3.41 \(\int \frac{(d+i c d x)^4 (a+b \tan ^{-1}(c x))}{x^7} \, dx\)

Optimal. Leaf size=168 \[ \frac{i c d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{30 x^5}-\frac{d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{16 i b c^4 d^4}{15 x^2}+\frac{5 b c^3 d^4}{9 x^3}-\frac{i b c^2 d^4}{5 x^4}-\frac{13 b c^5 d^4}{6 x}+\frac{32}{15} i b c^6 d^4 \log (x)-\frac{32}{15} i b c^6 d^4 \log (c x+i)-\frac{b c d^4}{30 x^5} \]

[Out]

-(b*c*d^4)/(30*x^5) - ((I/5)*b*c^2*d^4)/x^4 + (5*b*c^3*d^4)/(9*x^3) + (((16*I)/15)*b*c^4*d^4)/x^2 - (13*b*c^5*
d^4)/(6*x) - (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(6*x^6) + ((I/30)*c*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x])
)/x^5 + ((32*I)/15)*b*c^6*d^4*Log[x] - ((32*I)/15)*b*c^6*d^4*Log[I + c*x]

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Rubi [A]  time = 0.113769, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217, Rules used = {45, 37, 4872, 12, 148} \[ \frac{i c d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{30 x^5}-\frac{d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{16 i b c^4 d^4}{15 x^2}+\frac{5 b c^3 d^4}{9 x^3}-\frac{i b c^2 d^4}{5 x^4}-\frac{13 b c^5 d^4}{6 x}+\frac{32}{15} i b c^6 d^4 \log (x)-\frac{32}{15} i b c^6 d^4 \log (c x+i)-\frac{b c d^4}{30 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^7,x]

[Out]

-(b*c*d^4)/(30*x^5) - ((I/5)*b*c^2*d^4)/x^4 + (5*b*c^3*d^4)/(9*x^3) + (((16*I)/15)*b*c^4*d^4)/x^2 - (13*b*c^5*
d^4)/(6*x) - (d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x]))/(6*x^6) + ((I/30)*c*d^4*(1 + I*c*x)^5*(a + b*ArcTan[c*x])
)/x^5 + ((32*I)/15)*b*c^6*d^4*Log[x] - ((32*I)/15)*b*c^6*d^4*Log[I + c*x]

Rule 45

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

Rule 37

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

Rule 4872

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTan[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/(1 + c^2*x^
2), x], x], x]] /; FreeQ[{a, b, c, d, e, f, q}, x] && NeQ[q, -1] && IntegerQ[2*m] && ((IGtQ[m, 0] && IGtQ[q, 0
]) || (ILtQ[m + q + 1, 0] && LtQ[m*q, 0]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 148

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x), x], x] /; FreeQ[{a, b, c, d, e, f, g
, h, m}, x] && (IntegersQ[m, n, p] || (IGtQ[n, 0] && IGtQ[p, 0]))

Rubi steps

\begin{align*} \int \frac{(d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )}{x^7} \, dx &=-\frac{d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{i c d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{30 x^5}-(b c) \int \frac{d^4 (i-c x)^4 (-5 i-c x)}{30 x^6 (i+c x)} \, dx\\ &=-\frac{d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{i c d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{30 x^5}-\frac{1}{30} \left (b c d^4\right ) \int \frac{(i-c x)^4 (-5 i-c x)}{x^6 (i+c x)} \, dx\\ &=-\frac{d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{i c d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{30 x^5}-\frac{1}{30} \left (b c d^4\right ) \int \left (-\frac{5}{x^6}-\frac{24 i c}{x^5}+\frac{50 c^2}{x^4}+\frac{64 i c^3}{x^3}-\frac{65 c^4}{x^2}-\frac{64 i c^5}{x}+\frac{64 i c^6}{i+c x}\right ) \, dx\\ &=-\frac{b c d^4}{30 x^5}-\frac{i b c^2 d^4}{5 x^4}+\frac{5 b c^3 d^4}{9 x^3}+\frac{16 i b c^4 d^4}{15 x^2}-\frac{13 b c^5 d^4}{6 x}-\frac{d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{6 x^6}+\frac{i c d^4 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{30 x^5}+\frac{32}{15} i b c^6 d^4 \log (x)-\frac{32}{15} i b c^6 d^4 \log (i+c x)\\ \end{align*}

