3.30 \(\int \frac {(d+c d x)^3 (a+b \tanh ^{-1}(c x))}{x^7} \, dx\)

Optimal. Leaf size=196 \[ -\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (c x+1)-\frac {11 b c^5 d^3}{12 x}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^3 d^3}{36 x^3}-\frac {3 b c^2 d^3}{20 x^4}-\frac {b c d^3}{30 x^5} \]

[Out]

-1/30*b*c*d^3/x^5-3/20*b*c^2*d^3/x^4-11/36*b*c^3*d^3/x^3-7/15*b*c^4*d^3/x^2-11/12*b*c^5*d^3/x-1/6*d^3*(a+b*arc
tanh(c*x))/x^6-3/5*c*d^3*(a+b*arctanh(c*x))/x^5-3/4*c^2*d^3*(a+b*arctanh(c*x))/x^4-1/3*c^3*d^3*(a+b*arctanh(c*
x))/x^3+14/15*b*c^6*d^3*ln(x)-37/40*b*c^6*d^3*ln(-c*x+1)-1/120*b*c^6*d^3*ln(c*x+1)

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Rubi [A]  time = 0.18, antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {43, 5936, 12, 1802} \[ -\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^3 d^3}{36 x^3}-\frac {3 b c^2 d^3}{20 x^4}-\frac {11 b c^5 d^3}{12 x}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (c x+1)-\frac {b c d^3}{30 x^5} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]

[Out]

-(b*c*d^3)/(30*x^5) - (3*b*c^2*d^3)/(20*x^4) - (11*b*c^3*d^3)/(36*x^3) - (7*b*c^4*d^3)/(15*x^2) - (11*b*c^5*d^
3)/(12*x) - (d^3*(a + b*ArcTanh[c*x]))/(6*x^6) - (3*c*d^3*(a + b*ArcTanh[c*x]))/(5*x^5) - (3*c^2*d^3*(a + b*Ar
cTanh[c*x]))/(4*x^4) - (c^3*d^3*(a + b*ArcTanh[c*x]))/(3*x^3) + (14*b*c^6*d^3*Log[x])/15 - (37*b*c^6*d^3*Log[1
- c*x])/40 - (b*c^6*d^3*Log[1 + c*x])/120

Rule 12

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

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 1802

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a + b*x
^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 5936

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_.) + (e_.)*(x_))^(q_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(d + e*x)^q, x]}, Dist[a + b*ArcTanh[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]))

Rubi steps

\begin {align*} \int \frac {(d+c d x)^3 \left (a+b \tanh ^{-1}(c x)\right )}{x^7} \, dx &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d^3 \left (-10-36 c x-45 c^2 x^2-20 c^3 x^3\right )}{60 x^6 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{60} \left (b c d^3\right ) \int \frac {-10-36 c x-45 c^2 x^2-20 c^3 x^3}{x^6 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{60} \left (b c d^3\right ) \int \left (-\frac {10}{x^6}-\frac {36 c}{x^5}-\frac {55 c^2}{x^4}-\frac {56 c^3}{x^3}-\frac {55 c^4}{x^2}-\frac {56 c^5}{x}+\frac {111 c^6}{2 (-1+c x)}+\frac {c^6}{2 (1+c x)}\right ) \, dx\\ &=-\frac {b c d^3}{30 x^5}-\frac {3 b c^2 d^3}{20 x^4}-\frac {11 b c^3 d^3}{36 x^3}-\frac {7 b c^4 d^3}{15 x^2}-\frac {11 b c^5 d^3}{12 x}-\frac {d^3 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {3 c d^3 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {3 c^2 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{4 x^4}-\frac {c^3 d^3 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {14}{15} b c^6 d^3 \log (x)-\frac {37}{40} b c^6 d^3 \log (1-c x)-\frac {1}{120} b c^6 d^3 \log (1+c x)\\ \end {align*}

