3.152 \(\int (c+d x)^m (a+a \cosh (e+f x))^2 \, dx\)

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

[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)

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Rubi [A]  time = 0.341582, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3318, 3312, 3307, 2181} \[ \frac{a^2 2^{-m-3} e^{2 e-\frac{2 c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )}{f}+\frac{a^2 e^{e-\frac{c f}{d}} (c+d x)^m \left (-\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )}{f}-\frac{a^2 e^{\frac{c f}{d}-e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )}{f}-\frac{a^2 2^{-m-3} e^{\frac{2 c f}{d}-2 e} (c+d x)^m \left (\frac{f (c+d x)}{d}\right )^{-m} \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )}{f}+\frac{3 a^2 (c+d x)^{m+1}}{2 d (m+1)} \]

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 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]

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*}

Mathematica [A]  time = 1.0451, size = 302, normalized size = 1.15 \[ -\frac{a^2 2^{-m-5} e^{-2 \left (\frac{c f}{d}+e\right )} (c+d x)^m (\cosh (e+f x)+1)^2 \text{sech}^4\left (\frac{1}{2} (e+f x)\right ) \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^{-m} \left (-d e^{4 e} (m+1) \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{2 f (c+d x)}{d}\right )-d 2^{m+3} (m+1) e^{\frac{c f}{d}+3 e} \left (f \left (\frac{c}{d}+x\right )\right )^m \text{Gamma}\left (m+1,-\frac{f (c+d x)}{d}\right )+d 2^{m+3} (m+1) e^{\frac{3 c f}{d}+e} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{f (c+d x)}{d}\right )+d (m+1) e^{\frac{4 c f}{d}} \left (-\frac{f (c+d x)}{d}\right )^m \text{Gamma}\left (m+1,\frac{2 f (c+d x)}{d}\right )-3 f 2^{m+2} (c+d x) e^{2 \left (\frac{c f}{d}+e\right )} \left (-\frac{f^2 (c+d x)^2}{d^2}\right )^m\right )}{d f (m+1)} \]

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 = 0.113, size = 0, normalized size = 0. \begin{align*} \int \left ( dx+c \right ) ^{m} \left ( a+a\cosh \left ( fx+e \right ) \right ) ^{2}\, dx \end{align*}

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 [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+a*cosh(f*x+e))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.29729, size = 1139, normalized size = 4.33 \begin{align*} -\frac{{\left (a^{2} d m + a^{2} d\right )} \cosh \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 ) + 8 \,{\left (a^{2} d m + a^{2} d\right )} \cosh \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 ) - 8 \,{\left (a^{2} d m + a^{2} d\right )} \cosh \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 (a^{2} d m + a^{2} d\right )} \cosh \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 (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 \, d e - 2 \, c f}{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 ) + d e - c f}{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 ) - d e + c f}{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 \, d e + 2 \, c f}{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{align*}

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*d*e - 2*c*f)/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) + d*e - c*f)/d)*gamma(m + 1, (d*f*x + c*f)/d) - 8*(a^2*d*m + a^2*d)*cosh((d*m*log(
-f/d) - d*e + c*f)/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*d*e + 2*c*f
)/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*d*e - 2*c*f)/d) - 8*(a^2*d*m + a^2*d)*gamma(m + 1, (d*f*x + c*f)/d)*sinh((d*m*log(f/d) + d*e - c*f)/d) +
8*(a^2*d*m + a^2*d)*gamma(m + 1, -(d*f*x + c*f)/d)*sinh((d*m*log(-f/d) - d*e + c*f)/d) + (a^2*d*m + a^2*d)*gam
ma(m + 1, -2*(d*f*x + c*f)/d)*sinh((d*m*log(-2*f/d) - 2*d*e + 2*c*f)/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., 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+a*cosh(f*x+e))**2,x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \cosh \left (f x + e\right ) + a\right )}^{2}{\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+a*cosh(f*x+e))^2,x, algorithm="giac")

[Out]

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