3.1.28 \(\int \frac {a+b \tanh ^{-1}(c x^2)}{(d+e x)^3} \, dx\) [28]
Optimal. Leaf size=226 \[ -\frac {b c d e}{\left (c^2 d^4-e^4\right ) (d+e x)}+\frac {b c^{3/2} d \text {ArcTan}\left (\sqrt {c} x\right )}{\left (c d^2+e^2\right )^2}-\frac {b c^{3/2} d \tanh ^{-1}\left (\sqrt {c} x\right )}{\left (c d^2-e^2\right )^2}-\frac {a+b \tanh ^{-1}\left (c x^2\right )}{2 e (d+e x)^2}+\frac {b c e \left (3 c^2 d^4+e^4\right ) \log (d+e x)}{\left (c^2 d^4-e^4\right )^2}-\frac {b c \left (c d^2+e^2\right ) \log \left (1-c x^2\right )}{4 e \left (c d^2-e^2\right )^2}+\frac {b c \left (c d^2-e^2\right ) \log \left (1+c x^2\right )}{4 e \left (c d^2+e^2\right )^2} \]
[Out]
-b*c*d*e/(c^2*d^4-e^4)/(e*x+d)+b*c^(3/2)*d*arctan(x*c^(1/2))/(c*d^2+e^2)^2+1/2*(-a-b*arctanh(c*x^2))/e/(e*x+d)
^2-b*c^(3/2)*d*arctanh(x*c^(1/2))/(c*d^2-e^2)^2+b*c*e*(3*c^2*d^4+e^4)*ln(e*x+d)/(c^2*d^4-e^4)^2-1/4*b*c*(c*d^2
+e^2)*ln(-c*x^2+1)/e/(c*d^2-e^2)^2+1/4*b*c*(c*d^2-e^2)*ln(c*x^2+1)/e/(c*d^2+e^2)^2
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Rubi [A]
time = 0.26, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6071, 6857,
649, 212, 266, 209} \begin {gather*} -\frac {a+b \tanh ^{-1}\left (c x^2\right )}{2 e (d+e x)^2}+\frac {b c^{3/2} d \text {ArcTan}\left (\sqrt {c} x\right )}{\left (c d^2+e^2\right )^2}-\frac {b c^{3/2} d \tanh ^{-1}\left (\sqrt {c} x\right )}{\left (c d^2-e^2\right )^2}-\frac {b c d e}{\left (c^2 d^4-e^4\right ) (d+e x)}+\frac {b c e \left (3 c^2 d^4+e^4\right ) \log (d+e x)}{\left (c^2 d^4-e^4\right )^2}-\frac {b c \left (c d^2+e^2\right ) \log \left (1-c x^2\right )}{4 e \left (c d^2-e^2\right )^2}+\frac {b c \left (c d^2-e^2\right ) \log \left (c x^2+1\right )}{4 e \left (c d^2+e^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*ArcTanh[c*x^2])/(d + e*x)^3,x]
[Out]
-((b*c*d*e)/((c^2*d^4 - e^4)*(d + e*x))) + (b*c^(3/2)*d*ArcTan[Sqrt[c]*x])/(c*d^2 + e^2)^2 - (b*c^(3/2)*d*ArcT
anh[Sqrt[c]*x])/(c*d^2 - e^2)^2 - (a + b*ArcTanh[c*x^2])/(2*e*(d + e*x)^2) + (b*c*e*(3*c^2*d^4 + e^4)*Log[d +
e*x])/(c^2*d^4 - e^4)^2 - (b*c*(c*d^2 + e^2)*Log[1 - c*x^2])/(4*e*(c*d^2 - e^2)^2) + (b*c*(c*d^2 - e^2)*Log[1
+ c*x^2])/(4*e*(c*d^2 + e^2)^2)
Rule 209
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])
Rule 212
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 266
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 649
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[(-a)*c]
Rule 6071
Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(
(a + b*ArcTanh[c*x^n])/(e*(m + 1))), x] - Dist[b*c*(n/(e*(m + 1))), Int[x^(n - 1)*((d + e*x)^(m + 1)/(1 - c^2*
x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[m, -1]
Rule 6857
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]
Rubi steps
