3.1025 \(\int \frac {(a+b \coth (x))^3 \text {csch}^2(x)}{c+d \coth (x)} \, dx\)

Optimal. Leaf size=78 \[ \frac {(b c-a d)^3 \log (c+d \coth (x))}{d^4}-\frac {b \coth (x) (b c-a d)^2}{d^3}+\frac {(b c-a d) (a+b \coth (x))^2}{2 d^2}-\frac {(a+b \coth (x))^3}{3 d} \]

[Out]

-b*(-a*d+b*c)^2*coth(x)/d^3+1/2*(-a*d+b*c)*(a+b*coth(x))^2/d^2-1/3*(a+b*coth(x))^3/d+(-a*d+b*c)^3*ln(c+d*coth(
x))/d^4

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Rubi [A]  time = 0.15, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {4344, 43} \[ -\frac {b \coth (x) (b c-a d)^2}{d^3}+\frac {(b c-a d) (a+b \coth (x))^2}{2 d^2}+\frac {(b c-a d)^3 \log (c+d \coth (x))}{d^4}-\frac {(a+b \coth (x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*Coth[x])^3*Csch[x]^2)/(c + d*Coth[x]),x]

[Out]

-((b*(b*c - a*d)^2*Coth[x])/d^3) + ((b*c - a*d)*(a + b*Coth[x])^2)/(2*d^2) - (a + b*Coth[x])^3/(3*d) + ((b*c -
a*d)^3*Log[c + d*Coth[x]])/d^4

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 4344

Int[(u_)*(F_)[(c_.)*((a_.) + (b_.)*(x_))]^2, x_Symbol] :> With[{d = FreeFactors[Cot[c*(a + b*x)], x]}, -Dist[d
/(b*c), Subst[Int[SubstFor[1, Cot[c*(a + b*x)]/d, u, x], x], x, Cot[c*(a + b*x)]/d], x] /; FunctionOfQ[Cot[c*(
a + b*x)]/d, u, x, True]] /; FreeQ[{a, b, c}, x] && NonsumQ[u] && (EqQ[F, Csc] || EqQ[F, csc])

Rubi steps

\begin {align*} \int \frac {(a+b \coth (x))^3 \text {csch}^2(x)}{c+d \coth (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {(a+b x)^3}{c+d x} \, dx,x,\coth (x)\right )\\ &=-\operatorname {Subst}\left (\int \left (\frac {b (b c-a d)^2}{d^3}-\frac {b (b c-a d) (a+b x)}{d^2}+\frac {b (a+b x)^2}{d}+\frac {(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx,x,\coth (x)\right )\\ &=-\frac {b (b c-a d)^2 \coth (x)}{d^3}+\frac {(b c-a d) (a+b \coth (x))^2}{2 d^2}-\frac {(a+b \coth (x))^3}{3 d}+\frac {(b c-a d)^3 \log (c+d \coth (x))}{d^4}\\ \end {align*}

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Mathematica [A]  time = 1.28, size = 136, normalized size = 1.74 \[ \frac {(a+b \coth (x))^3 (c \sinh (x)+d \cosh (x)) \left (-b d \left (\sinh (2 x) \left (9 a^2 d^2-9 a b c d+b^2 \left (3 c^2+d^2\right )\right )-3 b d (b c-3 a d)\right )-6 \sinh ^2(x) (b c-a d)^3 (\log (\sinh (x))-\log (c \sinh (x)+d \cosh (x)))-2 b^3 d^3 \coth (x)\right )}{6 d^4 (c+d \coth (x)) (a \sinh (x)+b \cosh (x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*Coth[x])^3*Csch[x]^2)/(c + d*Coth[x]),x]

[Out]

((a + b*Coth[x])^3*(d*Cosh[x] + c*Sinh[x])*(-2*b^3*d^3*Coth[x] - 6*(b*c - a*d)^3*(Log[Sinh[x]] - Log[d*Cosh[x]
+ c*Sinh[x]])*Sinh[x]^2 - b*d*(-3*b*d*(b*c - 3*a*d) + (-9*a*b*c*d + 9*a^2*d^2 + b^2*(3*c^2 + d^2))*Sinh[2*x])
))/(6*d^4*(c + d*Coth[x])*(b*Cosh[x] + a*Sinh[x])^3)

