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

Optimal. Leaf size=151 \[ -\frac {d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}+\frac {c d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}+\frac {32}{15} b c^6 d^4 \log (x)-\frac {32}{15} b c^6 d^4 \log (1-c x)-\frac {13 b c^5 d^4}{6 x}-\frac {16 b c^4 d^4}{15 x^2}-\frac {5 b c^3 d^4}{9 x^3}-\frac {b c^2 d^4}{5 x^4}-\frac {b c d^4}{30 x^5} \]

[Out]

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

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Rubi [A]  time = 0.13, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {45, 37, 5936, 12, 148} \[ \frac {c d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{30 x^5}-\frac {d^4 (c x+1)^5 \left (a+b \tanh ^{-1}(c x)\right )}{6 x^6}-\frac {16 b c^4 d^4}{15 x^2}-\frac {5 b c^3 d^4}{9 x^3}-\frac {b c^2 d^4}{5 x^4}-\frac {13 b c^5 d^4}{6 x}+\frac {32}{15} b c^6 d^4 \log (x)-\frac {32}{15} b c^6 d^4 \log (1-c x)-\frac {b c d^4}{30 x^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

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 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 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]))

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

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Mathematica [A]  time = 0.15, size = 166, normalized size = 1.10 \[ -\frac {d^4 \left (90 a c^4 x^4+240 a c^3 x^3+270 a c^2 x^2+144 a c x+30 a-384 b c^6 x^6 \log (x)+387 b c^6 x^6 \log (1-c x)-3 b c^6 x^6 \log (c x+1)+390 b c^5 x^5+192 b c^4 x^4+100 b c^3 x^3+36 b c^2 x^2+6 b \left (15 c^4 x^4+40 c^3 x^3+45 c^2 x^2+24 c x+5\right ) \tanh ^{-1}(c x)+6 b c x\right )}{180 x^6} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/180*(d^4*(30*a + 144*a*c*x + 6*b*c*x + 270*a*c^2*x^2 + 36*b*c^2*x^2 + 240*a*c^3*x^3 + 100*b*c^3*x^3 + 90*a*
c^4*x^4 + 192*b*c^4*x^4 + 390*b*c^5*x^5 + 6*b*(5 + 24*c*x + 45*c^2*x^2 + 40*c^3*x^3 + 15*c^4*x^4)*ArcTanh[c*x]
- 384*b*c^6*x^6*Log[x] + 387*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 = 0.47, size = 208, normalized size = 1.38 \[ \frac {3 \, b c^{6} d^{4} x^{6} \log \left (c x + 1\right ) - 387 \, b c^{6} d^{4} x^{6} \log \left (c x - 1\right ) + 384 \, b c^{6} d^{4} x^{6} \log \relax (x) - 390 \, b c^{5} d^{4} x^{5} - 6 \, {\left (15 \, a + 32 \, b\right )} c^{4} d^{4} x^{4} - 20 \, {\left (12 \, a + 5 \, b\right )} c^{3} d^{4} x^{3} - 18 \, {\left (15 \, a + 2 \, b\right )} c^{2} d^{4} x^{2} - 6 \, {\left (24 \, a + b\right )} c d^{4} x - 30 \, a d^{4} - 3 \, {\left (15 \, b c^{4} d^{4} x^{4} + 40 \, b c^{3} d^{4} x^{3} + 45 \, b c^{2} d^{4} x^{2} + 24 \, b c d^{4} x + 5 \, b d^{4}\right )} \log \left (-\frac {c x + 1}{c x - 1}\right )}{180 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/180*(3*b*c^6*d^4*x^6*log(c*x + 1) - 387*b*c^6*d^4*x^6*log(c*x - 1) + 384*b*c^6*d^4*x^6*log(x) - 390*b*c^5*d^
4*x^5 - 6*(15*a + 32*b)*c^4*d^4*x^4 - 20*(12*a + 5*b)*c^3*d^4*x^3 - 18*(15*a + 2*b)*c^2*d^4*x^2 - 6*(24*a + b)
*c*d^4*x - 30*a*d^4 - 3*(15*b*c^4*d^4*x^4 + 40*b*c^3*d^4*x^3 + 45*b*c^2*d^4*x^2 + 24*b*c*d^4*x + 5*b*d^4)*log(
-(c*x + 1)/(c*x - 1)))/x^6

