3.42 \(\int \frac {(d+c d x)^4 (a+b \tanh ^{-1}(c x))}{x^8} \, dx\)

Optimal. Leaf size=229 \[ -\frac {c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}+\frac {176}{105} b c^7 d^4 \log (x)-\frac {117}{70} b c^7 d^4 \log (1-c x)-\frac {1}{210} b c^7 d^4 \log (c x+1)-\frac {5 b c^6 d^4}{3 x}-\frac {88 b c^5 d^4}{105 x^2}-\frac {5 b c^4 d^4}{9 x^3}-\frac {47 b c^3 d^4}{140 x^4}-\frac {2 b c^2 d^4}{15 x^5}-\frac {b c d^4}{42 x^6} \]

[Out]

-1/42*b*c*d^4/x^6-2/15*b*c^2*d^4/x^5-47/140*b*c^3*d^4/x^4-5/9*b*c^4*d^4/x^3-88/105*b*c^5*d^4/x^2-5/3*b*c^6*d^4
/x-1/7*d^4*(a+b*arctanh(c*x))/x^7-2/3*c*d^4*(a+b*arctanh(c*x))/x^6-6/5*c^2*d^4*(a+b*arctanh(c*x))/x^5-c^3*d^4*
(a+b*arctanh(c*x))/x^4-1/3*c^4*d^4*(a+b*arctanh(c*x))/x^3+176/105*b*c^7*d^4*ln(x)-117/70*b*c^7*d^4*ln(-c*x+1)-
1/210*b*c^7*d^4*ln(c*x+1)

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Rubi [A]  time = 0.20, antiderivative size = 229, 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} \[ -\frac {c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac {88 b c^5 d^4}{105 x^2}-\frac {5 b c^4 d^4}{9 x^3}-\frac {47 b c^3 d^4}{140 x^4}-\frac {2 b c^2 d^4}{15 x^5}-\frac {5 b c^6 d^4}{3 x}+\frac {176}{105} b c^7 d^4 \log (x)-\frac {117}{70} b c^7 d^4 \log (1-c x)-\frac {1}{210} b c^7 d^4 \log (c x+1)-\frac {b c d^4}{42 x^6} \]

Antiderivative was successfully verified.

[In]

Int[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^8,x]

[Out]

-(b*c*d^4)/(42*x^6) - (2*b*c^2*d^4)/(15*x^5) - (47*b*c^3*d^4)/(140*x^4) - (5*b*c^4*d^4)/(9*x^3) - (88*b*c^5*d^
4)/(105*x^2) - (5*b*c^6*d^4)/(3*x) - (d^4*(a + b*ArcTanh[c*x]))/(7*x^7) - (2*c*d^4*(a + b*ArcTanh[c*x]))/(3*x^
6) - (6*c^2*d^4*(a + b*ArcTanh[c*x]))/(5*x^5) - (c^3*d^4*(a + b*ArcTanh[c*x]))/x^4 - (c^4*d^4*(a + b*ArcTanh[c
*x]))/(3*x^3) + (176*b*c^7*d^4*Log[x])/105 - (117*b*c^7*d^4*Log[1 - c*x])/70 - (b*c^7*d^4*Log[1 + c*x])/210