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Mathematica [A]  time = 0.13, size = 149, normalized size = 0.76 \[ -\frac {d^3 \left (120 a c^3 x^3+270 a c^2 x^2+216 a c x+60 a-336 b c^6 x^6 \log (x)+333 b c^6 x^6 \log (1-c x)+3 b c^6 x^6 \log (c x+1)+330 b c^5 x^5+168 b c^4 x^4+110 b c^3 x^3+54 b c^2 x^2+6 b \left (20 c^3 x^3+45 c^2 x^2+36 c x+10\right ) \tanh ^{-1}(c x)+12 b c x\right )}{360 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^3*(a + b*ArcTanh[c*x]))/x^7,x]

[Out]

-1/360*(d^3*(60*a + 216*a*c*x + 12*b*c*x + 270*a*c^2*x^2 + 54*b*c^2*x^2 + 120*a*c^3*x^3 + 110*b*c^3*x^3 + 168*
b*c^4*x^4 + 330*b*c^5*x^5 + 6*b*(10 + 36*c*x + 45*c^2*x^2 + 20*c^3*x^3)*ArcTanh[c*x] - 336*b*c^6*x^6*Log[x] +
333*b*c^6*x^6*Log[1 - c*x] + 3*b*c^6*x^6*Log[1 + c*x]))/x^6

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fricas [A]  time = 0.48, size = 188, normalized size = 0.96 \[ -\frac {3 \, b c^{6} d^{3} x^{6} \log \left (c x + 1\right ) + 333 \, b c^{6} d^{3} x^{6} \log \left (c x - 1\right ) - 336 \, b c^{6} d^{3} x^{6} \log \relax (x) + 330 \, b c^{5} d^{3} x^{5} + 168 \, b c^{4} d^{3} x^{4} + 10 \, {\left (12 \, a + 11 \, b\right )} c^{3} d^{3} x^{3} + 54 \, {\left (5 \, a + b\right )} c^{2} d^{3} x^{2} + 12 \, {\left (18 \, a + b\right )} c d^{3} x + 60 \, a d^{3} + 3 \, {\left (20 \, b c^{3} d^{3} x^{3} + 45 \, b c^{2} d^{3} x^{2} + 36 \, b c d^{3} x + 10 \, b d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{360 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="fricas")

[Out]

-1/360*(3*b*c^6*d^3*x^6*log(c*x + 1) + 333*b*c^6*d^3*x^6*log(c*x - 1) - 336*b*c^6*d^3*x^6*log(x) + 330*b*c^5*d
^3*x^5 + 168*b*c^4*d^3*x^4 + 10*(12*a + 11*b)*c^3*d^3*x^3 + 54*(5*a + b)*c^2*d^3*x^2 + 12*(18*a + b)*c*d^3*x +
60*a*d^3 + 3*(20*b*c^3*d^3*x^3 + 45*b*c^2*d^3*x^2 + 36*b*c*d^3*x + 10*b*d^3)*log(-(c*x + 1)/(c*x - 1)))/x^6

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giac [B]  time = 0.16, size = 634, normalized size = 3.23 \[ \frac {1}{45} \, {\left (42 \, b c^{5} d^{3} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 42 \, b c^{5} d^{3} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {6 \, {\left (\frac {60 \, {\left (c x + 1\right )}^{5} b c^{5} d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {90 \, {\left (c x + 1\right )}^{4} b c^{5} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {140 \, {\left (c x + 1\right )}^{3} b c^{5} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {105 \, {\left (c x + 1\right )}^{2} b c^{5} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {42 \, {\left (c x + 1\right )} b c^{5} d^{3}}{c x - 1} + 7 \, b c^{5} d^{3}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{6}}{{\left (c x - 1\right )}^{6}} + \frac {6 \, {\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {20 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {6 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {720 \, {\left (c x + 1\right )}^{5} a c^{5} d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {1080 \, {\left (c x + 1\right )}^{4} a c^{5} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {1680 \, {\left (c x + 1\right )}^{3} a c^{5} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {1260 \, {\left (c x + 1\right )}^{2} a c^{5} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {504 \, {\left (c x + 1\right )} a c^{5} d^{3}}{c x - 1} + 84 \, a c^{5} d^{3} + \frac {318 \, {\left (c x + 1\right )}^{5} b c^{5} d^{3}}{{\left (c x - 1\right )}^{5}} + \frac {1119 \, {\left (c x + 1\right )}^{4} b c^{5} d^{3}}{{\left (c x - 1\right )}^{4}} + \frac {1742 \, {\left (c x + 1\right )}^{3} b c^{5} d^{3}}{{\left (c x - 1\right )}^{3}} + \frac {1464 \, {\left (c x + 1\right )}^{2} b c^{5} d^{3}}{{\left (c x - 1\right )}^{2}} + \frac {636 \, {\left (c x + 1\right )} b c^{5} d^{3}}{c x - 1} + 113 \, b c^{5} d^{3}}{\frac {{\left (c x + 1\right )}^{6}}{{\left (c x - 1\right )}^{6}} + \frac {6 \, {\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {15 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {20 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {15 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {6 \, {\left (c x + 1\right )}}{c x - 1} + 1}\right )} c \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="giac")