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giac [B]  time = 0.56, size = 634, normalized size = 4.20 \[ \frac {8}{45} \, {\left (12 \, b c^{5} d^{4} \log \left (-\frac {c x + 1}{c x - 1} - 1\right ) - 12 \, b c^{5} d^{4} \log \left (-\frac {c x + 1}{c x - 1}\right ) + \frac {6 \, {\left (\frac {15 \, {\left (c x + 1\right )}^{5} b c^{5} d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {30 \, {\left (c x + 1\right )}^{4} b c^{5} d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {40 \, {\left (c x + 1\right )}^{3} b c^{5} d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {30 \, {\left (c x + 1\right )}^{2} b c^{5} d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {12 \, {\left (c x + 1\right )} b c^{5} d^{4}}{c x - 1} + 2 \, b c^{5} d^{4}\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 {180 \, {\left (c x + 1\right )}^{5} a c^{5} d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {360 \, {\left (c x + 1\right )}^{4} a c^{5} d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {480 \, {\left (c x + 1\right )}^{3} a c^{5} d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {360 \, {\left (c x + 1\right )}^{2} a c^{5} d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {144 \, {\left (c x + 1\right )} a c^{5} d^{4}}{c x - 1} + 24 \, a c^{5} d^{4} + \frac {78 \, {\left (c x + 1\right )}^{5} b c^{5} d^{4}}{{\left (c x - 1\right )}^{5}} + \frac {294 \, {\left (c x + 1\right )}^{4} b c^{5} d^{4}}{{\left (c x - 1\right )}^{4}} + \frac {472 \, {\left (c x + 1\right )}^{3} b c^{5} d^{4}}{{\left (c x - 1\right )}^{3}} + \frac {399 \, {\left (c x + 1\right )}^{2} b c^{5} d^{4}}{{\left (c x - 1\right )}^{2}} + \frac {174 \, {\left (c x + 1\right )} b c^{5} d^{4}}{c x - 1} + 31 \, b c^{5} d^{4}}{\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)^4*(a+b*arctanh(c*x))/x^7,x, algorithm="giac")

[Out]

8/45*(12*b*c^5*d^4*log(-(c*x + 1)/(c*x - 1) - 1) - 12*b*c^5*d^4*log(-(c*x + 1)/(c*x - 1)) + 6*(15*(c*x + 1)^5*
b*c^5*d^4/(c*x - 1)^5 + 30*(c*x + 1)^4*b*c^5*d^4/(c*x - 1)^4 + 40*(c*x + 1)^3*b*c^5*d^4/(c*x - 1)^3 + 30*(c*x
+ 1)^2*b*c^5*d^4/(c*x - 1)^2 + 12*(c*x + 1)*b*c^5*d^4/(c*x - 1) + 2*b*c^5*d^4)*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) + (180*(c*x + 1)^5*a*c^5*d^4/(c*x - 1)^5 + 360*(c*x + 1)
^4*a*c^5*d^4/(c*x - 1)^4 + 480*(c*x + 1)^3*a*c^5*d^4/(c*x - 1)^3 + 360*(c*x + 1)^2*a*c^5*d^4/(c*x - 1)^2 + 144
*(c*x + 1)*a*c^5*d^4/(c*x - 1) + 24*a*c^5*d^4 + 78*(c*x + 1)^5*b*c^5*d^4/(c*x - 1)^5 + 294*(c*x + 1)^4*b*c^5*d
^4/(c*x - 1)^4 + 472*(c*x + 1)^3*b*c^5*d^4/(c*x - 1)^3 + 399*(c*x + 1)^2*b*c^5*d^4/(c*x - 1)^2 + 174*(c*x + 1)
*b*c^5*d^4/(c*x - 1) + 31*b*c^5*d^4)/((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 = 233, normalized size = 1.54 \[ -\frac {4 c^{3} d^{4} a}{3 x^{3}}-\frac {c^{4} d^{4} a}{2 x^{2}}-\frac {d^{4} a}{6 x^{6}}-\frac {3 c^{2} d^{4} a}{2 x^{4}}-\frac {4 c \,d^{4} a}{5 x^{5}}-\frac {4 c^{3} d^{4} b \arctanh \left (c x \right )}{3 x^{3}}-\frac {c^{4} d^{4} b \arctanh \left (c x \right )}{2 x^{2}}-\frac {d^{4} b \arctanh \left (c x \right )}{6 x^{6}}-\frac {3 c^{2} d^{4} b \arctanh \left (c x \right )}{2 x^{4}}-\frac {4 c \,d^{4} b \arctanh \left (c x \right )}{5 x^{5}}-\frac {b c \,d^{4}}{30 x^{5}}-\frac {b \,c^{2} d^{4}}{5 x^{4}}-\frac {5 b \,c^{3} d^{4}}{9 x^{3}}-\frac {16 b \,c^{4} d^{4}}{15 x^{2}}-\frac {13 b \,c^{5} d^{4}}{6 x}+\frac {32 c^{6} d^{4} b \ln \left (c x \right )}{15}-\frac {43 c^{6} d^{4} b \ln \left (c x -1\right )}{20}+\frac {c^{6} d^{4} b \ln \left (c x +1\right )}{60} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/3*c^3*d^4*a/x^3-1/2*c^4*d^4*a/x^2-1/6*d^4*a/x^6-3/2*c^2*d^4*a/x^4-4/5*c*d^4*a/x^5-4/3*c^3*d^4*b*arctanh(c*x
)/x^3-1/2*c^4*d^4*b*arctanh(c*x)/x^2-1/6*d^4*b*arctanh(c*x)/x^6-3/2*c^2*d^4*b*arctanh(c*x)/x^4-4/5*c*d^4*b*arc
tanh(c*x)/x^5-1/30*b*c*d^4/x^5-1/5*b*c^2*d^4/x^4-5/9*b*c^3*d^4/x^3-16/15*b*c^4*d^4/x^2-13/6*b*c^5*d^4/x+32/15*
c^6*d^4*b*ln(c*x)-43/20*c^6*d^4*b*ln(c*x-1)+1/60*c^6*d^4*b*ln(c*x+1)

