{2} d m + a^{2} 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 ) + 8 \, {\left (a^{2} d m + a^{2} 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 (a^{2} d m + a^{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 ) - 12 \, {\left (a^{2} d f x + a^{2} c f\right )} \cosh \left (m \log \left (d x + c\right )\right ) - 12 \, {\left (a^{2} d f x + a^{2} c f\right )} \sinh \left (m \log \left (d x + c\right )\right )}{8 \, {\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+a*cosh(f*x+e))^2,x, algorithm="fricas")

[Out]

-1/8*((a^2*d*m + a^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) + 8*(a^2*d*m + a^2*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
) - 8*(a^2*d*m + a^2*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) -
(a^2*d*m + a^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) - (a^2*d*m + a^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) - 8*(a^2*d*m + a^2*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
) + 8*(a^2*d*m + a^2*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) +
(a^2*d*m + a^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) - 12*(a^2*d*f*x + a^2*c*f)*cosh(m*log(d*x + c)) - 12*(a^2*d*f*x + a^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+a*cosh(f*x+e))**2,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+a*cosh(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((a*cosh(f*x + e) + a)^2*(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+a\,\mathrm {cosh}\left (e+f\,x\right )\right )}^2\,{\left (c+d\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*cosh(e + f*x))^2*(c + d*x)^m,x)

[Out]

int((a + a*cosh(e + f*x))^2*(c + d*x)^m, x)

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