\begin {align*} \int \frac {a+b \tanh ^{-1}\left (c x^2\right )}{(d+e x)^3} \, dx &=\int \left (\frac {a}{(d+e x)^3}+\frac {b \tanh ^{-1}\left (c x^2\right )}{(d+e x)^3}\right ) \, dx\\ &=-\frac {a}{2 e (d+e x)^2}+b \int \frac {\tanh ^{-1}\left (c x^2\right )}{(d+e x)^3} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.46, size = 379, normalized size = 1.68 \begin {gather*} \frac {1}{4} \left (-\frac {2 a}{e (d+e x)^2}-\frac {4 b c d e}{\left (c^2 d^4-e^4\right ) (d+e x)}+\frac {4 b c^{3/2} d \text {ArcTan}\left (\sqrt {c} x\right )}{\left (c d^2+e^2\right )^2}-\frac {2 b \tanh ^{-1}\left (c x^2\right )}{e (d+e x)^2}-\frac {b c^{3/2} d \left (c^{5/2} d^5-2 c^2 d^4 e-4 c d^2 e^3+3 \sqrt {c} d e^4-2 e^5\right ) \log \left (1-\sqrt {c} x\right )}{e \left (-c^2 d^4+e^4\right )^2}-\frac {b c^{3/2} d \left (c^{5/2} d^5+2 c^2 d^4 e+4 c d^2 e^3+3 \sqrt {c} d e^4+2 e^5\right ) \log \left (1+\sqrt {c} x\right )}{e \left (-c^2 d^4+e^4\right )^2}+\frac {4 b c e \left (3 c^2 d^4+e^4\right ) \log (d+e x)}{\left (-c^2 d^4+e^4\right )^2}+\frac {b c^2 \left (c^2 d^6+3 d^2 e^4\right ) \log \left (1+c x^2\right )}{e \left (-c^2 d^4+e^4\right )^2}-\frac {b c e \left (3 c^2 d^4+e^4\right ) \log \left (1-c^2 x^4\right )}{\left (-c^2 d^4+e^4\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*ArcTanh[c*x^2])/(d + e*x)^3,x]
[Out]
((-2*a)/(e*(d + e*x)^2) - (4*b*c*d*e)/((c^2*d^4 - e^4)*(d + e*x)) + (4*b*c^(3/2)*d*ArcTan[Sqrt[c]*x])/(c*d^2 +
e^2)^2 - (2*b*ArcTanh[c*x^2])/(e*(d + e*x)^2) - (b*c^(3/2)*d*(c^(5/2)*d^5 - 2*c^2*d^4*e - 4*c*d^2*e^3 + 3*Sqr
t[c]*d*e^4 - 2*e^5)*Log[1 - Sqrt[c]*x])/(e*(-(c^2*d^4) + e^4)^2) - (b*c^(3/2)*d*(c^(5/2)*d^5 + 2*c^2*d^4*e + 4
*c*d^2*e^3 + 3*Sqrt[c]*d*e^4 + 2*e^5)*Log[1 + Sqrt[c]*x])/(e*(-(c^2*d^4) + e^4)^2) + (4*b*c*e*(3*c^2*d^4 + e^4
)*Log[d + e*x])/(-(c^2*d^4) + e^4)^2 + (b*c^2*(c^2*d^6 + 3*d^2*e^4)*Log[1 + c*x^2])/(e*(-(c^2*d^4) + e^4)^2) -
(b*c*e*(3*c^2*d^4 + e^4)*Log[1 - c^2*x^4])/(-(c^2*d^4) + e^4)^2)/4
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Maple [A]
time = 0.34, size = 310, normalized size = 1.37
| | |
| method |
result |
size |
| | |
| default |
\(-\frac {a}{2 \left (e x +d \right )^{2} e}-\frac {b \arctanh \left (c \,x^{2}\right )}{2 \left (e x +d \right )^{2} e}-\frac {b \,c^{2} \ln \left (c \,x^{2}-1\right ) d^{2}}{4 e \left (c \,d^{2}-e^{2}\right )^{2}}-\frac {b e c \ln \left (c \,x^{2}-1\right )}{4 \left (c \,d^{2}-e^{2}\right )^{2}}-\frac {b \,c^{\frac {3}{2}} d \arctanh \left (x \sqrt {c}\right )}{\left (c \,d^{2}-e^{2}\right )^{2}}+\frac {b \,c^{2} \ln \left (c \,x^{2}+1\right ) d^{2}}{4 e \left (c \,d^{2}+e^{2}\right )^{2}}-\frac {b e c \ln \left (c \,x^{2}+1\right )}{4 \left (c \,d^{2}+e^{2}\right )^{2}}+\frac {b \,c^{\frac {3}{2}} d \arctan \left (x \sqrt {c}\right )}{\left (c \,d^{2}+e^{2}\right )^{2}}-\frac {b e c d}{\left (c \,d^{2}-e^{2}\right ) \left (c \,d^{2}+e^{2}\right ) \left (e x +d \right )}+\frac {3 b e \,c^{3} \ln \left (e x +d \right ) d^{4}}{\left (c \,d^{2}-e^{2}\right )^{2} \left (c \,d^{2}+e^{2}\right )^{2}}+\frac {b \,e^{5} c \ln \left (e x +d \right )}{\left (c \,d^{2}-e^{2}\right )^{2} \left (c \,d^{2}+e^{2}\right )^{2}}\) |
\(310\) |
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Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a+b*arctanh(c*x^2))/(e*x+d)^3,x,method=_RETURNVERBOSE)
[Out]
-1/2*a/(e*x+d)^2/e-1/2*b/(e*x+d)^2/e*arctanh(c*x^2)-1/4*b/e*c^2/(c*d^2-e^2)^2*ln(c*x^2-1)*d^2-1/4*b*e*c/(c*d^2
-e^2)^2*ln(c*x^2-1)-b*c^(3/2)*d*arctanh(x*c^(1/2))/(c*d^2-e^2)^2+1/4*b/e*c^2/(c*d^2+e^2)^2*ln(c*x^2+1)*d^2-1/4
*b*e*c/(c*d^2+e^2)^2*ln(c*x^2+1)+b*c^(3/2)*d*arctan(x*c^(1/2))/(c*d^2+e^2)^2-b*e*c*d/(c*d^2-e^2)/(c*d^2+e^2)/(
e*x+d)+3*b*e*c^3/(c*d^2-e^2)^2/(c*d^2+e^2)^2*ln(e*x+d)*d^4+b*e^5*c/(c*d^2-e^2)^2/(c*d^2+e^2)^2*ln(e*x+d)
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Maxima [A]
time = 0.48, size = 300, normalized size = 1.33 \begin {gather*} \frac {1}{4} \, {\left ({\left (\frac {4 \, \sqrt {c} d \arctan \left (\sqrt {c} x\right )}{c^{2} d^{4} + 2 \, c d^{2} e^{2} + e^{4}} + \frac {2 \, \sqrt {c} d \log \left (\frac {c x - \sqrt {c}}{c x + \sqrt {c}}\right )}{c^{2} d^{4} - 2 \, c d^{2} e^{2} + e^{4}} - \frac {4 \, d e}{c^{2} d^{5} + {\left (c^{2} d^{4} e - e^{5}\right )} x - d e^{4}} + \frac {{\left (c d^{2} - e^{2}\right )} \log \left (c x^{2} + 1\right )}{c^{2} d^{4} e + 2 \, c d^{2} e^{3} + e^{5}} - \frac {{\left (c d^{2} + e^{2}\right )} \log \left (c x^{2} - 1\right )}{c^{2} d^{4} e - 2 \, c d^{2} e^{3} + e^{5}} + \frac {4 \, {\left (3 \, c^{2} d^{4} e + e^{5}\right )} \log \left (x e + d\right )}{c^{4} d^{8} - 2 \, c^{2} d^{4} e^{4} + e^{8}}\right )} c - \frac {2 \, \operatorname {artanh}\left (c x^{2}\right )}{x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e}\right )} b - \frac {a}{2 \, {\left (x^{2} e^{3} + 2 \, d x e^{2} + d^{2} e\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^2))/(e*x+d)^3,x, algorithm="maxima")
[Out]
1/4*((4*sqrt(c)*d*arctan(sqrt(c)*x)/(c^2*d^4 + 2*c*d^2*e^2 + e^4) + 2*sqrt(c)*d*log((c*x - sqrt(c))/(c*x + sqr
t(c)))/(c^2*d^4 - 2*c*d^2*e^2 + e^4) - 4*d*e/(c^2*d^5 + (c^2*d^4*e - e^5)*x - d*e^4) + (c*d^2 - e^2)*log(c*x^2
+ 1)/(c^2*d^4*e + 2*c*d^2*e^3 + e^5) - (c*d^2 + e^2)*log(c*x^2 - 1)/(c^2*d^4*e - 2*c*d^2*e^3 + e^5) + 4*(3*c^
2*d^4*e + e^5)*log(x*e + d)/(c^4*d^8 - 2*c^2*d^4*e^4 + e^8))*c - 2*arctanh(c*x^2)/(x^2*e^3 + 2*d*x*e^2 + d^2*e
))*b - 1/2*a/(x^2*e^3 + 2*d*x*e^2 + d^2*e)
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 4393 vs.