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fricas [B]  time = 0.50, size = 1980, normalized size = 25.38 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x, algorithm="fricas")

[Out]

-1/3*(6*b^3*c^2*d - 18*a*b^2*c*d^2 + 6*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cos
h(x)^4 + 24*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cosh(x)*sinh(x)^3 + 6*(b^3*c^2
*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*sinh(x)^4 + 2*(9*a^2*b + b^3)*d^3 - 6*(2*b^3*c^2*d
- (6*a*b^2 + b^3)*c*d^2 + 3*(2*a^2*b + a*b^2)*d^3)*cosh(x)^2 - 6*(2*b^3*c^2*d - (6*a*b^2 + b^3)*c*d^2 + 3*(2*
a^2*b + a*b^2)*d^3 - 6*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cosh(x)^2)*sinh(x)^
2 - 3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^6 + 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*cosh(x)*sinh(x)^5 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sinh(x)^6 - b^3*c^3 + 3
*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^4 - 3*(
b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh
(x)^2)*sinh(x)^4 + 4*(5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^3 - 3*(b^3*c^3 - 3*a*b^2*c
^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x)^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cos
h(x)^2 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 + 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a
^3*d^3)*cosh(x)^4 - 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^2)*sinh(x)^2 + 6*((b^3*c^3 -
3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^5 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*co
sh(x)^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x))*log(2*(d*cosh(x) + c*sinh(x))/
(cosh(x) - sinh(x))) + 3*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^6 + 6*(b^3*c^3 - 3*a*b^2
*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)*sinh(x)^5 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*sinh
(x)^6 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2 + a^3*d^3 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d
^3)*cosh(x)^4 - 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 - 5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*
d^2 - a^3*d^3)*cosh(x)^2)*sinh(x)^4 + 4*(5*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^3 - 3*(
b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x)^3 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c
*d^2 - a^3*d^3)*cosh(x)^2 + 3*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3 + 5*(b^3*c^3 - 3*a*b^2*c^2*d
+ 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^4 - 6*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^2)*sinh(x
)^2 + 6*((b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x)^5 - 2*(b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*
c*d^2 - a^3*d^3)*cosh(x)^3 + (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*cosh(x))*sinh(x))*log(2*sinh(
x)/(cosh(x) - sinh(x))) + 12*(2*(b^3*c^2*d - (3*a*b^2 + b^3)*c*d^2 + (3*a^2*b + 3*a*b^2 + b^3)*d^3)*cosh(x)^3
- (2*b^3*c^2*d - (6*a*b^2 + b^3)*c*d^2 + 3*(2*a^2*b + a*b^2)*d^3)*cosh(x))*sinh(x))/(d^4*cosh(x)^6 + 6*d^4*cos
h(x)*sinh(x)^5 + d^4*sinh(x)^6 - 3*d^4*cosh(x)^4 + 3*d^4*cosh(x)^2 + 3*(5*d^4*cosh(x)^2 - d^4)*sinh(x)^4 - d^4
+ 4*(5*d^4*cosh(x)^3 - 3*d^4*cosh(x))*sinh(x)^3 + 3*(5*d^4*cosh(x)^4 - 6*d^4*cosh(x)^2 + d^4)*sinh(x)^2 + 6*(
d^4*cosh(x)^5 - 2*d^4*cosh(x)^3 + d^4*cosh(x))*sinh(x))