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)^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^8} \, dx &=-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac {d^4 \left (-15-70 c x-126 c^2 x^2-105 c^3 x^3-35 c^4 x^4\right )}{105 x^7 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{105} \left (b c d^4\right ) \int \frac {-15-70 c x-126 c^2 x^2-105 c^3 x^3-35 c^4 x^4}{x^7 \left (1-c^2 x^2\right )} \, dx\\ &=-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}-\frac {1}{105} \left (b c d^4\right ) \int \left (-\frac {15}{x^7}-\frac {70 c}{x^6}-\frac {141 c^2}{x^5}-\frac {175 c^3}{x^4}-\frac {176 c^4}{x^3}-\frac {175 c^5}{x^2}-\frac {176 c^6}{x}+\frac {351 c^7}{2 (-1+c x)}+\frac {c^7}{2 (1+c x)}\right ) \, dx\\ &=-\frac {b c d^4}{42 x^6}-\frac {2 b c^2 d^4}{15 x^5}-\frac {47 b c^3 d^4}{140 x^4}-\frac {5 b c^4 d^4}{9 x^3}-\frac {88 b c^5 d^4}{105 x^2}-\frac {5 b c^6 d^4}{3 x}-\frac {d^4 \left (a+b \tanh ^{-1}(c x)\right )}{7 x^7}-\frac {2 c d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^6}-\frac {6 c^2 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{5 x^5}-\frac {c^3 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{x^4}-\frac {c^4 d^4 \left (a+b \tanh ^{-1}(c x)\right )}{3 x^3}+\frac {176}{105} b c^7 d^4 \log (x)-\frac {117}{70} b c^7 d^4 \log (1-c x)-\frac {1}{210} b c^7 d^4 \log (1+c x)\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 175, normalized size = 0.76 \[ -\frac {d^4 \left (420 a c^4 x^4+1260 a c^3 x^3+1512 a c^2 x^2+840 a c x+180 a-2112 b c^7 x^7 \log (x)+2106 b c^7 x^7 \log (1-c x)+6 b c^7 x^7 \log (c x+1)+2100 b c^6 x^6+1056 b c^5 x^5+700 b c^4 x^4+423 b c^3 x^3+168 b c^2 x^2+12 b \left (35 c^4 x^4+105 c^3 x^3+126 c^2 x^2+70 c x+15\right ) \tanh ^{-1}(c x)+30 b c x\right )}{1260 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + c*d*x)^4*(a + b*ArcTanh[c*x]))/x^8,x]

[Out]

-1/1260*(d^4*(180*a + 840*a*c*x + 30*b*c*x + 1512*a*c^2*x^2 + 168*b*c^2*x^2 + 1260*a*c^3*x^3 + 423*b*c^3*x^3 +
420*a*c^4*x^4 + 700*b*c^4*x^4 + 1056*b*c^5*x^5 + 2100*b*c^6*x^6 + 12*b*(15 + 70*c*x + 126*c^2*x^2 + 105*c^3*x
^3 + 35*c^4*x^4)*ArcTanh[c*x] - 2112*b*c^7*x^7*Log[x] + 2106*b*c^7*x^7*Log[1 - c*x] + 6*b*c^7*x^7*Log[1 + c*x]
))/x^7

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fricas [A]  time = 0.60, size = 218, normalized size = 0.95 \[ -\frac {6 \, b c^{7} d^{4} x^{7} \log \left (c x + 1\right ) + 2106 \, b c^{7} d^{4} x^{7} \log \left (c x - 1\right ) - 2112 \, b c^{7} d^{4} x^{7} \log \relax (x) + 2100 \, b c^{6} d^{4} x^{6} + 1056 \, b c^{5} d^{4} x^{5} + 140 \, {\left (3 \, a + 5 \, b\right )} c^{4} d^{4} x^{4} + 9 \, {\left (140 \, a + 47 \, b\right )} c^{3} d^{4} x^{3} + 168 \, {\left (9 \, a + b\right )} c^{2} d^{4} x^{2} + 30 \, {\left (28 \, a + b\right )} c d^{4} x + 180 \, a d^{4} + 6 \, {\left (35 \, b c^{4} d^{4} x^{4} + 105 \, b c^{3} d^{4} x^{3} + 126 \, b c^{2} d^{4} x^{2} + 70 \, b c d^{4} x + 15 \, b d^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{1260 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^4*(a+b*arctanh(c*x))/x^8,x, algorithm="fricas")

[Out]

-1/1260*(6*b*c^7*d^4*x^7*log(c*x + 1) + 2106*b*c^7*d^4*x^7*log(c*x - 1) - 2112*b*c^7*d^4*x^7*log(x) + 2100*b*c
^6*d^4*x^6 + 1056*b*c^5*d^4*x^5 + 140*(3*a + 5*b)*c^4*d^4*x^4 + 9*(140*a + 47*b)*c^3*d^4*x^3 + 168*(9*a + b)*c
^2*d^4*x^2 + 30*(28*a + b)*c*d^4*x + 180*a*d^4 + 6*(35*b*c^4*d^4*x^4 + 105*b*c^3*d^4*x^3 + 126*b*c^2*d^4*x^2 +
70*b*c*d^4*x + 15*b*d^4)*log(-(c*x + 1)/(c*x - 1)))/x^7

