Optimal. Leaf size=110 \[ -\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}+x (a+b)^4-\frac {b^4 \coth ^7(c+d x)}{7 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3661, 390, 206} \[ -\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}+x (a+b)^4-\frac {b^4 \coth ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 390
Rule 3661
Rubi steps
\begin {align*} \int \left (a+b \coth ^2(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^4}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b (2 a+b) \left (2 a^2+2 a b+b^2\right )-b^2 \left (6 a^2+4 a b+b^2\right ) x^2-b^3 (4 a+b) x^4-b^4 x^6+\frac {(a+b)^4}{1-x^2}\right ) \, dx,x,\coth (c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d}+\frac {(a+b)^4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=(a+b)^4 x-\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \coth (c+d x)}{d}-\frac {b^2 \left (6 a^2+4 a b+b^2\right ) \coth ^3(c+d x)}{3 d}-\frac {b^3 (4 a+b) \coth ^5(c+d x)}{5 d}-\frac {b^4 \coth ^7(c+d x)}{7 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.08, size = 127, normalized size = 1.15 \[ -\frac {\coth (c+d x) \left (b \left (35 b \left (6 a^2+4 a b+b^2\right ) \coth ^2(c+d x)+105 \left (4 a^3+6 a^2 b+4 a b^2+b^3\right )+21 b^2 (4 a+b) \coth ^4(c+d x)+15 b^3 \coth ^6(c+d x)\right )-105 (a+b)^4 \tanh ^{-1}\left (\sqrt {\tanh ^2(c+d x)}\right ) \sqrt {\tanh ^2(c+d x)}\right )}{105 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.46, size = 1164, normalized size = 10.58 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.23, size = 447, normalized size = 4.06 \[ \frac {105 \, {\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (d x + c\right )} - \frac {8 \, {\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 315 \, a b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 105 \, b^{4} e^{\left (12 \, d x + 12 \, c\right )} - 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} - 1575 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} - 1260 \, a b^{3} e^{\left (10 \, d x + 10 \, c\right )} - 315 \, b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 3360 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 2555 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 770 \, b^{4} e^{\left (8 \, d x + 8 \, c\right )} - 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} - 3990 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} - 3080 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} - 770 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 2835 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2121 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 609 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} - 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} - 1155 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 812 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} - 203 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b + 210 \, a^{2} b^{2} + 161 \, a b^{3} + 44 \, b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{7}}}{105 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.03, size = 344, normalized size = 3.13 \[ -\frac {\left (\coth ^{5}\left (d x +c \right )\right ) b^{4}}{5 d}-\frac {b^{4} \left (\coth ^{3}\left (d x +c \right )\right )}{3 d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) a^{4}}{2 d}+\frac {2 \ln \left (\coth \left (d x +c \right )+1\right ) a^{3} b}{d}+\frac {3 \ln \left (\coth \left (d x +c \right )+1\right ) a^{2} b^{2}}{d}+\frac {2 \ln \left (\coth \left (d x +c \right )+1\right ) a \,b^{3}}{d}+\frac {\ln \left (\coth \left (d x +c \right )+1\right ) b^{4}}{2 d}-\frac {\coth \left (d x +c \right ) b^{4}}{d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) a^{4}}{2 d}-\frac {2 \ln \left (\coth \left (d x +c \right )-1\right ) a^{3} b}{d}-\frac {3 \ln \left (\coth \left (d x +c \right )-1\right ) a^{2} b^{2}}{d}-\frac {2 \ln \left (\coth \left (d x +c \right )-1\right ) a \,b^{3}}{d}-\frac {\ln \left (\coth \left (d x +c \right )-1\right ) b^{4}}{2 d}-\frac {b^{4} \left (\coth ^{7}\left (d x +c \right )\right )}{7 d}-\frac {4 \left (\coth ^{5}\left (d x +c \right )\right ) a \,b^{3}}{5 d}-\frac {2 \left (\coth ^{3}\left (d x +c \right )\right ) a^{2} b^{2}}{d}-\frac {4 \left (\coth ^{3}\left (d x +c \right )\right ) a \,b^{3}}{3 d}-\frac {4 a \,b^{3} \coth \left (d x +c \right )}{d}-\frac {4 a^{3} b \coth \left (d x +c \right )}{d}-\frac {6 a^{2} b^{2} \coth \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.34, size = 410, normalized size = 3.73 \[ \frac {1}{105} \, b^{4} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} - 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} - 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} - 105 \, e^{\left (-12 \, d x - 12 \, c\right )} - 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} - 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} - 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} - 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} - 1\right )}}\right )} + \frac {4}{15} \, a b^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} - 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} - 45 \, e^{\left (-8 \, d x - 8 \, c\right )} - 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} - 1\right )}}\right )} + 2 \, a^{2} b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} - 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + 4 \, a^{3} b {\left (x + \frac {c}{d} + \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}}\right )} + a^{4} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.31, size = 111, normalized size = 1.01 \[ x\,{\left (a+b\right )}^4-\frac {{\mathrm {coth}\left (c+d\,x\right )}^3\,\left (6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{3\,d}-\frac {{\mathrm {coth}\left (c+d\,x\right )}^5\,\left (b^4+4\,a\,b^3\right )}{5\,d}-\frac {b^4\,{\mathrm {coth}\left (c+d\,x\right )}^7}{7\,d}-\frac {b\,\mathrm {coth}\left (c+d\,x\right )\,\left (4\,a^3+6\,a^2\,b+4\,a\,b^2+b^3\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 119.40, size = 264, normalized size = 2.40 \[ \begin {cases} a^{4} x + \tilde {\infty } a^{3} b x + \tilde {\infty } a^{2} b^{2} x + \tilde {\infty } a b^{3} x + \tilde {\infty } b^{4} x & \text {for}\: c = \log {\left (- e^{- d x} \right )} \vee c = \log {\left (e^{- d x} \right )} \\x \left (a + b \coth ^{2}{\relax (c )}\right )^{4} & \text {for}\: d = 0 \\a^{4} x + 4 a^{3} b x - \frac {4 a^{3} b}{d \tanh {\left (c + d x \right )}} + 6 a^{2} b^{2} x - \frac {6 a^{2} b^{2}}{d \tanh {\left (c + d x \right )}} - \frac {2 a^{2} b^{2}}{d \tanh ^{3}{\left (c + d x \right )}} + 4 a b^{3} x - \frac {4 a b^{3}}{d \tanh {\left (c + d x \right )}} - \frac {4 a b^{3}}{3 d \tanh ^{3}{\left (c + d x \right )}} - \frac {4 a b^{3}}{5 d \tanh ^{5}{\left (c + d x \right )}} + b^{4} x - \frac {b^{4}}{d \tanh {\left (c + d x \right )}} - \frac {b^{4}}{3 d \tanh ^{3}{\left (c + d x \right )}} - \frac {b^{4}}{5 d \tanh ^{5}{\left (c + d x \right )}} - \frac {b^{4}}{7 d \tanh ^{7}{\left (c + d x \right )}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________