[Out]

1/45*(42*b*c^5*d^3*log(-(c*x + 1)/(c*x - 1) - 1) - 42*b*c^5*d^3*log(-(c*x + 1)/(c*x - 1)) + 6*(60*(c*x + 1)^5*
b*c^5*d^3/(c*x - 1)^5 + 90*(c*x + 1)^4*b*c^5*d^3/(c*x - 1)^4 + 140*(c*x + 1)^3*b*c^5*d^3/(c*x - 1)^3 + 105*(c*
x + 1)^2*b*c^5*d^3/(c*x - 1)^2 + 42*(c*x + 1)*b*c^5*d^3/(c*x - 1) + 7*b*c^5*d^3)*log(-(c*x + 1)/(c*x - 1))/((c
*x + 1)^6/(c*x - 1)^6 + 6*(c*x + 1)^5/(c*x - 1)^5 + 15*(c*x + 1)^4/(c*x - 1)^4 + 20*(c*x + 1)^3/(c*x - 1)^3 +
15*(c*x + 1)^2/(c*x - 1)^2 + 6*(c*x + 1)/(c*x - 1) + 1) + (720*(c*x + 1)^5*a*c^5*d^3/(c*x - 1)^5 + 1080*(c*x +
1)^4*a*c^5*d^3/(c*x - 1)^4 + 1680*(c*x + 1)^3*a*c^5*d^3/(c*x - 1)^3 + 1260*(c*x + 1)^2*a*c^5*d^3/(c*x - 1)^2
+ 504*(c*x + 1)*a*c^5*d^3/(c*x - 1) + 84*a*c^5*d^3 + 318*(c*x + 1)^5*b*c^5*d^3/(c*x - 1)^5 + 1119*(c*x + 1)^4*
b*c^5*d^3/(c*x - 1)^4 + 1742*(c*x + 1)^3*b*c^5*d^3/(c*x - 1)^3 + 1464*(c*x + 1)^2*b*c^5*d^3/(c*x - 1)^2 + 636*
(c*x + 1)*b*c^5*d^3/(c*x - 1) + 113*b*c^5*d^3)/((c*x + 1)^6/(c*x - 1)^6 + 6*(c*x + 1)^5/(c*x - 1)^5 + 15*(c*x
+ 1)^4/(c*x - 1)^4 + 20*(c*x + 1)^3/(c*x - 1)^3 + 15*(c*x + 1)^2/(c*x - 1)^2 + 6*(c*x + 1)/(c*x - 1) + 1))*c