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giac [B]  time = 0.33, size = 735, normalized size = 3.21 \[ \frac {4}{315} \, {\left (132 \, b c^{6} d^{4} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 132 \, b c^{6} d^{4} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {12 \, {\left (\frac {105 \, {\left (c x + 1\right )}^{6} b c^{6} d^{4}}{{\left (c x - 1\right )}^{6}} + \frac {210 \, {\left (c x + 1\right )}^{5} b c^{6} d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {385 \, {\left (c x + 1\right )}^{4} b c^{6} d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {385 \, {\left (c x + 1\right )}^{3} b c^{6} d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {231 \, {\left (c x + 1\right )}^{2} b c^{6} d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {77 \, {\left (c x + 1\right )} b c^{6} d^{4}}{c x - 1} + 11 \, b c^{6} d^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{\frac {{\left (c x + 1\right )}^{7}}{{\left (c x - 1\right )}^{7}} + \frac {7 \, {\left (c x + 1\right )}^{6}}{{\left (c x - 1\right )}^{6}} + \frac {21 \, {\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {35 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {35 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {21 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\left (c x + 1\right )}}{c x - 1} + 1} + \frac {\frac {2520 \, {\left (c x + 1\right )}^{6} a c^{6} d^{4}}{{\left (c x - 1\right )}^{6}} + \frac {5040 \, {\left (c x + 1\right )}^{5} a c^{6} d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {9240 \, {\left (c x + 1\right )}^{4} a c^{6} d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {9240 \, {\left (c x + 1\right )}^{3} a c^{6} d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {5544 \, {\left (c x + 1\right )}^{2} a c^{6} d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {1848 \, {\left (c x + 1\right )} a c^{6} d^{4}}{c x - 1} + 264 \, a c^{6} d^{4} + \frac {1128 \, {\left (c x + 1\right )}^{6} b c^{6} d^{4}}{{\left (c x - 1\right )}^{6}} + \frac {4812 \, {\left (c x + 1\right )}^{5} b c^{6} d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {9476 \, {\left (c x + 1\right )}^{4} b c^{6} d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {10631 \, {\left (c x + 1\right )}^{3} b c^{6} d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {6933 \, {\left (c x + 1\right )}^{2} b c^{6} d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {2465 \, {\left (c x + 1\right )} b c^{6} d^{4}}{c x - 1} + 371 \, b c^{6} d^{4}}{\frac {{\left (c x + 1\right )}^{7}}{{\left (c x - 1\right )}^{7}} + \frac {7 \, {\left (c x + 1\right )}^{6}}{{\left (c x - 1\right )}^{6}} + \frac {21 \, {\left (c x + 1\right )}^{5}}{{\left (c x - 1\right )}^{5}} + \frac {35 \, {\left (c x + 1\right )}^{4}}{{\left (c x - 1\right )}^{4}} + \frac {35 \, {\left (c x + 1\right )}^{3}}{{\left (c x - 1\right )}^{3}} + \frac {21 \, {\left (c x + 1\right )}^{2}}{{\left (c x - 1\right )}^{2}} + \frac {7 \, {\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)^4*(a+b*arctanh(c*x))/x^8,x, algorithm="giac")

[Out]