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maxima [B]  time = 0.33, size = 329, normalized size = 2.18 \[ \frac {1}{4} \, {\left ({\left (c \log \left (c x + 1\right ) - c \log \left (c x - 1\right ) - \frac {2}{x}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{2}}\right )} b c^{4} d^{4} - \frac {2}{3} \, {\left ({\left (c^{2} \log \left (c^{2} x^{2} - 1\right ) - c^{2} \log \left (x^{2}\right ) + \frac {1}{x^{2}}\right )} c + \frac {2 \, \operatorname {artanh}\left (c x\right )}{x^{3}}\right )} b c^{3} d^{4} + \frac {1}{4} \, {\left ({\left (3 \, c^{3} \log \left (c x + 1\right ) - 3 \, c^{3} \log \left (c x - 1\right ) - \frac {2 \, {\left (3 \, c^{2} x^{2} + 1\right )}}{x^{3}}\right )} c - \frac {6 \, \operatorname {artanh}\left (c x\right )}{x^{4}}\right )} b c^{2} d^{4} - \frac {1}{5} \, {\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} - 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) + \frac {2 \, c^{2} x^{2} + 1}{x^{4}}\right )} c + \frac {4 \, \operatorname {artanh}\left (c x\right )}{x^{5}}\right )} b c d^{4} - \frac {a c^{4} d^{4}}{2 \, x^{2}} + \frac {1}{180} \, {\left ({\left (15 \, c^{5} \log \left (c x + 1\right ) - 15 \, c^{5} \log \left (c x - 1\right ) - \frac {2 \, {\left (15 \, c^{4} x^{4} + 5 \, c^{2} x^{2} + 3\right )}}{x^{5}}\right )} c - \frac {30 \, \operatorname {artanh}\left (c x\right )}{x^{6}}\right )} b d^{4} - \frac {4 \, a c^{3} d^{4}}{3 \, x^{3}} - \frac {3 \, a c^{2} d^{4}}{2 \, x^{4}} - \frac {4 \, a c d^{4}}{5 \, x^{5}} - \frac {a d^{4}}{6 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.24, size = 248, normalized size = 1.64 \[ \frac {32\,b\,c^6\,d^4\,\ln \relax (x)}{15}-\frac {16\,b\,c^6\,d^4\,\ln \left (c^2\,x^2-1\right )}{15}-\frac {3\,a\,c^2\,d^4}{2\,x^4}-\frac {4\,a\,c^3\,d^4}{3\,x^3}-\frac {a\,c^4\,d^4}{2\,x^2}-\frac {b\,c^2\,d^4}{5\,x^4}-\frac {5\,b\,c^3\,d^4}{9\,x^3}-\frac {16\,b\,c^4\,d^4}{15\,x^2}-\frac {13\,b\,c^5\,d^4}{6\,x}-\frac {a\,d^4}{6\,x^6}-\frac {4\,a\,c\,d^4}{5\,x^5}-\frac {b\,c\,d^4}{30\,x^5}-\frac {b\,d^4\,\mathrm {atanh}\left (c\,x\right )}{6\,x^6}-\frac {13\,b\,c^7\,d^4\,\mathrm {atan}\left (\frac {c^2\,x}{\sqrt {-c^2}}\right )}{6\,\sqrt {-c^2}}-\frac {4\,b\,c\,d^4\,\mathrm {atanh}\left (c\,x\right )}{5\,x^5}-\frac {3\,b\,c^2\,d^4\,\mathrm {atanh}\left (c\,x\right )}{2\,x^4}-\frac {4\,b\,c^3\,d^4\,\mathrm {atanh}\left (c\,x\right )}{3\,x^3}-\frac {b\,c^4\,d^4\,\mathrm {atanh}\left (c\,x\right )}{2\,x^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(32*b*c^6*d^4*log(x))/15 - (16*b*c^6*d^4*log(c^2*x^2 - 1))/15 - (3*a*c^2*d^4)/(2*x^4) - (4*a*c^3*d^4)/(3*x^3)
- (a*c^4*d^4)/(2*x^2) - (b*c^2*d^4)/(5*x^4) - (5*b*c^3*d^4)/(9*x^3) - (16*b*c^4*d^4)/(15*x^2) - (13*b*c^5*d^4)
/(6*x) - (a*d^4)/(6*x^6) - (4*a*c*d^4)/(5*x^5) - (b*c*d^4)/(30*x^5) - (b*d^4*atanh(c*x))/(6*x^6) - (13*b*c^7*d
^4*atan((c^2*x)/(-c^2)^(1/2)))/(6*(-c^2)^(1/2)) - (4*b*c*d^4*atanh(c*x))/(5*x^5) - (3*b*c^2*d^4*atanh(c*x))/(2
*x^4) - (4*b*c^3*d^4*atanh(c*x))/(3*x^3) - (b*c^4*d^4*atanh(c*x))/(2*x^2)