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maple [A]  time = 0.04, size = 205, normalized size = 1.05 \[ -\frac {c^{3} d^{3} a}{3 x^{3}}-\frac {d^{3} a}{6 x^{6}}-\frac {3 c^{2} d^{3} a}{4 x^{4}}-\frac {3 c \,d^{3} a}{5 x^{5}}-\frac {c^{3} d^{3} b \arctanh \left (c x \right )}{3 x^{3}}-\frac {d^{3} b \arctanh \left (c x \right )}{6 x^{6}}-\frac {3 c^{2} d^{3} b \arctanh \left (c x \right )}{4 x^{4}}-\frac {3 c \,d^{3} b \arctanh \left (c x \right )}{5 x^{5}}-\frac {b c \,d^{3}}{30 x^{5}}-\frac {3 b \,c^{2} d^{3}}{20 x^{4}}-\frac {11 b \,c^{3} d^{3}}{36 x^{3}}-\frac {7 b \,c^{4} d^{3}}{15 x^{2}}-\frac {11 b \,c^{5} d^{3}}{12 x}+\frac {14 c^{6} d^{3} b \ln \left (c x \right )}{15}-\frac {37 c^{6} d^{3} b \ln \left (c x -1\right )}{40}-\frac {b \,c^{6} d^{3} \ln \left (c x +1\right )}{120} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x)

[Out]

-1/3*c^3*d^3*a/x^3-1/6*d^3*a/x^6-3/4*c^2*d^3*a/x^4-3/5*c*d^3*a/x^5-1/3*c^3*d^3*b*arctanh(c*x)/x^3-1/6*d^3*b*ar
ctanh(c*x)/x^6-3/4*c^2*d^3*b*arctanh(c*x)/x^4-3/5*c*d^3*b*arctanh(c*x)/x^5-1/30*b*c*d^3/x^5-3/20*b*c^2*d^3/x^4
-11/36*b*c^3*d^3/x^3-7/15*b*c^4*d^3/x^2-11/12*b*c^5*d^3/x+14/15*c^6*d^3*b*ln(c*x)-37/40*c^6*d^3*b*ln(c*x-1)-1/
120*b*c^6*d^3*ln(c*x+1)

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maxima [A]  time = 0.33, size = 273, normalized size = 1.39 \[ -\frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c^{3} d^{3} + \frac {1}{8} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b c^{2} d^{3} - \frac {3}{20} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b c d^{3} + \frac {1}{180} \, {\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c x\right )}{x^{6}}\right )} b d^{3} - \frac {a c^{3} d^{3}}{3 \, x^{3}} - \frac {3 \, a c^{2} d^{3}}{4 \, x^{4}} - \frac {3 \, a c d^{3}}{5 \, x^{5}} - \frac {a d^{3}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^3*(a+b*arctanh(c*x))/x^7,x, algorithm="maxima")

[Out]

-1/6*((c^2*log(c^2*x^2 - 1) - c^2*log(x^2) + 1/x^2)*c + 2*arctanh(c*x)/x^3)*b*c^3*d^3 + 1/8*((3*c^3*log(c*x +
1) - 3*c^3*log(c*x - 1) - 2*(3*c^2*x^2 + 1)/x^3)*c - 6*arctanh(c*x)/x^4)*b*c^2*d^3 - 3/20*((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*arctanh(c*x)/x^5)*b*c*d^3 + 1/180*((15*c^5*log(c*x + 1) - 1
5*c^5*log(c*x - 1) - 2*(15*c^4*x^4 + 5*c^2*x^2 + 3)/x^5)*c - 30*arctanh(c*x)/x^6)*b*d^3 - 1/3*a*c^3*d^3/x^3 -
3/4*a*c^2*d^3/x^4 - 3/5*a*c*d^3/x^5 - 1/6*a*d^3/x^6

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mupad [B]  time = 1.07, size = 220, normalized size = 1.12 \[ \frac {14\,b\,c^6\,d^3\,\ln \relax (x)}{15}-\frac {7\,b\,c^6\,d^3\,\ln \left (c^2\,x^2-1\right )}{15}-\frac {3\,a\,c^2\,d^3}{4\,x^4}-\frac {a\,c^3\,d^3}{3\,x^3}-\frac {3\,b\,c^2\,d^3}{20\,x^4}-\frac {11\,b\,c^3\,d^3}{36\,x^3}-\frac {7\,b\,c^4\,d^3}{15\,x^2}-\frac {11\,b\,c^5\,d^3}{12\,x}-\frac {a\,d^3}{6\,x^6}-\frac {3\,a\,c\,d^3}{5\,x^5}-\frac {b\,c\,d^3}{30\,x^5}-\frac {b\,d^3\,\mathrm {atanh}\left (c\,x\right )}{6\,x^6}-\frac {11\,b\,c^7\,d^3\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{12\,\sqrt {-c^2}}-\frac {3\,b\,c\,d^3\,\mathrm {atanh}\left (c\,x\right )}{5\,x^5}-\frac {3\,b\,c^2\,d^3\,\mathrm {atanh}\left (c\,x\right )}{4\,x^4}-\frac {b\,c^3\,d^3\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atanh(c*x))*(d + c*d*x)^3)/x^7,x)