4/315*(132*b*c^6*d^4*log(-(c*x + 1)/(c*x - 1) - 1) - 132*b*c^6*d^4*log(-(c*x + 1)/(c*x - 1)) + 12*(105*(c*x +
1)^6*b*c^6*d^4/(c*x - 1)^6 + 210*(c*x + 1)^5*b*c^6*d^4/(c*x - 1)^5 + 385*(c*x + 1)^4*b*c^6*d^4/(c*x - 1)^4 + 3
85*(c*x + 1)^3*b*c^6*d^4/(c*x - 1)^3 + 231*(c*x + 1)^2*b*c^6*d^4/(c*x - 1)^2 + 77*(c*x + 1)*b*c^6*d^4/(c*x - 1
) + 11*b*c^6*d^4)*log(-(c*x + 1)/(c*x - 1))/((c*x + 1)^7/(c*x - 1)^7 + 7*(c*x + 1)^6/(c*x - 1)^6 + 21*(c*x + 1
)^5/(c*x - 1)^5 + 35*(c*x + 1)^4/(c*x - 1)^4 + 35*(c*x + 1)^3/(c*x - 1)^3 + 21*(c*x + 1)^2/(c*x - 1)^2 + 7*(c*
x + 1)/(c*x - 1) + 1) + (2520*(c*x + 1)^6*a*c^6*d^4/(c*x - 1)^6 + 5040*(c*x + 1)^5*a*c^6*d^4/(c*x - 1)^5 + 924
0*(c*x + 1)^4*a*c^6*d^4/(c*x - 1)^4 + 9240*(c*x + 1)^3*a*c^6*d^4/(c*x - 1)^3 + 5544*(c*x + 1)^2*a*c^6*d^4/(c*x
- 1)^2 + 1848*(c*x + 1)*a*c^6*d^4/(c*x - 1) + 264*a*c^6*d^4 + 1128*(c*x + 1)^6*b*c^6*d^4/(c*x - 1)^6 + 4812*(
c*x + 1)^5*b*c^6*d^4/(c*x - 1)^5 + 9476*(c*x + 1)^4*b*c^6*d^4/(c*x - 1)^4 + 10631*(c*x + 1)^3*b*c^6*d^4/(c*x -
1)^3 + 6933*(c*x + 1)^2*b*c^6*d^4/(c*x - 1)^2 + 2465*(c*x + 1)*b*c^6*d^4/(c*x - 1) + 371*b*c^6*d^4)/((c*x + 1
)^7/(c*x - 1)^7 + 7*(c*x + 1)^6/(c*x - 1)^6 + 21*(c*x + 1)^5/(c*x - 1)^5 + 35*(c*x + 1)^4/(c*x - 1)^4 + 35*(c*
x + 1)^3/(c*x - 1)^3 + 21*(c*x + 1)^2/(c*x - 1)^2 + 7*(c*x + 1)/(c*x - 1) + 1))*c

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maple [A]  time = 0.04, size = 245, normalized size = 1.07 \[ -\frac {c^{4} d^{4} a}{3 x^{3}}-\frac {d^{4} a}{7 x^{7}}-\frac {2 c \,d^{4} a}{3 x^{6}}-\frac {c^{3} d^{4} a}{x^{4}}-\frac {6 c^{2} d^{4} a}{5 x^{5}}-\frac {c^{4} d^{4} b \arctanh \left (c x \right )}{3 x^{3}}-\frac {d^{4} b \arctanh \left (c x \right )}{7 x^{7}}-\frac {2 c \,d^{4} b \arctanh \left (c x \right )}{3 x^{6}}-\frac {c^{3} d^{4} b \arctanh \left (c x \right )}{x^{4}}-\frac {6 c^{2} d^{4} b \arctanh \left (c x \right )}{5 x^{5}}-\frac {b c \,d^{4}}{42 x^{6}}-\frac {2 b \,c^{2} d^{4}}{15 x^{5}}-\frac {47 b \,c^{3} d^{4}}{140 x^{4}}-\frac {5 b \,c^{4} d^{4}}{9 x^{3}}-\frac {88 b \,c^{5} d^{4}}{105 x^{2}}-\frac {5 b \,c^{6} d^{4}}{3 x}+\frac {176 c^{7} d^{4} b \ln \left (c x \right )}{105}-\frac {117 c^{7} d^{4} b \ln \left (c x -1\right )}{70}-\frac {b \,c^{7} d^{4} \ln \left (c x +1\right )}{210} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*d*x+d)^4*(a+b*arctanh(c*x))/x^8,x)

[Out]

