Optimal. Leaf size=907 \[ -\frac {2 \text {sech}(c+d x) \tanh (c+d x) a^2}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {4 \tanh (c+d x) a}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 \left (8 a^2-5 b^2\right ) \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} a}{3 b^4 \sqrt {a+b} d}+\frac {4 \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} a}{b^2 \sqrt {a+b} d}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {2 (2 a+b) (4 a+b) \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{3 b^3 \sqrt {a+b} d}+\frac {4 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{b \sqrt {a+b} d}+\frac {2 b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)} a}-\frac {2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{\sqrt {a+b} d a}+\frac {2 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{\sqrt {a+b} d a}+\frac {2 \sqrt {a+b} \coth (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{d a^2} \]
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Rubi [A] time = 1.37, antiderivative size = 907, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3895, 3785, 4058, 3921, 3784, 3832, 4004, 3836, 4005, 3845, 4082} \[ -\frac {2 \text {sech}(c+d x) \tanh (c+d x) a^2}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {4 \tanh (c+d x) a}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 \left (8 a^2-5 b^2\right ) \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} a}{3 b^4 \sqrt {a+b} d}+\frac {4 \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} a}{b^2 \sqrt {a+b} d}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {2 (2 a+b) (4 a+b) \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{3 b^3 \sqrt {a+b} d}+\frac {4 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{b \sqrt {a+b} d}+\frac {2 b^2 \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)} a}-\frac {2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{\sqrt {a+b} d a}+\frac {2 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{\sqrt {a+b} d a}+\frac {2 \sqrt {a+b} \coth (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}}}{d a^2} \]
Antiderivative was successfully verified.
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Rule 3784
Rule 3785
Rule 3832
Rule 3836
Rule 3845
Rule 3895
Rule 3921
Rule 4004
Rule 4005
Rule 4058
Rule 4082
Rubi steps
\begin {align*} \int \frac {\tanh ^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx &=\int \left (\frac {1}{(a+b \text {sech}(c+d x))^{3/2}}-\frac {2 \text {sech}^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}}+\frac {\text {sech}^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}}\right ) \, dx\\ &=-\left (2 \int \frac {\text {sech}^2(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\right )+\int \frac {1}{(a+b \text {sech}(c+d x))^{3/2}} \, dx+\int \frac {\text {sech}^4(c+d x)}{(a+b \text {sech}(c+d x))^{3/2}} \, dx\\ &=-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {4 \int \frac {\text {sech}(c+d x) \left (-\frac {b}{2}-\frac {1}{2} a \text {sech}(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a^2-b^2}-\frac {2 \int \frac {\frac {1}{2} \left (-a^2+b^2\right )+\frac {1}{2} a b \text {sech}(c+d x)+\frac {1}{2} b^2 \text {sech}^2(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}-\frac {2 \int \frac {\text {sech}(c+d x) \left (a^2-\frac {1}{2} a b \text {sech}(c+d x)-\frac {1}{2} \left (4 a^2-b^2\right ) \text {sech}^2(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{b \left (a^2-b^2\right )}\\ &=-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}-\frac {2 \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a+b}-\frac {2 \int \frac {\frac {1}{2} \left (-a^2+b^2\right )+\left (\frac {a b}{2}-\frac {b^2}{2}\right ) \text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}+\frac {(2 a) \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a^2-b^2}-\frac {4 \int \frac {\text {sech}(c+d x) \left (\frac {1}{4} b \left (2 a^2+b^2\right )+\frac {1}{4} a \left (8 a^2-5 b^2\right ) \text {sech}(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )}-\frac {b^2 \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a \left (a^2-b^2\right )}\\ &=-\frac {2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 a \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b^2 \sqrt {a+b} d}+\frac {4 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b \sqrt {a+b} d}-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}+\frac {\int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a}-\frac {b \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{a (a+b)}+\frac {((2 a+b) (4 a+b)) \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{3 b^2 (a+b)}-\frac {\left (a \left (8 a^2-5 b^2\right )\right ) \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{3 b^2 \left (a^2-b^2\right )}\\ &=-\frac {2 \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 a \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b^2 \sqrt {a+b} d}-\frac {2 a \left (8 a^2-5 b^2\right ) \coth (c+d x) E\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^4 \sqrt {a+b} d}+\frac {2 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a \sqrt {a+b} d}+\frac {4 \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b \sqrt {a+b} d}-\frac {2 (2 a+b) (4 a+b) \coth (c+d x) F\left (\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{3 b^3 \sqrt {a+b} d}+\frac {2 \sqrt {a+b} \coth (c+d x) \Pi \left (\frac {a+b}{a};\sin ^{-1}\left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a^2 d}-\frac {4 a \tanh (c+d x)}{\left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 b^2 \tanh (c+d x)}{a \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}-\frac {2 a^2 \text {sech}(c+d x) \tanh (c+d x)}{b \left (a^2-b^2\right ) d \sqrt {a+b \text {sech}(c+d x)}}+\frac {2 \left (4 a^2-b^2\right ) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{3 b^2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [F] time = 180.00, size = 0, normalized size = 0.00 \[ \text {\$Aborted} \]
Verification is Not applicable to the result.
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fricas [F] time = 7.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {b \operatorname {sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )^{4}}{b^{2} \operatorname {sech}\left (d x + c\right )^{2} + 2 \, a b \operatorname {sech}\left (d x + c\right ) + a^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.61, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{4}\left (d x +c \right )}{\left (a +b \,\mathrm {sech}\left (d x +c \right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh \left (d x + c\right )^{4}}{{\left (b \operatorname {sech}\left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^4}{{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{4}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}{\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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