3.146 \(\int (e+f x)^m (a+b \cot ^{-1}(c+d x)) \, dx\)

Optimal. Leaf size=177 \[ \frac{(e+f x)^{m+1} \left (a+b \cot ^{-1}(c+d x)\right )}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)} \]

[Out]

((e + f*x)^(1 + m)*(a + b*ArcCot[c + d*x]))/(f*(1 + m)) + ((I/2)*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2
+ m, 3 + m, (d*(e + f*x))/(d*e + I*f - c*f)])/(f*(d*e + (I - c)*f)*(1 + m)*(2 + m)) - ((I/2)*b*d*(e + f*x)^(2
+ m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)])/(f*(d*e - (I + c)*f)*(1 + m)*(2 + m)
)

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Rubi [A]  time = 0.243636, antiderivative size = 177, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {5048, 4863, 712, 68} \[ \frac{(e+f x)^{m+1} \left (a+b \cot ^{-1}(c+d x)\right )}{f (m+1)}+\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-c f+i f}\right )}{2 f (m+1) (m+2) (d e+(-c+i) f)}-\frac{i b d (e+f x)^{m+2} \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right )}{2 f (m+1) (m+2) (d e-(c+i) f)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)^(1 + m)*(a + b*ArcCot[c + d*x]))/(f*(1 + m)) + ((I/2)*b*d*(e + f*x)^(2 + m)*Hypergeometric2F1[1, 2
+ m, 3 + m, (d*(e + f*x))/(d*e + I*f - c*f)])/(f*(d*e + (I - c)*f)*(1 + m)*(2 + m)) - ((I/2)*b*d*(e + f*x)^(2
+ m)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*x))/(d*e - (I + c)*f)])/(f*(d*e - (I + c)*f)*(1 + m)*(2 + m)
)

Rule 5048

Int[((a_.) + ArcCot[(c_) + (d_.)*(x_)]*(b_.))^(p_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcCot[x])^p, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x]
&& IGtQ[p, 0]

Rule 4863

Int[((a_.) + ArcCot[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1)*(a + b*
ArcCot[c*x]))/(e*(q + 1)), x] + Dist[(b*c)/(e*(q + 1)), Int[(d + e*x)^(q + 1)/(1 + c^2*x^2), x], x] /; FreeQ[{
a, b, c, d, e, q}, x] && NeQ[q, -1]

Rule 712

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m, 1/(a + c*x^2
), x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[m]

Rule 68

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((b*c - a*d)^n*(a + b*x)^(m + 1)*Hype
rgeometric2F1[-n, m + 1, m + 2, -((d*(a + b*x))/(b*c - a*d))])/(b^(n + 1)*(m + 1)), x] /; FreeQ[{a, b, c, d, m
}, x] && NeQ[b*c - a*d, 0] &&  !IntegerQ[m] && IntegerQ[n]

Rubi steps

\begin{align*} \int (e+f x)^m \left (a+b \cot ^{-1}(c+d x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^m \left (a+b \cot ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{b \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{1+x^2} \, dx,x,c+d x\right )}{f (1+m)}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{b \operatorname{Subst}\left (\int \left (\frac{i \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{2 (i-x)}+\frac{i \left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{2 (i+x)}\right ) \, dx,x,c+d x\right )}{f (1+m)}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{i-x} \, dx,x,c+d x\right )}{2 f (1+m)}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\left (\frac{d e-c f}{d}+\frac{f x}{d}\right )^{1+m}}{i+x} \, dx,x,c+d x\right )}{2 f (1+m)}\\ &=\frac{(e+f x)^{1+m} \left (a+b \cot ^{-1}(c+d x)\right )}{f (1+m)}+\frac{i b d (e+f x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{d (e+f x)}{d e+i f-c f}\right )}{2 f (d e+(i-c) f) (1+m) (2+m)}-\frac{i b d (e+f x)^{2+m} \, _2F_1\left (1,2+m;3+m;\frac{d (e+f x)}{d e-(i+c) f}\right )}{2 f (d e-(i+c) f) (1+m) (2+m)}\\ \end{align*}

Mathematica [A]  time = 0.326997, size = 162, normalized size = 0.92 \[ \frac{(e+f x)^{m+1} \left (2 \left (a+b \cot ^{-1}(c+d x)\right )+\frac{b d (e+f x) \left ((d e-(c+i) f) \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c-i) f}\right )+(-d e+(c-i) f) \, _2F_1\left (1,m+2;m+3;\frac{d (e+f x)}{d e-(c+i) f}\right )\right )}{(m+2) (i c f-i d e+f) (d e-(c+i) f)}\right )}{2 f (m+1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((e + f*x)^(1 + m)*(2*(a + b*ArcCot[c + d*x]) + (b*d*(e + f*x)*((d*e - (I + c)*f)*Hypergeometric2F1[1, 2 + m,
3 + m, (d*(e + f*x))/(d*e - (-I + c)*f)] + (-(d*e) + (-I + c)*f)*Hypergeometric2F1[1, 2 + m, 3 + m, (d*(e + f*
x))/(d*e - (I + c)*f)]))/(((-I)*d*e + f + I*c*f)*(d*e - (I + c)*f)*(2 + m))))/(2*f*(1 + m))

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Maple [F]  time = 1.395, size = 0, normalized size = 0. \begin{align*} \int \left ( fx+e \right ) ^{m} \left ( a+b{\rm arccot} \left (dx+c\right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((f*x+e)^m*(a+b*arccot(d*x+c)),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((f*x+e)^m*(a+b*arccot(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}{\left (f x + e\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arccot(d*x+c)),x, algorithm="fricas")

[Out]

integral((b*arccot(d*x + c) + a)*(f*x + e)^m, x)

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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((f*x+e)**m*(a+b*acot(d*x+c)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arccot}\left (d x + c\right ) + a\right )}{\left (f x + e\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*(a+b*arccot(d*x+c)),x, algorithm="giac")

[Out]

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