-1/3*c^4*d^4*a/x^3-1/7*d^4*a/x^7-2/3*c*d^4*a/x^6-c^3*d^4*a/x^4-6/5*c^2*d^4*a/x^5-1/3*c^4*d^4*b*arctanh(c*x)/x^
3-1/7*d^4*b*arctanh(c*x)/x^7-2/3*c*d^4*b*arctanh(c*x)/x^6-c^3*d^4*b*arctanh(c*x)/x^4-6/5*c^2*d^4*b*arctanh(c*x
)/x^5-1/42*b*c*d^4/x^6-2/15*b*c^2*d^4/x^5-47/140*b*c^3*d^4/x^4-5/9*b*c^4*d^4/x^3-88/105*b*c^5*d^4/x^2-5/3*b*c^
6*d^4/x+176/105*c^7*d^4*b*ln(c*x)-117/70*c^7*d^4*b*ln(c*x-1)-1/210*b*c^7*d^4*ln(c*x+1)

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maxima [A]  time = 0.33, size = 353, normalized size = 1.54 \[ -\frac {1}{6} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c^{4} d^{4} + \frac {1}{6} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b c^{3} d^{4} - \frac {3}{10} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b c^{2} d^{4} + \frac {1}{45} \, {\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c x\right )}{x^{6}}\right )} b c d^{4} - \frac {1}{84} \, {\left ({\left (6 \, c^{6} \log \left (c^{2} x^{2} - 1\right ) - 6 \, c^{6} \log \left (x^{2}\right ) + \frac {6 \, c^{4} x^{4} + 3 \, c^{2} x^{2} + 2}{x^{6}}\right )} c + \frac {12 \, \operatorname {artanh}\left (c x\right )}{x^{7}}\right )} b d^{4} - \frac {a c^{4} d^{4}}{3 \, x^{3}} - \frac {a c^{3} d^{4}}{x^{4}} - \frac {6 \, a c^{2} d^{4}}{5 \, x^{5}} - \frac {2 \, a c d^{4}}{3 \, x^{6}} - \frac {a d^{4}}{7 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)^4*(a+b*arctanh(c*x))/x^8,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^4*d^4 + 1/6*((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^3*d^4 - 3/10*((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^2*d^4 + 1/45*((15*c^5*log(c*x + 1) -
15*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*c*d^4 - 1/84*((6*c^6*log(
c^2*x^2 - 1) - 6*c^6*log(x^2) + (6*c^4*x^4 + 3*c^2*x^2 + 2)/x^6)*c + 12*arctanh(c*x)/x^7)*b*d^4 - 1/3*a*c^4*d^
4/x^3 - a*c^3*d^4/x^4 - 6/5*a*c^2*d^4/x^5 - 2/3*a*c*d^4/x^6 - 1/7*a*d^4/x^7

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mupad [B]  time = 1.21, size = 260, normalized size = 1.14 \[ \frac {176\,b\,c^7\,d^4\,\ln \relax (x)}{105}-\frac {88\,b\,c^7\,d^4\,\ln \left (c^2\,x^2-1\right )}{105}-\frac {6\,a\,c^2\,d^4}{5\,x^5}-\frac {a\,c^3\,d^4}{x^4}-\frac {a\,c^4\,d^4}{3\,x^3}-\frac {2\,b\,c^2\,d^4}{15\,x^5}-\frac {47\,b\,c^3\,d^4}{140\,x^4}-\frac {5\,b\,c^4\,d^4}{9\,x^3}-\frac {88\,b\,c^5\,d^4}{105\,x^2}-\frac {5\,b\,c^6\,d^4}{3\,x}-\frac {a\,d^4}{7\,x^7}-\frac {2\,a\,c\,d^4}{3\,x^6}-\frac {b\,c\,d^4}{42\,x^6}-\frac {b\,d^4\,\mathrm {atanh}\left (c\,x\right )}{7\,x^7}-\frac {5\,b\,c^8\,d^4\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{3\,\sqrt {-c^2}}-\frac {2\,b\,c\,d^4\,\mathrm {atanh}\left (c\,x\right )}{3\,x^6}-\frac {6\,b\,c^2\,d^4\,\mathrm {atanh}\left (c\,x\right )}{5\,x^5}-\frac {b\,c^3\,d^4\,\mathrm {atanh}\left (c\,x\right )}{x^4}-\frac {b\,c^4\,d^4\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*atanh(c*x))*(d + c*d*x)^4)/x^8,x)