\(2 (205) = 410\).
time = 19.41, size = 8795, normalized size = 38.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^2))/(e*x+d)^3,x, algorithm="fricas")
[Out]
[-1/4*(2*a*c^4*d^8 + 4*b*c^3*d^5*x*cosh(1)^3 + 4*b*c^3*d^6*cosh(1)^2 - 4*a*c^2*d^4*cosh(1)^4 - 4*b*c*d*x*cosh(
1)^7 - 4*b*c*d^2*cosh(1)^6 + 2*a*cosh(1)^8 + 2*a*sinh(1)^8 - 4*(b*c*d*x - 4*a*cosh(1))*sinh(1)^7 - 4*(7*b*c*d*
x*cosh(1) + b*c*d^2 - 14*a*cosh(1)^2)*sinh(1)^6 - 4*(21*b*c*d*x*cosh(1)^2 + 6*b*c*d^2*cosh(1) - 28*a*cosh(1)^3
)*sinh(1)^5 - 4*(a*c^2*d^4 + 35*b*c*d*x*cosh(1)^3 + 15*b*c*d^2*cosh(1)^2 - 35*a*cosh(1)^4)*sinh(1)^4 + 4*(b*c^
3*d^5*x - 4*a*c^2*d^4*cosh(1) - 35*b*c*d*x*cosh(1)^4 - 20*b*c*d^2*cosh(1)^3 + 28*a*cosh(1)^5)*sinh(1)^3 + 4*(3
*b*c^3*d^5*x*cosh(1) + b*c^3*d^6 - 6*a*c^2*d^4*cosh(1)^2 - 21*b*c*d*x*cosh(1)^5 - 15*b*c*d^2*cosh(1)^4 + 14*a*
cosh(1)^6)*sinh(1)^2 - 4*(2*b*c^3*d^6*x*cosh(1)^2 + b*c^3*d^7*cosh(1) + 4*b*c^2*d^4*x*cosh(1)^4 + b*c*d*x^2*co
sh(1)^7 + b*c*d*x^2*sinh(1)^7 + 2*b*c*d^2*x*cosh(1)^6 + (7*b*c*d*x^2*cosh(1) + 2*b*c*d^2*x)*sinh(1)^6 + (2*b*c
^2*d^3*x^2 + b*c*d^3)*cosh(1)^5 + (2*b*c^2*d^3*x^2 + 21*b*c*d*x^2*cosh(1)^2 + 12*b*c*d^2*x*cosh(1) + b*c*d^3)*
sinh(1)^5 + (4*b*c^2*d^4*x + 35*b*c*d*x^2*cosh(1)^3 + 30*b*c*d^2*x*cosh(1)^2 + 5*(2*b*c^2*d^3*x^2 + b*c*d^3)*c
osh(1))*sinh(1)^4 + (b*c^3*d^5*x^2 + 2*b*c^2*d^5)*cosh(1)^3 + (b*c^3*d^5*x^2 + 16*b*c^2*d^4*x*cosh(1) + 35*b*c
*d*x^2*cosh(1)^4 + 2*b*c^2*d^5 + 40*b*c*d^2*x*cosh(1)^3 + 10*(2*b*c^2*d^3*x^2 + b*c*d^3)*cosh(1)^2)*sinh(1)^3
+ (2*b*c^3*d^6*x + 24*b*c^2*d^4*x*cosh(1)^2 + 21*b*c*d*x^2*cosh(1)^5 + 30*b*c*d^2*x*cosh(1)^4 + 10*(2*b*c^2*d^
3*x^2 + b*c*d^3)*cosh(1)^3 + 3*(b*c^3*d^5*x^2 + 2*b*c^2*d^5)*cosh(1))*sinh(1)^2 + (4*b*c^3*d^6*x*cosh(1) + b*c
^3*d^7 + 16*b*c^2*d^4*x*cosh(1)^3 + 7*b*c*d*x^2*cosh(1)^6 + 12*b*c*d^2*x*cosh(1)^5 + 5*(2*b*c^2*d^3*x^2 + b*c*
d^3)*cosh(1)^4 + 3*(b*c^3*d^5*x^2 + 2*b*c^2*d^5)*cosh(1)^2)*sinh(1))*sqrt(-c)*arctan(sqrt(-c)*x) - 2*(2*b*c^3*
d^6*x*cosh(1)^2 + b*c^3*d^7*cosh(1) - 4*b*c^2*d^4*x*cosh(1)^4 + b*c*d*x^2*cosh(1)^7 + b*c*d*x^2*sinh(1)^7 + 2*
b*c*d^2*x*cosh(1)^6 + (7*b*c*d*x^2*cosh(1) + 2*b*c*d^2*x)*sinh(1)^6 - (2*b*c^2*d^3*x^2 - b*c*d^3)*cosh(1)^5 -
(2*b*c^2*d^3*x^2 - 21*b*c*d*x^2*cosh(1)^2 - 12*b*c*d^2*x*cosh(1) - b*c*d^3)*sinh(1)^5 - (4*b*c^2*d^4*x - 35*b*