Mathematica [C]  time = 0.119118, size = 235, normalized size = 1.4 \[ -\frac{d^4 \left (15 b c^5 x^5 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )-15 b c^3 x^3 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )+b c x \text{Hypergeometric2F1}\left (-\frac{5}{2},1,-\frac{3}{2},-c^2 x^2\right )+15 a c^4 x^4-40 i a c^3 x^3-45 a c^2 x^2+24 i a c x+5 a-32 i b c^4 x^4+6 i b c^2 x^2-64 i b c^6 x^6 \log (x)+32 i b c^6 x^6 \log \left (c^2 x^2+1\right )+15 b c^4 x^4 \tan ^{-1}(c x)-40 i b c^3 x^3 \tan ^{-1}(c x)-45 b c^2 x^2 \tan ^{-1}(c x)+24 i b c x \tan ^{-1}(c x)+5 b \tan ^{-1}(c x)\right )}{30 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + I*c*d*x)^4*(a + b*ArcTan[c*x]))/x^7,x]

[Out]

-(d^4*(5*a + (24*I)*a*c*x - 45*a*c^2*x^2 + (6*I)*b*c^2*x^2 - (40*I)*a*c^3*x^3 + 15*a*c^4*x^4 - (32*I)*b*c^4*x^
4 + 5*b*ArcTan[c*x] + (24*I)*b*c*x*ArcTan[c*x] - 45*b*c^2*x^2*ArcTan[c*x] - (40*I)*b*c^3*x^3*ArcTan[c*x] + 15*
b*c^4*x^4*ArcTan[c*x] + b*c*x*Hypergeometric2F1[-5/2, 1, -3/2, -(c^2*x^2)] - 15*b*c^3*x^3*Hypergeometric2F1[-3
/2, 1, -1/2, -(c^2*x^2)] + 15*b*c^5*x^5*Hypergeometric2F1[-1/2, 1, 1/2, -(c^2*x^2)] - (64*I)*b*c^6*x^6*Log[x]
+ (32*I)*b*c^6*x^6*Log[1 + c^2*x^2]))/(30*x^6)

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Maple [A]  time = 0.039, size = 243, normalized size = 1.5 \begin{align*} -{\frac{{d}^{4}{c}^{4}a}{2\,{x}^{2}}}+{\frac{3\,{c}^{2}{d}^{4}a}{2\,{x}^{4}}}-{\frac{{d}^{4}a}{6\,{x}^{6}}}+{\frac{{\frac{4\,i}{3}}{c}^{3}{d}^{4}a}{{x}^{3}}}-{\frac{{\frac{4\,i}{5}}c{d}^{4}b\arctan \left ( cx \right ) }{{x}^{5}}}-{\frac{b{c}^{4}{d}^{4}\arctan \left ( cx \right ) }{2\,{x}^{2}}}+{\frac{3\,{c}^{2}{d}^{4}b\arctan \left ( cx \right ) }{2\,{x}^{4}}}-{\frac{b{d}^{4}\arctan \left ( cx \right ) }{6\,{x}^{6}}}-{\frac{{\frac{4\,i}{5}}c{d}^{4}a}{{x}^{5}}}-{\frac{{\frac{i}{5}}b{c}^{2}{d}^{4}}{{x}^{4}}}-{\frac{16\,i}{15}}{c}^{6}{d}^{4}b\ln \left ({c}^{2}{x}^{2}+1 \right ) -{\frac{13\,{c}^{6}{d}^{4}b\arctan \left ( cx \right ) }{6}}+{\frac{32\,i}{15}}{c}^{6}{d}^{4}b\ln \left ( cx \right ) +{\frac{{\frac{16\,i}{15}}b{c}^{4}{d}^{4}}{{x}^{2}}}+{\frac{{\frac{4\,i}{3}}{c}^{3}{d}^{4}b\arctan \left ( cx \right ) }{{x}^{3}}}-{\frac{bc{d}^{4}}{30\,{x}^{5}}}+{\frac{5\,b{c}^{3}{d}^{4}}{9\,{x}^{3}}}-{\frac{13\,b{c}^{5}{d}^{4}}{6\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^7,x)