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giac [B]  time = 0.14, size = 544, normalized size = 6.97 \[ \frac {{\left (b^{3} c^{4} - 3 \, a b^{2} c^{3} d + b^{3} c^{3} d + 3 \, a^{2} b c^{2} d^{2} - 3 \, a b^{2} c^{2} d^{2} - a^{3} c d^{3} + 3 \, a^{2} b c d^{3} - a^{3} d^{4}\right )} \log \left ({\left | c e^{\left (2 \, x\right )} + d e^{\left (2 \, x\right )} - c + d \right |}\right )}{c d^{4} + d^{5}} - \frac {{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \log \left ({\left | e^{\left (2 \, x\right )} - 1 \right |}\right )}{d^{4}} + \frac {11 \, b^{3} c^{3} e^{\left (6 \, x\right )} - 33 \, a b^{2} c^{2} d e^{\left (6 \, x\right )} + 33 \, a^{2} b c d^{2} e^{\left (6 \, x\right )} - 11 \, a^{3} d^{3} e^{\left (6 \, x\right )} - 33 \, b^{3} c^{3} e^{\left (4 \, x\right )} + 99 \, a b^{2} c^{2} d e^{\left (4 \, x\right )} - 12 \, b^{3} c^{2} d e^{\left (4 \, x\right )} - 99 \, a^{2} b c d^{2} e^{\left (4 \, x\right )} + 36 \, a b^{2} c d^{2} e^{\left (4 \, x\right )} + 12 \, b^{3} c d^{2} e^{\left (4 \, x\right )} + 33 \, a^{3} d^{3} e^{\left (4 \, x\right )} - 36 \, a^{2} b d^{3} e^{\left (4 \, x\right )} - 36 \, a b^{2} d^{3} e^{\left (4 \, x\right )} - 12 \, b^{3} d^{3} e^{\left (4 \, x\right )} + 33 \, b^{3} c^{3} e^{\left (2 \, x\right )} - 99 \, a b^{2} c^{2} d e^{\left (2 \, x\right )} + 24 \, b^{3} c^{2} d e^{\left (2 \, x\right )} + 99 \, a^{2} b c d^{2} e^{\left (2 \, x\right )} - 72 \, a b^{2} c d^{2} e^{\left (2 \, x\right )} - 12 \, b^{3} c d^{2} e^{\left (2 \, x\right )} - 33 \, a^{3} d^{3} e^{\left (2 \, x\right )} + 72 \, a^{2} b d^{3} e^{\left (2 \, x\right )} + 36 \, a b^{2} d^{3} e^{\left (2 \, x\right )} - 11 \, b^{3} c^{3} + 33 \, a b^{2} c^{2} d - 12 \, b^{3} c^{2} d - 33 \, a^{2} b c d^{2} + 36 \, a b^{2} c d^{2} + 11 \, a^{3} d^{3} - 36 \, a^{2} b d^{3} - 4 \, b^{3} d^{3}}{6 \, d^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x, algorithm="giac")

[Out]

(b^3*c^4 - 3*a*b^2*c^3*d + b^3*c^3*d + 3*a^2*b*c^2*d^2 - 3*a*b^2*c^2*d^2 - a^3*c*d^3 + 3*a^2*b*c*d^3 - a^3*d^4
)*log(abs(c*e^(2*x) + d*e^(2*x) - c + d))/(c*d^4 + d^5) - (b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*
log(abs(e^(2*x) - 1))/d^4 + 1/6*(11*b^3*c^3*e^(6*x) - 33*a*b^2*c^2*d*e^(6*x) + 33*a^2*b*c*d^2*e^(6*x) - 11*a^3
*d^3*e^(6*x) - 33*b^3*c^3*e^(4*x) + 99*a*b^2*c^2*d*e^(4*x) - 12*b^3*c^2*d*e^(4*x) - 99*a^2*b*c*d^2*e^(4*x) + 3
6*a*b^2*c*d^2*e^(4*x) + 12*b^3*c*d^2*e^(4*x) + 33*a^3*d^3*e^(4*x) - 36*a^2*b*d^3*e^(4*x) - 36*a*b^2*d^3*e^(4*x
) - 12*b^3*d^3*e^(4*x) + 33*b^3*c^3*e^(2*x) - 99*a*b^2*c^2*d*e^(2*x) + 24*b^3*c^2*d*e^(2*x) + 99*a^2*b*c*d^2*e
^(2*x) - 72*a*b^2*c*d^2*e^(2*x) - 12*b^3*c*d^2*e^(2*x) - 33*a^3*d^3*e^(2*x) + 72*a^2*b*d^3*e^(2*x) + 36*a*b^2*
d^3*e^(2*x) - 11*b^3*c^3 + 33*a*b^2*c^2*d - 12*b^3*c^2*d - 33*a^2*b*c*d^2 + 36*a*b^2*c*d^2 + 11*a^3*d^3 - 36*a
^2*b*d^3 - 4*b^3*d^3)/(d^4*(e^(2*x) - 1)^3)