c*d*x^2*cosh(1)^3 - 30*b*c*d^2*x*cosh(1)^2 + 5*(2*b*c^2*d^3*x^2 - b*c*d^3)*cosh(1))*sinh(1)^4 + (b*c^3*d^5*x^2
- 2*b*c^2*d^5)*cosh(1)^3 + (b*c^3*d^5*x^2 - 16*b*c^2*d^4*x*cosh(1) + 35*b*c*d*x^2*cosh(1)^4 - 2*b*c^2*d^5 + 4
0*b*c*d^2*x*cosh(1)^3 - 10*(2*b*c^2*d^3*x^2 - b*c*d^3)*cosh(1)^2)*sinh(1)^3 + (2*b*c^3*d^6*x - 24*b*c^2*d^4*x*
cosh(1)^2 + 21*b*c*d*x^2*cosh(1)^5 + 30*b*c*d^2*x*cosh(1)^4 - 10*(2*b*c^2*d^3*x^2 - b*c*d^3)*cosh(1)^3 + 3*(b*
c^3*d^5*x^2 - 2*b*c^2*d^5)*cosh(1))*sinh(1)^2 + (4*b*c^3*d^6*x*cosh(1) + b*c^3*d^7 - 16*b*c^2*d^4*x*cosh(1)^3
+ 7*b*c*d*x^2*cosh(1)^6 + 12*b*c*d^2*x*cosh(1)^5 - 5*(2*b*c^2*d^3*x^2 - b*c*d^3)*cosh(1)^4 + 3*(b*c^3*d^5*x^2
- 2*b*c^2*d^5)*cosh(1)^2)*sinh(1))*sqrt(-c)*log((c*x^2 + 2*sqrt(-c)*x - 1)/(c*x^2 + 1)) - (2*b*c^4*d^7*x*cosh(
1) + b*c^4*d^8 - 6*b*c^3*d^5*x*cosh(1)^3 + 6*b*c^2*d^3*x*cosh(1)^5 - b*c*x^2*cosh(1)^8 - b*c*x^2*sinh(1)^8 - 2
*b*c*d*x*cosh(1)^7 - 2*(4*b*c*x^2*cosh(1) + b*c*d*x)*sinh(1)^7 + (3*b*c^2*d^2*x^2 - b*c*d^2)*cosh(1)^6 + (3*b*
c^2*d^2*x^2 - 28*b*c*x^2*cosh(1)^2 - 14*b*c*d*x*cosh(1) - b*c*d^2)*sinh(1)^6 + 2*(3*b*c^2*d^3*x - 28*b*c*x^2*c
osh(1)^3 - 21*b*c*d*x*cosh(1)^2 + 3*(3*b*c^2*d^2*x^2 - b*c*d^2)*cosh(1))*sinh(1)^5 - 3*(b*c^3*d^4*x^2 - b*c^2*
d^4)*cosh(1)^4 - (3*b*c^3*d^4*x^2 - 30*b*c^2*d^3*x*cosh(1) + 70*b*c*x^2*cosh(1)^4 - 3*b*c^2*d^4 + 70*b*c*d*x*c
osh(1)^3 - 15*(3*b*c^2*d^2*x^2 - b*c*d^2)*cosh(1)^2)*sinh(1)^4 - 2*(3*b*c^3*d^5*x - 30*b*c^2*d^3*x*cosh(1)^2 +
28*b*c*x^2*cosh(1)^5 + 35*b*c*d*x*cosh(1)^4 - 10*(3*b*c^2*d^2*x^2 - b*c*d^2)*cosh(1)^3 + 6*(b*c^3*d^4*x^2 - b
*c^2*d^4)*cosh(1))*sinh(1)^3 + (b*c^4*d^6*x^2 - 3*b*c^3*d^6)*cosh(1)^2 + (b*c^4*d^6*x^2 - 18*b*c^3*d^5*x*cosh(
1) - 3*b*c^3*d^6 + 60*b*c^2*d^3*x*cosh(1)^3 - 28*b*c*x^2*cosh(1)^6 - 42*b*c*d*x*cosh(1)^5 + 15*(3*b*c^2*d^2*x^
2 - b*c*d^2)*cosh(1)^4 - 18*(b*c^3*d^4*x^2 - b*c^2*d^4)*cosh(1)^2)*sinh(1)^2 + 2*(b*c^4*d^7*x - 9*b*c^3*d^5*x*
cosh(1)^2 + 15*b*c^2*d^3*x*cosh(1)^4 - 4*b*c*x^2*cosh(1)^7 - 7*b*c*d*x*cosh(1)^6 + 3*(3*b*c^2*d^2*x^2 - b*c*d^
2)*cosh(1)^5 - 6*(b*c^3*d^4*x^2 - b*c^2*d^4)*cosh(1)^3 + (b*c^4*d^6*x^2 - 3*b*c^3*d^6)*cosh(1))*sinh(1))*log(c
*x^2 + 1) + (2*b*c^4*d^7*x*cosh(1) + b*c^4*d^8 + 6*b*c^3*d^5*x*cosh(1)^3 + 6*b*c^2*d^3*x*cosh(1)^5 + b*c*x^2*c