[Out]

-1/2*c^4*d^4*a/x^2+3/2*c^2*d^4*a/x^4-1/6*d^4*a/x^6+4/3*I*c^3*d^4*a/x^3-4/5*I*c*d^4*b*arctan(c*x)/x^5-1/2*c^4*d
^4*b*arctan(c*x)/x^2+3/2*c^2*d^4*b*arctan(c*x)/x^4-1/6*d^4*b*arctan(c*x)/x^6-4/5*I*c*d^4*a/x^5-1/5*I*b*c^2*d^4
/x^4-16/15*I*c^6*d^4*b*ln(c^2*x^2+1)-13/6*c^6*d^4*b*arctan(c*x)+32/15*I*c^6*d^4*b*ln(c*x)+16/15*I*b*c^4*d^4/x^
2+4/3*I*c^3*d^4*b*arctan(c*x)/x^3-1/30*b*c*d^4/x^5+5/9*b*c^3*d^4/x^3-13/6*b*c^5*d^4/x

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Maxima [B]  time = 1.48864, size = 392, normalized size = 2.33 \begin{align*} -\frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b c^{4} d^{4} - \frac{2}{3} i \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{x^{3}}\right )} b c^{3} d^{4} - \frac{1}{2} \,{\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac{3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac{3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c^{2} d^{4} - \frac{1}{5} i \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac{2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac{4 \, \arctan \left (c x\right )}{x^{5}}\right )} b c d^{4} - \frac{a c^{4} d^{4}}{2 \, x^{2}} - \frac{1}{90} \,{\left ({\left (15 \, c^{5} \arctan \left (c x\right ) + \frac{15 \, c^{4} x^{4} - 5 \, c^{2} x^{2} + 3}{x^{5}}\right )} c + \frac{15 \, \arctan \left (c x\right )}{x^{6}}\right )} b d^{4} + \frac{4 i \, a c^{3} d^{4}}{3 \, x^{3}} + \frac{3 \, a c^{2} d^{4}}{2 \, x^{4}} - \frac{4 i \, a c d^{4}}{5 \, x^{5}} - \frac{a d^{4}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^7,x, algorithm="maxima")

[Out]

-1/2*((c*arctan(c*x) + 1/x)*c + arctan(c*x)/x^2)*b*c^4*d^4 - 2/3*I*((c^2*log(c^2*x^2 + 1) - c^2*log(x^2) - 1/x
^2)*c - 2*arctan(c*x)/x^3)*b*c^3*d^4 - 1/2*((3*c^3*arctan(c*x) + (3*c^2*x^2 - 1)/x^3)*c - 3*arctan(c*x)/x^4)*b
*c^2*d^4 - 1/5*I*((2*c^4*log(c^2*x^2 + 1) - 2*c^4*log(x^2) - (2*c^2*x^2 - 1)/x^4)*c + 4*arctan(c*x)/x^5)*b*c*d
^4 - 1/2*a*c^4*d^4/x^2 - 1/90*((15*c^5*arctan(c*x) + (15*c^4*x^4 - 5*c^2*x^2 + 3)/x^5)*c + 15*arctan(c*x)/x^6)
*b*d^4 + 4/3*I*a*c^3*d^4/x^3 + 3/2*a*c^2*d^4/x^4 - 4/5*I*a*c*d^4/x^5 - 1/6*a*d^4/x^6