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maple [B]  time = 0.24, size = 378, normalized size = 4.85 \[ -\frac {b^{3} \left (\tanh ^{3}\left (\frac {x}{2}\right )\right )}{24 d}-\frac {3 b^{2} \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) a}{8 d}+\frac {b^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) c}{8 d^{2}}-\frac {3 b \,a^{2} \tanh \left (\frac {x}{2}\right )}{2 d}+\frac {3 b^{2} c a \tanh \left (\frac {x}{2}\right )}{2 d^{2}}-\frac {b^{3} c^{2} \tanh \left (\frac {x}{2}\right )}{2 d^{3}}-\frac {b^{3} \tanh \left (\frac {x}{2}\right )}{8 d}-\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d +2 c \tanh \left (\frac {x}{2}\right )+d \right ) a^{3}}{d}+\frac {3 \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d +2 c \tanh \left (\frac {x}{2}\right )+d \right ) a^{2} b c}{d^{2}}-\frac {3 \ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d +2 c \tanh \left (\frac {x}{2}\right )+d \right ) c^{2} b^{2} a}{d^{3}}+\frac {\ln \left (\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) d +2 c \tanh \left (\frac {x}{2}\right )+d \right ) c^{3} b^{3}}{d^{4}}-\frac {b^{3}}{24 d \tanh \left (\frac {x}{2}\right )^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) a^{3}}{d}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) a^{2} b c}{d^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) c^{2} b^{2} a}{d^{3}}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) c^{3} b^{3}}{d^{4}}-\frac {3 b \,a^{2}}{2 d \tanh \left (\frac {x}{2}\right )}+\frac {3 b^{2} c a}{2 d^{2} \tanh \left (\frac {x}{2}\right )}-\frac {b^{3} c^{2}}{2 d^{3} \tanh \left (\frac {x}{2}\right )}-\frac {b^{3}}{8 d \tanh \left (\frac {x}{2}\right )}-\frac {3 b^{2} a}{8 d \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b^{3} c}{8 d^{2} \tanh \left (\frac {x}{2}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x)

[Out]

-1/24*b^3/d*tanh(1/2*x)^3-3/8*b^2/d*tanh(1/2*x)^2*a+1/8*b^3/d^2*tanh(1/2*x)^2*c-3/2*b/d*a^2*tanh(1/2*x)+3/2*b^
2/d^2*c*a*tanh(1/2*x)-1/2*b^3/d^3*c^2*tanh(1/2*x)-1/8*b^3/d*tanh(1/2*x)-1/d*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)
+d)*a^3+3/d^2*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)+d)*a^2*b*c-3/d^3*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)+d)*c^2*b^
2*a+1/d^4*ln(tanh(1/2*x)^2*d+2*c*tanh(1/2*x)+d)*c^3*b^3-1/24*b^3/d/tanh(1/2*x)^3+1/d*ln(tanh(1/2*x))*a^3-3/d^2
*ln(tanh(1/2*x))*a^2*b*c+3/d^3*ln(tanh(1/2*x))*c^2*b^2*a-1/d^4*ln(tanh(1/2*x))*c^3*b^3-3/2*b/d/tanh(1/2*x)*a^2
+3/2*b^2/d^2/tanh(1/2*x)*c*a-1/2*b^3/d^3/tanh(1/2*x)*c^2-1/8*b^3/d/tanh(1/2*x)-3/8*b^2/d/tanh(1/2*x)^2*a+1/8*b
^3/d^2/tanh(1/2*x)^2*c