osh(1)^8 + b*c*x^2*sinh(1)^8 + 2*b*c*d*x*cosh(1)^7 + 2*(4*b*c*x^2*cosh(1) + b*c*d*x)*sinh(1)^7 + (3*b*c^2*d^2*
x^2 + b*c*d^2)*cosh(1)^6 + (3*b*c^2*d^2*x^2 + 28*b*c*x^2*cosh(1)^2 + 14*b*c*d*x*cosh(1) + b*c*d^2)*sinh(1)^6 +
2*(3*b*c^2*d^3*x + 28*b*c*x^2*cosh(1)^3 + 21*b*c*d*x*cosh(1)^2 + 3*(3*b*c^2*d^2*x^2 + b*c*d^2)*cosh(1))*sinh(
1)^5 + 3*(b*c^3*d^4*x^2 + b*c^2*d^4)*cosh(1)^4 + (3*b*c^3*d^4*x^2 + 30*b*c^2*d^3*x*cosh(1) + 70*b*c*x^2*cosh(1
)^4 + 3*b*c^2*d^4 + 70*b*c*d*x*cosh(1)^3 + 15*(3*b*c^2*d^2*x^2 + b*c*d^2)*cosh(1)^2)*sinh(1)^4 + 2*(3*b*c^3*d^
5*x + 30*b*c^2*d^3*x*cosh(1)^2 + 28*b*c*x^2*cosh(1)^5 + 35*b*c*d*x*cosh(1)^4 + 10*(3*b*c^2*d^2*x^2 + b*c*d^2)*
cosh(1)^3 + 6*(b*c^3*d^4*x^2 + b*c^2*d^4)*cosh(...
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*atanh(c*x**2))/(e*x+d)**3,x)
[Out]
Timed out
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Giac [A]
time = 22.47, size = 362, normalized size = 1.60 \begin {gather*} \frac {b c^{\frac {3}{2}} d \arctan \left (\sqrt {c} x\right )}{c^{2} d^{4} + 2 \, c d^{2} e^{2} + e^{4}} + \frac {b c^{2} d \arctan \left (\frac {c x}{\sqrt {-c}}\right )}{{\left (c^{2} d^{4} - 2 \, c d^{2} e^{2} + e^{4}\right )} \sqrt {-c}} + \frac {{\left (b c^{2} d^{2} - b c e^{2}\right )} \log \left (c x^{2} + 1\right )}{4 \, {\left (c^{2} d^{4} e + 2 \, c d^{2} e^{3} + e^{5}\right )}} - \frac {{\left (b c^{2} d^{2} + b c e^{2}\right )} \log \left (-c x^{2} + 1\right )}{4 \, {\left (c^{2} d^{4} e - 2 \, c d^{2} e^{3} + e^{5}\right )}} + \frac {{\left (3 \, b c^{3} d^{4} e + b c e^{5}\right )} \log \left (e x + d\right )}{c^{4} d^{8} - 2 \, c^{2} d^{4} e^{4} + e^{8}} - \frac {b \log \left (-\frac {c x^{2} + 1}{c x^{2} - 1}\right )}{4 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} - \frac {a c^{2} d^{4} + 2 \, b c d e^{3} x + 2 \, b c d^{2} e^{2} - a e^{4}}{2 \, {\left (c^{2} d^{4} e^{3} x^{2} + 2 \, c^{2} d^{5} e^{2} x + c^{2} d^{6} e - e^{7} x^{2} - 2 \, d e^{6} x - d^{2} e^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((a+b*arctanh(c*x^2))/(e*x+d)^3,x, algorithm="giac")
[Out]
b*c^(3/2)*d*arctan(sqrt(c)*x)/(c^2*d^4 + 2*c*d^2*e^2 + e^4) + b*c^2*d*arctan(c*x/sqrt(-c))/((c^2*d^4 - 2*c*d^2
*e^2 + e^4)*sqrt(-c)) + 1/4*(b*c^2*d^2 - b*c*e^2)*log(c*x^2 + 1)/(c^2*d^4*e + 2*c*d^2*e^3 + e^5) - 1/4*(b*c^2*