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Fricas [A]  time = 2.40283, size = 527, normalized size = 3.14 \begin{align*} \frac{384 i \, b c^{6} d^{4} x^{6} \log \left (x\right ) - 387 i \, b c^{6} d^{4} x^{6} \log \left (\frac{c x + i}{c}\right ) + 3 i \, b c^{6} d^{4} x^{6} \log \left (\frac{c x - i}{c}\right ) - 390 \, b c^{5} d^{4} x^{5} - 6 \,{\left (15 \, a - 32 i \, b\right )} c^{4} d^{4} x^{4} +{\left (240 i \, a + 100 \, b\right )} c^{3} d^{4} x^{3} + 18 \,{\left (15 \, a - 2 i \, b\right )} c^{2} d^{4} x^{2} +{\left (-144 i \, a - 6 \, b\right )} c d^{4} x - 30 \, a d^{4} +{\left (-45 i \, b c^{4} d^{4} x^{4} - 120 \, b c^{3} d^{4} x^{3} + 135 i \, b c^{2} d^{4} x^{2} + 72 \, b c d^{4} x - 15 i \, b d^{4}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{180 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^7,x, algorithm="fricas")

[Out]

1/180*(384*I*b*c^6*d^4*x^6*log(x) - 387*I*b*c^6*d^4*x^6*log((c*x + I)/c) + 3*I*b*c^6*d^4*x^6*log((c*x - I)/c)
- 390*b*c^5*d^4*x^5 - 6*(15*a - 32*I*b)*c^4*d^4*x^4 + (240*I*a + 100*b)*c^3*d^4*x^3 + 18*(15*a - 2*I*b)*c^2*d^
4*x^2 + (-144*I*a - 6*b)*c*d^4*x - 30*a*d^4 + (-45*I*b*c^4*d^4*x^4 - 120*b*c^3*d^4*x^3 + 135*I*b*c^2*d^4*x^2 +
72*b*c*d^4*x - 15*I*b*d^4)*log(-(c*x + I)/(c*x - I)))/x^6

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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((d+I*c*d*x)**4*(a+b*atan(c*x))/x**7,x)

[Out]

Timed out

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Giac [A]  time = 1.56953, size = 329, normalized size = 1.96 \begin{align*} \frac{3 \, b c^{6} d^{4} i x^{6} \log \left (c i x + 1\right ) - 387 \, b c^{6} d^{4} i x^{6} \log \left (-c i x + 1\right ) + 384 \, b c^{6} d^{4} i x^{6} \log \left (x\right ) - 390 \, b c^{5} d^{4} x^{5} + 192 \, b c^{4} d^{4} i x^{4} - 90 \, b c^{4} d^{4} x^{4} \arctan \left (c x\right ) - 90 \, a c^{4} d^{4} x^{4} + 240 \, b c^{3} d^{4} i x^{3} \arctan \left (c x\right ) + 240 \, a c^{3} d^{4} i x^{3} + 100 \, b c^{3} d^{4} x^{3} - 36 \, b c^{2} d^{4} i x^{2} + 270 \, b c^{2} d^{4} x^{2} \arctan \left (c x\right ) + 270 \, a c^{2} d^{4} x^{2} - 144 \, b c d^{4} i x \arctan \left (c x\right ) - 144 \, a c d^{4} i x - 6 \, b c d^{4} x - 30 \, b d^{4} \arctan \left (c x\right ) - 30 \, a d^{4}}{180 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^4*(a+b*arctan(c*x))/x^7,x, algorithm="giac")

[Out]

1/180*(3*b*c^6*d^4*i*x^6*log(c*i*x + 1) - 387*b*c^6*d^4*i*x^6*log(-c*i*x + 1) + 384*b*c^6*d^4*i*x^6*log(x) - 3
90*b*c^5*d^4*x^5 + 192*b*c^4*d^4*i*x^4 - 90*b*c^4*d^4*x^4*arctan(c*x) - 90*a*c^4*d^4*x^4 + 240*b*c^3*d^4*i*x^3
*arctan(c*x) + 240*a*c^3*d^4*i*x^3 + 100*b*c^3*d^4*x^3 - 36*b*c^2*d^4*i*x^2 + 270*b*c^2*d^4*x^2*arctan(c*x) +
270*a*c^2*d^4*x^2 - 144*b*c*d^4*i*x*arctan(c*x) - 144*a*c*d^4*i*x - 6*b*c*d^4*x - 30*b*d^4*arctan(c*x) - 30*a*
d^4)/x^6