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]

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

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Rubi [A]  time = 0.24, antiderivative size = 177, normalized size of antiderivative = 1.00, 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 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]

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 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 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]

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*}

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Mathematica [A]  time = 0.36, 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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fricas [F]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \operatorname {arccot}\left (d x + c\right ) + a\right )} {\left (f x + e\right )}^{m}, x\right ) \]

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

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)

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maple [F]  time = 2.11, size = 0, normalized size = 0.00 \[ \int \left (f x +e \right )^{m} \left (a +b \,\mathrm {arccot}\left (d x +c \right )\right )\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\frac {1}{2} \, {\left (3 \, {\left (f x \arctan \left (1, d x + c\right ) + e \arctan \left (1, d x + c\right )\right )} {\left (f x + e\right )}^{m} + 2 \, {\left (f m + f\right )} \int \frac {{\left ({\left (c^{2} \arctan \left (1, d x + c\right ) + \arctan \left (1, d x + c\right )\right )} f m + {\left (d^{2} f m \arctan \left (1, d x + c\right ) + d^{2} f \arctan \left (1, d x + c\right )\right )} x^{2} + 3 \, d e + {\left (c^{2} \arctan \left (1, d x + c\right ) + \arctan \left (1, d x + c\right )\right )} f + {\left (2 \, c d f m \arctan \left (1, d x + c\right ) + {\left (2 \, c \arctan \left (1, d x + c\right ) + 3\right )} d f\right )} x\right )} {\left (f x + e\right )}^{m}}{2 \, {\left ({\left (c^{2} + 1\right )} f m + {\left (d^{2} f m + d^{2} f\right )} x^{2} + {\left (c^{2} + 1\right )} f + 2 \, {\left (c d f m + c d f\right )} x\right )}}\,{d x}\right )} b}{2 \, {\left (f m + f\right )}} + \frac {{\left (f x + e\right )}^{m + 1} a}{f {\left (m + 1\right )}} \]

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]

1/2*((f*x*arctan2(1, d*x + c) + e*arctan2(1, d*x + c))*(f*x + e)^m + 2*(f*m + f)*integrate(1/2*((c^2*arctan2(1
, d*x + c) + arctan2(1, d*x + c))*f*m + (d^2*f*m*arctan2(1, d*x + c) + d^2*f*arctan2(1, d*x + c))*x^2 + d*e +
(c^2*arctan2(1, d*x + c) + arctan2(1, d*x + c))*f + (2*c*d*f*m*arctan2(1, d*x + c) + (2*c*arctan2(1, d*x + c)
+ 1)*d*f)*x)*(f*x + e)^m/((c^2 + 1)*f*m + (d^2*f*m + d^2*f)*x^2 + (c^2 + 1)*f + 2*(c*d*f*m + c*d*f)*x), x))*b/
(f*m + f) + (f*x + e)^(m + 1)*a/(f*(m + 1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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