d^2 + b*c*e^2)*log(-c*x^2 + 1)/(c^2*d^4*e - 2*c*d^2*e^3 + e^5) + (3*b*c^3*d^4*e + b*c*e^5)*log(e*x + d)/(c^4*d
^8 - 2*c^2*d^4*e^4 + e^8) - 1/4*b*log(-(c*x^2 + 1)/(c*x^2 - 1))/(e^3*x^2 + 2*d*e^2*x + d^2*e) - 1/2*(a*c^2*d^4
+ 2*b*c*d*e^3*x + 2*b*c*d^2*e^2 - a*e^4)/(c^2*d^4*e^3*x^2 + 2*c^2*d^5*e^2*x + c^2*d^6*e - e^7*x^2 - 2*d*e^6*x
- d^2*e^5)
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Mupad [B]
time = 5.28, size = 2016, normalized size = 8.92 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*atanh(c*x^2))/(d + e*x)^3,x)
[Out]
((a*c^2*d^4 - a*e^4 + 2*b*c*d^2*e^2)/(2*(e^4 - c^2*d^4)) + (b*c*d*e^3*x)/(e^4 - c^2*d^4))/(d^2*e + e^3*x^2 + 2
*d*e^2*x) + (log(c*d^28*(-c^3)^(13/2) + c^21*d^28*x + 100*c^7*e^28*x - 100*c^2*e^28*(-c^3)^(3/2) + 496*d^2*e^2
6*(-c^3)^(5/2) - 7398*d^8*e^20*(-c^3)^(7/2) + 17176*d^14*e^14*(-c^3)^(9/2) - 2703*d^20*e^8*(-c^3)^(11/2) - 20*
d^26*e^2*(-c^3)^(13/2) + 496*c^8*d^2*e^26*x + 1473*c^9*d^4*e^24*x + 3692*c^10*d^6*e^22*x + 7398*c^11*d^8*e^20*
x + 11868*c^12*d^10*e^18*x + 16015*c^13*d^12*e^16*x + 17176*c^14*d^14*e^14*x + 13192*c^15*d^16*e^12*x + 6984*c
^16*d^18*e^10*x + 2703*c^17*d^20*e^8*x + 764*c^18*d^22*e^6*x + 78*c^19*d^24*e^4*x - 20*c^20*d^26*e^2*x + 3692*
c^2*d^6*e^22*(-c^3)^(5/2) - 16015*c^2*d^12*e^16*(-c^3)^(7/2) + 6984*c^2*d^18*e^10*(-c^3)^(9/2) - 78*c^2*d^24*e
^4*(-c^3)^(11/2) + 1473*c*d^4*e^24*(-c^3)^(5/2) - 11868*c*d^10*e^18*(-c^3)^(7/2) + 13192*c*d^16*e^12*(-c^3)^(9
/2) - 764*c*d^22*e^6*(-c^3)^(11/2))*(b*c^2*d^2 - b*c*e^2 + 2*b*d*e*(-c^3)^(1/2)))/(4*(e^5 + 2*c*d^2*e^3 + c^2*
d^4*e)) - (log(c^21*d^28*x - c*d^28*(-c^3)^(13/2) + 100*c^7*e^28*x + 100*c^2*e^28*(-c^3)^(3/2) - 496*d^2*e^26*
(-c^3)^(5/2) + 7398*d^8*e^20*(-c^3)^(7/2) - 17176*d^14*e^14*(-c^3)^(9/2) + 2703*d^20*e^8*(-c^3)^(11/2) + 20*d^
26*e^2*(-c^3)^(13/2) + 496*c^8*d^2*e^26*x + 1473*c^9*d^4*e^24*x + 3692*c^10*d^6*e^22*x + 7398*c^11*d^8*e^20*x
+ 11868*c^12*d^10*e^18*x + 16015*c^13*d^12*e^16*x + 17176*c^14*d^14*e^14*x + 13192*c^15*d^16*e^12*x + 6984*c^1
6*d^18*e^10*x + 2703*c^17*d^20*e^8*x + 764*c^18*d^22*e^6*x + 78*c^19*d^24*e^4*x - 20*c^20*d^26*e^2*x - 3692*c^
2*d^6*e^22*(-c^3)^(5/2) + 16015*c^2*d^12*e^16*(-c^3)^(7/2) - 6984*c^2*d^18*e^10*(-c^3)^(9/2) + 78*c^2*d^24*e^4
