Optimal. Leaf size=74 \[ -\frac {a^3 \coth ^5(c+d x)}{5 d}-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {a^2 (a+3 b) \coth ^3(c+d x)}{3 d}+x (a+b)^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3670, 461, 207} \[ -\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {a^2 (a+3 b) \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}+x (a+b)^3 \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 461
Rule 3670
Rubi steps
\begin {align*} \int \coth ^6(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{x^6 \left (1-x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {a^3}{x^6}+\frac {a^2 (a+3 b)}{x^4}+\frac {a \left (a^2+3 a b+3 b^2\right )}{x^2}-\frac {(a+b)^3}{-1+x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {a^2 (a+3 b) \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}-\frac {(a+b)^3 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=(a+b)^3 x-\frac {a \left (a^2+3 a b+3 b^2\right ) \coth (c+d x)}{d}-\frac {a^2 (a+3 b) \coth ^3(c+d x)}{3 d}-\frac {a^3 \coth ^5(c+d x)}{5 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.78, size = 100, normalized size = 1.35 \[ \frac {(a+b)^3 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \tanh (c+d x)}{d \sqrt {\tanh ^2(c+d x)}}-\frac {a \coth (c+d x) \left (15 \left (a^2+3 a b+3 b^2\right )+3 a^2 \coth ^4(c+d x)+5 a (a+3 b) \coth ^2(c+d x)\right )}{15 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.44, size = 557, normalized size = 7.53 \[ -\frac {{\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2}\right )} \cosh \left (d x + c\right )^{5} + 5 \, {\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{4} - {\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{5} - 5 \, {\left (5 \, a^{3} + 24 \, a^{2} b + 27 \, a b^{2}\right )} \cosh \left (d x + c\right )^{3} + 5 \, {\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x - 2 \, {\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (2 \, {\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2}\right )} \cosh \left (d x + c\right )^{3} - 3 \, {\left (5 \, a^{3} + 24 \, a^{2} b + 27 \, a b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 10 \, {\left (5 \, a^{3} + 6 \, a^{2} b + 9 \, a b^{2}\right )} \cosh \left (d x + c\right ) - 5 \, {\left ({\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{4} + 46 \, a^{3} + 120 \, a^{2} b + 90 \, a b^{2} + 30 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x - 3 \, {\left (23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2} + 15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{15 \, {\left (d \sinh \left (d x + c\right )^{5} + 5 \, {\left (2 \, d \cosh \left (d x + c\right )^{2} - d\right )} \sinh \left (d x + c\right )^{3} + 5 \, {\left (d \cosh \left (d x + c\right )^{4} - 3 \, d \cosh \left (d x + c\right )^{2} + 2 \, d\right )} \sinh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.56, size = 241, normalized size = 3.26 \[ \frac {15 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} {\left (d x + c\right )} - \frac {2 \, {\left (45 \, a^{3} e^{\left (8 \, d x + 8 \, c\right )} + 90 \, a^{2} b e^{\left (8 \, d x + 8 \, c\right )} + 45 \, a b^{2} e^{\left (8 \, d x + 8 \, c\right )} - 90 \, a^{3} e^{\left (6 \, d x + 6 \, c\right )} - 270 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 180 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 140 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 330 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 270 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 70 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 210 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 180 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 23 \, a^{3} + 60 \, a^{2} b + 45 \, a b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{5}}}{15 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.28, size = 100, normalized size = 1.35 \[ \frac {a^{3} \left (d x +c -\coth \left (d x +c \right )-\frac {\left (\coth ^{3}\left (d x +c \right )\right )}{3}-\frac {\left (\coth ^{5}\left (d x +c \right )\right )}{5}\right )+3 a^{2} b \left (d x +c -\coth \left (d x +c \right )-\frac {\left (\coth ^{3}\left (d x +c \right )\right )}{3}\right )+3 a \,b^{2} \left (d x +c -\coth \left (d x +c \right )\right )+b^{3} \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 239, normalized size = 3.23 \[ \frac {1}{15} \, a^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + a^{2} b {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 3 \, a b^{2} {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + b^{3} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.30, size = 568, normalized size = 7.68 \[ x\,{\left (a+b\right )}^3-\frac {\frac {6\,\left (a^3+2\,a^2\,b+a\,b^2\right )}{5\,d}+\frac {6\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^3+2\,a^2\,b+a\,b^2\right )}{5\,d}-\frac {24\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}-\frac {24\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3+6\,a^2\,b+9\,a\,b^2\right )}{5\,d}}{5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1}-\frac {\frac {2\,\left (5\,a^3+6\,a^2\,b+9\,a\,b^2\right )}{15\,d}+\frac {6\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^3+2\,a^2\,b+a\,b^2\right )}{5\,d}-\frac {12\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1}+\frac {\frac {6\,\left (a^2\,b+a\,b^2\right )}{5\,d}-\frac {6\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^3+2\,a^2\,b+a\,b^2\right )}{5\,d}+\frac {18\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^2\,b+a\,b^2\right )}{5\,d}-\frac {2\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3+6\,a^2\,b+9\,a\,b^2\right )}{5\,d}}{6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}+\frac {\frac {6\,\left (a^2\,b+a\,b^2\right )}{5\,d}-\frac {6\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3+2\,a^2\,b+a\,b^2\right )}{5\,d}}{{\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {6\,\left (a^3+2\,a^2\,b+a\,b^2\right )}{5\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \coth ^{6}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________