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sympy [A]  time = 3.36, size = 291, normalized size = 1.93 \[ \begin {cases} - \frac {a c^{4} d^{4}}{2 x^{2}} - \frac {4 a c^{3} d^{4}}{3 x^{3}} - \frac {3 a c^{2} d^{4}}{2 x^{4}} - \frac {4 a c d^{4}}{5 x^{5}} - \frac {a d^{4}}{6 x^{6}} + \frac {32 b c^{6} d^{4} \log {\relax (x )}}{15} - \frac {32 b c^{6} d^{4} \log {\left (x - \frac {1}{c} \right )}}{15} + \frac {b c^{6} d^{4} \operatorname {atanh}{\left (c x \right )}}{30} - \frac {13 b c^{5} d^{4}}{6 x} - \frac {b c^{4} d^{4} \operatorname {atanh}{\left (c x \right )}}{2 x^{2}} - \frac {16 b c^{4} d^{4}}{15 x^{2}} - \frac {4 b c^{3} d^{4} \operatorname {atanh}{\left (c x \right )}}{3 x^{3}} - \frac {5 b c^{3} d^{4}}{9 x^{3}} - \frac {3 b c^{2} d^{4} \operatorname {atanh}{\left (c x \right )}}{2 x^{4}} - \frac {b c^{2} d^{4}}{5 x^{4}} - \frac {4 b c d^{4} \operatorname {atanh}{\left (c x \right )}}{5 x^{5}} - \frac {b c d^{4}}{30 x^{5}} - \frac {b d^{4} \operatorname {atanh}{\left (c x \right )}}{6 x^{6}} & \text {for}\: c \neq 0 \\- \frac {a d^{4}}{6 x^{6}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*d*x+d)**4*(a+b*atanh(c*x))/x**7,x)

[Out]

Piecewise((-a*c**4*d**4/(2*x**2) - 4*a*c**3*d**4/(3*x**3) - 3*a*c**2*d**4/(2*x**4) - 4*a*c*d**4/(5*x**5) - a*d
**4/(6*x**6) + 32*b*c**6*d**4*log(x)/15 - 32*b*c**6*d**4*log(x - 1/c)/15 + b*c**6*d**4*atanh(c*x)/30 - 13*b*c*
*5*d**4/(6*x) - b*c**4*d**4*atanh(c*x)/(2*x**2) - 16*b*c**4*d**4/(15*x**2) - 4*b*c**3*d**4*atanh(c*x)/(3*x**3)
- 5*b*c**3*d**4/(9*x**3) - 3*b*c**2*d**4*atanh(c*x)/(2*x**4) - b*c**2*d**4/(5*x**4) - 4*b*c*d**4*atanh(c*x)/(
5*x**5) - b*c*d**4/(30*x**5) - b*d**4*atanh(c*x)/(6*x**6), Ne(c, 0)), (-a*d**4/(6*x**6), True))

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