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maxima [B]  time = 0.34, size = 316, normalized size = 4.05 \[ \frac {1}{3} \, b^{3} {\left (\frac {2 \, {\left (3 \, c^{2} + d^{2} - 3 \, {\left (2 \, c^{2} + c d\right )} e^{\left (-2 \, x\right )} + 3 \, {\left (c^{2} + c d + d^{2}\right )} e^{\left (-4 \, x\right )}\right )}}{3 \, d^{3} e^{\left (-2 \, x\right )} - 3 \, d^{3} e^{\left (-4 \, x\right )} + d^{3} e^{\left (-6 \, x\right )} - d^{3}} + \frac {3 \, c^{3} \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{4}} - \frac {3 \, c^{3} \log \left (e^{\left (-x\right )} + 1\right )}{d^{4}} - \frac {3 \, c^{3} \log \left (e^{\left (-x\right )} - 1\right )}{d^{4}}\right )} + 3 \, a b^{2} {\left (\frac {2 \, {\left ({\left (c + d\right )} e^{\left (-2 \, x\right )} - c\right )}}{2 \, d^{2} e^{\left (-2 \, x\right )} - d^{2} e^{\left (-4 \, x\right )} - d^{2}} - \frac {c^{2} \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{3}} + \frac {c^{2} \log \left (e^{\left (-x\right )} + 1\right )}{d^{3}} + \frac {c^{2} \log \left (e^{\left (-x\right )} - 1\right )}{d^{3}}\right )} + 3 \, a^{2} b {\left (\frac {c \log \left (-{\left (c - d\right )} e^{\left (-2 \, x\right )} + c + d\right )}{d^{2}} - \frac {c \log \left (e^{\left (-x\right )} + 1\right )}{d^{2}} - \frac {c \log \left (e^{\left (-x\right )} - 1\right )}{d^{2}} + \frac {2}{d e^{\left (-2 \, x\right )} - d}\right )} - \frac {a^{3} \log \left (d \coth \relax (x) + c\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))^3*csch(x)^2/(c+d*coth(x)),x, algorithm="maxima")

[Out]

1/3*b^3*(2*(3*c^2 + d^2 - 3*(2*c^2 + c*d)*e^(-2*x) + 3*(c^2 + c*d + d^2)*e^(-4*x))/(3*d^3*e^(-2*x) - 3*d^3*e^(
-4*x) + d^3*e^(-6*x) - d^3) + 3*c^3*log(-(c - d)*e^(-2*x) + c + d)/d^4 - 3*c^3*log(e^(-x) + 1)/d^4 - 3*c^3*log
(e^(-x) - 1)/d^4) + 3*a*b^2*(2*((c + d)*e^(-2*x) - c)/(2*d^2*e^(-2*x) - d^2*e^(-4*x) - d^2) - c^2*log(-(c - d)
*e^(-2*x) + c + d)/d^3 + c^2*log(e^(-x) + 1)/d^3 + c^2*log(e^(-x) - 1)/d^3) + 3*a^2*b*(c*log(-(c - d)*e^(-2*x)
+ c + d)/d^2 - c*log(e^(-x) + 1)/d^2 - c*log(e^(-x) - 1)/d^2 + 2/(d*e^(-2*x) - d)) - a^3*log(d*coth(x) + c)/d

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mupad [B]  time = 2.70, size = 1346, normalized size = 17.26 \[ \frac {2\,\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{2\,x}\,\left (\frac {32\,c\,\left (2\,a^3\,c\,d^8-6\,a^2\,b\,c^2\,d^7+6\,a\,b^2\,c^3\,d^6-2\,b^3\,c^4\,d^5\right )\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}}{d^{16}\,\sqrt {{\left (a\,d-b\,c\right )}^6}\,\left (c+d\right )\,{\left (c-d\right )}^2\,\left (c^2+2\,c\,d+d^2\right )}-\frac {16\,\left (c^2\,\sqrt {-d^8}\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}+d^2\,\sqrt {-d^8}\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}\right )\,\left (c^2+d^2\right )\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}}{d^{13}\,\left (c+d\right )\,{\left (c-d\right )}^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {-d^8}\,\left (c^2+2\,c\,d+d^2\right )}\right )+\frac {32\,c\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}\,\left (-a^3\,c\,d^8+a^3\,d^9+3\,a^2\,b\,c^2\,d^7-3\,a^2\,b\,c\,d^8-3\,a\,b^2\,c^3\,d^6+3\,a\,b^2\,c^2\,d^7+b^3\,c^4\,d^5-b^3\,c^3\,d^6\right )}{d^{16}\,\sqrt {{\left (a\,d-b\,c\right )}^6}\,\left (c+d\right )\,{\left (c-d\right )}^2\,\left (c^2+2\,c\,d+d^2\right )}+\frac {16\,\left (c^2\,\sqrt {-d^8}\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}-c\,d\,\sqrt {-d^8}\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}\right )\,\left (c^2+d^2\right )\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}}{d^{13}\,\left (c+d\right )\,{\left (c-d\right )}^2\,{\left (a\,d-b\,c\right )}^3\,\sqrt {-d^8}\,\left (c^2+2\,c\,d+d^2\right )}\right )\,\left (d^{10}\,\sqrt {-d^8}+2\,c\,d^9\,\sqrt {-d^8}+c^2\,d^8\,\sqrt {-d^8}\right )}{16\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}}\right )\,\sqrt {a^6\,d^6-6\,a^5\,b\,c\,d^5+15\,a^4\,b^2\,c^2\,d^4-20\,a^3\,b^3\,c^3\,d^3+15\,a^2\,b^4\,c^4\,d^2-6\,a\,b^5\,c^5\,d+b^6\,c^6}}{\sqrt {-d^8}}-\frac {2\,\left (2\,b^3\,d-b^3\,c+3\,a\,b^2\,d\right )}{d^2\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {8\,b^3}{3\,d\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {2\,\left (3\,a^2\,b\,d^2-3\,a\,b^2\,c\,d+3\,a\,b^2\,d^2+b^3\,c^2-b^3\,c\,d+b^3\,d^2\right )}{d^3\,\left ({\mathrm {e}}^{2\,x}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*coth(x))^3/(sinh(x)^2*(c + d*coth(x))),x)