[Out]

(176*b*c^7*d^4*log(x))/105 - (88*b*c^7*d^4*log(c^2*x^2 - 1))/105 - (6*a*c^2*d^4)/(5*x^5) - (a*c^3*d^4)/x^4 - (
a*c^4*d^4)/(3*x^3) - (2*b*c^2*d^4)/(15*x^5) - (47*b*c^3*d^4)/(140*x^4) - (5*b*c^4*d^4)/(9*x^3) - (88*b*c^5*d^4
)/(105*x^2) - (5*b*c^6*d^4)/(3*x) - (a*d^4)/(7*x^7) - (2*a*c*d^4)/(3*x^6) - (b*c*d^4)/(42*x^6) - (b*d^4*atanh(
c*x))/(7*x^7) - (5*b*c^8*d^4*atan((c^2*x)/(-c^2)^(1/2)))/(3*(-c^2)^(1/2)) - (2*b*c*d^4*atanh(c*x))/(3*x^6) - (
6*b*c^2*d^4*atanh(c*x))/(5*x^5) - (b*c^3*d^4*atanh(c*x))/x^4 - (b*c^4*d^4*atanh(c*x))/(3*x^3)

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sympy [A]  time = 4.19, size = 301, normalized size = 1.31 \[ \begin {cases} - \frac {a c^{4} d^{4}}{3 x^{3}} - \frac {a c^{3} d^{4}}{x^{4}} - \frac {6 a c^{2} d^{4}}{5 x^{5}} - \frac {2 a c d^{4}}{3 x^{6}} - \frac {a d^{4}}{7 x^{7}} + \frac {176 b c^{7} d^{4} \log {\relax (x )}}{105} - \frac {176 b c^{7} d^{4} \log {\left (x - \frac {1}{c} \right )}}{105} - \frac {b c^{7} d^{4} \operatorname {atanh}{\left (c x \right )}}{105} - \frac {5 b c^{6} d^{4}}{3 x} - \frac {88 b c^{5} d^{4}}{105 x^{2}} - \frac {b c^{4} d^{4} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} - \frac {5 b c^{4} d^{4}}{9 x^{3}} - \frac {b c^{3} d^{4} \operatorname {atanh}{\left (c x \right )}}{x^{4}} - \frac {47 b c^{3} d^{4}}{140 x^{4}} - \frac {6 b c^{2} d^{4} \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} - \frac {2 b c^{2} d^{4}}{15 x^{5}} - \frac {2 b c d^{4} \operatorname {atanh}{\left (c x \right )}}{3 x^{6}} - \frac {b c d^{4}}{42 x^{6}} - \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{7 x^{7}} & \text {for}\: c \neq 0 \\- \frac {a d^{4}}{7 x^{7}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**4*(a+b*atanh(c*x))/x**8,x)

[Out]

Piecewise((-a*c**4*d**4/(3*x**3) - a*c**3*d**4/x**4 - 6*a*c**2*d**4/(5*x**5) - 2*a*c*d**4/(3*x**6) - a*d**4/(7
*x**7) + 176*b*c**7*d**4*log(x)/105 - 176*b*c**7*d**4*log(x - 1/c)/105 - b*c**7*d**4*atanh(c*x)/105 - 5*b*c**6
*d**4/(3*x) - 88*b*c**5*d**4/(105*x**2) - b*c**4*d**4*atanh(c*x)/(3*x**3) - 5*b*c**4*d**4/(9*x**3) - b*c**3*d*
*4*atanh(c*x)/x**4 - 47*b*c**3*d**4/(140*x**4) - 6*b*c**2*d**4*atanh(c*x)/(5*x**5) - 2*b*c**2*d**4/(15*x**5) -
2*b*c*d**4*atanh(c*x)/(3*x**6) - b*c*d**4/(42*x**6) - b*d**4*atanh(c*x)/(7*x**7), Ne(c, 0)), (-a*d**4/(7*x**7
), True))

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