*(-c^3)^(11/2) - 1473*c*d^4*e^24*(-c^3)^(5/2) + 11868*c*d^10*e^18*(-c^3)^(7/2) - 13192*c*d^16*e^12*(-c^3)^(9/2
) + 764*c*d^22*e^6*(-c^3)^(11/2))*(b*c*e^2 - b*c^2*d^2 + 2*b*d*e*(-c^3)^(1/2)))/(4*(e^5 + 2*c*d^2*e^3 + c^2*d^
4*e)) - (log(100*e^28*(c^3)^(7/2) + c^2*d^28*(c^3)^(15/2) - 3692*d^6*e^22*(c^3)^(9/2) + 16015*d^12*e^16*(c^3)^
(11/2) - 6984*d^18*e^10*(c^3)^(13/2) + 78*d^24*e^4*(c^3)^(15/2) + c^25*d^28*x + 100*c^11*e^28*x - 496*c^12*d^2
*e^26*x + 1473*c^13*d^4*e^24*x - 3692*c^14*d^6*e^22*x + 7398*c^15*d^8*e^20*x - 11868*c^16*d^10*e^18*x + 16015*
c^17*d^12*e^16*x - 17176*c^18*d^14*e^14*x + 13192*c^19*d^16*e^12*x - 6984*c^20*d^18*e^10*x + 2703*c^21*d^20*e^
8*x - 764*c^22*d^22*e^6*x + 78*c^23*d^24*e^4*x + 20*c^24*d^26*e^2*x - 496*c*d^2*e^26*(c^3)^(7/2) + 7398*c*d^8*
e^20*(c^3)^(9/2) - 17176*c*d^14*e^14*(c^3)^(11/2) + 2703*c*d^20*e^8*(c^3)^(13/2) + 20*c*d^26*e^2*(c^3)^(15/2)
+ 1473*c^2*d^4*e^24*(c^3)^(7/2) - 11868*c^2*d^10*e^18*(c^3)^(9/2) + 13192*c^2*d^16*e^12*(c^3)^(11/2) - 764*c^2
*d^22*e^6*(c^3)^(13/2))*(b*c^2*d^2 + b*c*e^2 + 2*b*d*e*(c^3)^(1/2)))/(4*(e^5 - 2*c*d^2*e^3 + c^2*d^4*e)) - (lo
g(100*e^28*(c^3)^(7/2) + c^2*d^28*(c^3)^(15/2) - 3692*d^6*e^22*(c^3)^(9/2) + 16015*d^12*e^16*(c^3)^(11/2) - 69
84*d^18*e^10*(c^3)^(13/2) + 78*d^24*e^4*(c^3)^(15/2) - c^25*d^28*x - 100*c^11*e^28*x + 496*c^12*d^2*e^26*x - 1
473*c^13*d^4*e^24*x + 3692*c^14*d^6*e^22*x - 7398*c^15*d^8*e^20*x + 11868*c^16*d^10*e^18*x - 16015*c^17*d^12*e
^16*x + 17176*c^18*d^14*e^14*x - 13192*c^19*d^16*e^12*x + 6984*c^20*d^18*e^10*x - 2703*c^21*d^20*e^8*x + 764*c
^22*d^22*e^6*x - 78*c^23*d^24*e^4*x - 20*c^24*d^26*e^2*x - 496*c*d^2*e^26*(c^3)^(7/2) + 7398*c*d^8*e^20*(c^3)^
(9/2) - 17176*c*d^14*e^14*(c^3)^(11/2) + 2703*c*d^20*e^8*(c^3)^(13/2) + 20*c*d^26*e^2*(c^3)^(15/2) + 1473*c^2*
d^4*e^24*(c^3)^(7/2) - 11868*c^2*d^10*e^18*(c^3)^(9/2) + 13192*c^2*d^16*e^12*(c^3)^(11/2) - 764*c^2*d^22*e^6*(
c^3)^(13/2))*(b*c^2*d^2 + b*c*e^2 - 2*b*d*e*(c^3)^(1/2)))/(4*(e^5 - 2*c*d^2*e^3 + c^2*d^4*e)) + (log(d + e*x)*
(b*c*e^5 + 3*b*c^3*d^4*e))/(e^8 + c^4*d^8 - 2*c^2*d^4*e^4) - (b*log(c*x^2 + 1))/(4*e*(d^2 + e^2*x^2 + 2*d*e*x)
) + (b*log(1 - c*x^2))/(2*e*(2*d^2 + 2*e^2*x^2 + 4*d*e*x))
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