Optimal. Leaf size=114 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {a-b} d}+\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.07, antiderivative size = 114, 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 = {3270, 205, 214}
\begin {gather*} \frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} d \sqrt {a-b}}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 205
Rule 214
Rule 3270
Rubi steps
\begin {align*} \int \frac {\cosh ^4(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a-(a-b) x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \text {Subst}\left (\int \frac {1}{\left (a+(-a+b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 a^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^2 d}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {a-b} d}+\frac {\tanh (c+d x)}{4 a d \left (a-(a-b) \tanh ^2(c+d x)\right )^2}+\frac {3 \tanh (c+d x)}{8 a^2 d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.49, size = 102, normalized size = 0.89 \begin {gather*} \frac {\frac {3 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a-b}}+\frac {\sqrt {a} (8 a-3 b+(2 a+3 b) \cosh (2 (c+d x))) \sinh (2 (c+d x))}{(2 a-b+b \cosh (2 (c+d x)))^2}}{8 a^{5/2} d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs.
\(2(100)=200\).
time = 1.53, size = 300, normalized size = 2.63
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {3 \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a}}{d}\) | \(300\) |
default | \(\frac {-\frac {2 \left (-\frac {5 \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {3 \left (a +4 b \right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 a^{2}}-\frac {5 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {3 \left (\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 a}}{d}\) | \(300\) |
risch | \(-\frac {8 a^{2} b \,{\mathrm e}^{6 d x +6 c}-3 b^{3} {\mathrm e}^{6 d x +6 c}+16 a^{3} {\mathrm e}^{4 d x +4 c}+8 a^{2} b \,{\mathrm e}^{4 d x +4 c}-18 a \,b^{2} {\mathrm e}^{4 d x +4 c}+9 b^{3} {\mathrm e}^{4 d x +4 c}+8 a^{2} b \,{\mathrm e}^{2 d x +2 c}+16 a \,b^{2} {\mathrm e}^{2 d x +2 c}-9 b^{3} {\mathrm e}^{2 d x +2 c}+2 a \,b^{2}+3 b^{3}}{4 b^{2} a^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{16 \sqrt {a^{2}-a b}\, d \,a^{2}}-\frac {3 \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{16 \sqrt {a^{2}-a b}\, d \,a^{2}}\) | \(353\) |
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. 2115 vs.
\(2 (102) = 204\).
time = 0.43, size = 4486, normalized size = 39.35 \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. 244 vs.
\(2 (102) = 204\).
time = 1.75, size = 244, normalized size = 2.14 \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 )}{\sqrt {-a^{2} + a b} a^{2}} - \frac {2 \, {\left (8 \, a^{2} b e^{\left (6 \, d x + 6 \, c\right )} - 3 \, b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 9 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} + 16 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 9 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} + 3 \, b^{3}\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} a^{2} b^{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 {cosh}\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]
________________________________________________________________________________________