[Out]

(14*b*c^6*d^3*log(x))/15 - (7*b*c^6*d^3*log(c^2*x^2 - 1))/15 - (3*a*c^2*d^3)/(4*x^4) - (a*c^3*d^3)/(3*x^3) - (
3*b*c^2*d^3)/(20*x^4) - (11*b*c^3*d^3)/(36*x^3) - (7*b*c^4*d^3)/(15*x^2) - (11*b*c^5*d^3)/(12*x) - (a*d^3)/(6*
x^6) - (3*a*c*d^3)/(5*x^5) - (b*c*d^3)/(30*x^5) - (b*d^3*atanh(c*x))/(6*x^6) - (11*b*c^7*d^3*atan((c^2*x)/(-c^
2)^(1/2)))/(12*(-c^2)^(1/2)) - (3*b*c*d^3*atanh(c*x))/(5*x^5) - (3*b*c^2*d^3*atanh(c*x))/(4*x^4) - (b*c^3*d^3*
atanh(c*x))/(3*x^3)

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sympy [A]  time = 3.31, size = 257, normalized size = 1.31 \[ \begin {cases} - \frac {a c^{3} d^{3}}{3 x^{3}} - \frac {3 a c^{2} d^{3}}{4 x^{4}} - \frac {3 a c d^{3}}{5 x^{5}} - \frac {a d^{3}}{6 x^{6}} + \frac {14 b c^{6} d^{3} \log {\relax (x )}}{15} - \frac {14 b c^{6} d^{3} \log {\left (x - \frac {1}{c} \right )}}{15} - \frac {b c^{6} d^{3} \operatorname {atanh}{\left (c x \right )}}{60} - \frac {11 b c^{5} d^{3}}{12 x} - \frac {7 b c^{4} d^{3}}{15 x^{2}} - \frac {b c^{3} d^{3} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} - \frac {11 b c^{3} d^{3}}{36 x^{3}} - \frac {3 b c^{2} d^{3} \operatorname {atanh}{\left (c x \right )}}{4 x^{4}} - \frac {3 b c^{2} d^{3}}{20 x^{4}} - \frac {3 b c d^{3} \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} - \frac {b c d^{3}}{30 x^{5}} - \frac {b d^{3} \operatorname {atanh}{\left (c x \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a d^{3}}{6 x^{6}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**3*(a+b*atanh(c*x))/x**7,x)

[Out]

Piecewise((-a*c**3*d**3/(3*x**3) - 3*a*c**2*d**3/(4*x**4) - 3*a*c*d**3/(5*x**5) - a*d**3/(6*x**6) + 14*b*c**6*
d**3*log(x)/15 - 14*b*c**6*d**3*log(x - 1/c)/15 - b*c**6*d**3*atanh(c*x)/60 - 11*b*c**5*d**3/(12*x) - 7*b*c**4
*d**3/(15*x**2) - b*c**3*d**3*atanh(c*x)/(3*x**3) - 11*b*c**3*d**3/(36*x**3) - 3*b*c**2*d**3*atanh(c*x)/(4*x**
4) - 3*b*c**2*d**3/(20*x**4) - 3*b*c*d**3*atanh(c*x)/(5*x**5) - b*c*d**3/(30*x**5) - b*d**3*atanh(c*x)/(6*x**6
), Ne(c, 0)), (-a*d**3/(6*x**6), True))

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