Optimal. Leaf size=124 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a-b)^{5/2} d}+\frac {\tanh ^3(c+d x)}{4 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {3 \tanh (c+d x)}{8 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.09, antiderivative size = 124, 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 = {3266, 294, 214}
\begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} d (a-b)^{5/2}}-\frac {3 \tanh (c+d x)}{8 d (a-b)^2 \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\tanh ^3(c+d x)}{4 d (a-b) \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 214
Rule 294
Rule 3266
Rubi steps
\begin {align*} \int \frac {\sinh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh ^3(c+d x)}{4 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {3 \text {Subst}\left (\int \frac {x^2}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 (a-b) d}\\ &=\frac {\tanh ^3(c+d x)}{4 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {3 \tanh (c+d x)}{8 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {3 \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a-b)^2 d}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 \sqrt {a} (a-b)^{5/2} d}+\frac {\tanh ^3(c+d x)}{4 (a-b) d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}-\frac {3 \tanh (c+d x)}{8 (a-b)^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.93, size = 104, normalized size = 0.84 \begin {gather*} \frac {\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{5/2}}+\frac {(-8 a+5 b+(2 a-5 b) \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}}{8 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(356\) vs.
\(2(110)=220\).
time = 1.02, size = 357, normalized size = 2.88
method | result | size |
derivativedivides | \(\frac {-\frac {32 \left (\frac {3 a \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (11 a -20 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (11 a -20 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {3 a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(357\) |
default | \(\frac {-\frac {32 \left (\frac {3 a \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (11 a -20 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (11 a -20 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}+\frac {3 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{128 \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {3 a \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(357\) |
risch | \(-\frac {8 a^{2} b \,{\mathrm e}^{6 d x +6 c}-16 a \,b^{2} {\mathrm e}^{6 d x +6 c}+5 b^{3} {\mathrm e}^{6 d x +6 c}+16 a^{3} {\mathrm e}^{4 d x +4 c}-56 a^{2} b \,{\mathrm e}^{4 d x +4 c}+46 a \,b^{2} {\mathrm e}^{4 d x +4 c}-15 b^{3} {\mathrm e}^{4 d x +4 c}+8 a^{2} b \,{\mathrm e}^{2 d x +2 c}-32 a \,b^{2} {\mathrm e}^{2 d x +2 c}+15 b^{3} {\mathrm e}^{2 d x +2 c}+2 a \,b^{2}-5 b^{3}}{4 b^{2} d \left (a -b \right )^{2} \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}\) | \(380\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2465 vs.
\(2 (112) = 224\).
time = 0.65, size = 5186, normalized size = 41.82 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 282 vs.
\(2 (112) = 224\).
time = 3.80, size = 282, normalized size = 2.27 \begin {gather*} \frac {\frac {3 \, \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a^{2} + a b}} - \frac {2 \, {\left (8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 16 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 5 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 56 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 46 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 15 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 32 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 15 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} - 5 \, b^{3}\right )}}{{\left (a^{2} b^{2} - 2 \, a b^{3} + b^{4}\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^4}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________