[Out]

(2*atan(((exp(2*x)*((32*c*(2*a^3*c*d^8 - 2*b^3*c^4*d^5 + 6*a*b^2*c^3*d^6 - 6*a^2*b*c^2*d^7)*(a^6*d^6 + b^6*c^6
+ 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)^(1/2))/(d^16*
((a*d - b*c)^6)^(1/2)*(c + d)*(c - d)^2*(2*c*d + c^2 + d^2)) - (16*(c^2*(-d^8)^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a
^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)^(1/2) + d^2*(-d^8)^(
1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5
*b*c*d^5)^(1/2))*(c^2 + d^2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4
- 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)^(1/2))/(d^13*(c + d)*(c - d)^2*(a*d - b*c)^3*(-d^8)^(1/2)*(2*c*d + c^2 + d^2
))) + (32*c*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d
- 6*a^5*b*c*d^5)^(1/2)*(a^3*d^9 - a^3*c*d^8 - b^3*c^3*d^6 + b^3*c^4*d^5 + 3*a*b^2*c^2*d^7 - 3*a*b^2*c^3*d^6 +
3*a^2*b*c^2*d^7 - 3*a^2*b*c*d^8))/(d^16*((a*d - b*c)^6)^(1/2)*(c + d)*(c - d)^2*(2*c*d + c^2 + d^2)) + (16*(c^
2*(-d^8)^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5
*d - 6*a^5*b*c*d^5)^(1/2) - c*d*(-d^8)^(1/2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15
*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)^(1/2))*(c^2 + d^2)*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 -
20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5*b*c*d^5)^(1/2))/(d^13*(c + d)*(c - d)^2*(a*d
- b*c)^3*(-d^8)^(1/2)*(2*c*d + c^2 + d^2)))*(d^10*(-d^8)^(1/2) + 2*c*d^9*(-d^8)^(1/2) + c^2*d^8*(-d^8)^(1/2)))
/(16*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*c^5*d - 6*a^5
*b*c*d^5)^(1/2)))*(a^6*d^6 + b^6*c^6 + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a*b^5*
c^5*d - 6*a^5*b*c*d^5)^(1/2))/(-d^8)^(1/2) - (2*(2*b^3*d - b^3*c + 3*a*b^2*d))/(d^2*(exp(4*x) - 2*exp(2*x) + 1
)) - (8*b^3)/(3*d*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (2*(b^3*c^2 + b^3*d^2 + 3*a*b^2*d^2 + 3*a^2*b*d^
2 - b^3*c*d - 3*a*b^2*c*d))/(d^3*(exp(2*x) - 1))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \coth {\relax (x )}\right )^{3} \operatorname {csch}^{2}{\relax (x )}}{c + d \coth {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*coth(x))**3*csch(x)**2/(c+d*coth(x)),x)

[Out]

Integral((a + b*coth(x))**3*csch(x)**2/(c + d*coth(x)), x)

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