Integrand size = 23, antiderivative size = 610 \[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=-\frac {4 (a-b) \sqrt {a+b} \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b^2 d}+\frac {2 (a-b) \sqrt {a+b} \left (8 a^2+9 b^2\right ) \coth (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{15 b^4 d}-\frac {4 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{b d}+\frac {2 \sqrt {a+b} \left (8 a^2-2 a b+9 b^2\right ) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{15 b^3 d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (1+\text {sech}(c+d x))}{a-b}}}{a d}-\frac {8 a \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{15 b^2 d}+\frac {2 \text {sech}(c+d x) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{5 b d} \]
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Time = 0.57 (sec) , antiderivative size = 610, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {3980, 3869, 3922, 3917, 4089, 3945, 4167, 4090} \[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\frac {2 (a-b) \sqrt {a+b} \left (8 a^2+9 b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{15 b^4 d}+\frac {2 \sqrt {a+b} \left (8 a^2-2 a b+9 b^2\right ) \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{15 b^3 d}-\frac {4 (a-b) \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b^2 d}-\frac {4 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{b d}+\frac {2 \sqrt {a+b} \coth (c+d x) \sqrt {\frac {b (1-\text {sech}(c+d x))}{a+b}} \sqrt {-\frac {b (\text {sech}(c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \text {sech}(c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {8 a \tanh (c+d x) \sqrt {a+b \text {sech}(c+d x)}}{15 b^2 d}+\frac {2 \tanh (c+d x) \text {sech}(c+d x) \sqrt {a+b \text {sech}(c+d x)}}{5 b d} \]
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Rule 3869
Rule 3917
Rule 3922
Rule 3945
Rule 3980
Rule 4089
Rule 4090
Rule 4167
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\sqrt {a+b \text {sech}(c+d x)}}-\frac {2 \text {sech}^2(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}+\frac {\text {sech}^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}}\right ) \, dx \\ & = -\left (2 \int \frac {\text {sech}^2(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx\right )+\int \frac {1}{\sqrt {a+b \text {sech}(c+d x)}} \, dx+\int \frac {\text {sech}^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx \\ & = \frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \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 d}+\frac {2 \text {sech}(c+d x) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{5 b d}+2 \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx-2 \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx+\frac {\int \frac {\text {sech}(c+d x) \left (2 a+3 b \text {sech}(c+d x)-4 a \text {sech}^2(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{5 b} \\ & = -\frac {4 (a-b) \sqrt {a+b} \coth (c+d x) E\left (\arcsin \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 d}-\frac {4 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \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 d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \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 d}-\frac {8 a \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{15 b^2 d}+\frac {2 \text {sech}(c+d x) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{5 b d}+\frac {2 \int \frac {\text {sech}(c+d x) \left (a b+\frac {1}{2} \left (8 a^2+9 b^2\right ) \text {sech}(c+d x)\right )}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{15 b^2} \\ & = -\frac {4 (a-b) \sqrt {a+b} \coth (c+d x) E\left (\arcsin \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 d}-\frac {4 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \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 d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \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 d}-\frac {8 a \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{15 b^2 d}+\frac {2 \text {sech}(c+d x) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{5 b d}+\frac {1}{15} \left (9+\frac {8 a^2}{b^2}\right ) \int \frac {\text {sech}(c+d x) (1+\text {sech}(c+d x))}{\sqrt {a+b \text {sech}(c+d x)}} \, dx-\frac {\left (8 a^2-2 a b+9 b^2\right ) \int \frac {\text {sech}(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx}{15 b^2} \\ & = -\frac {4 (a-b) \sqrt {a+b} \coth (c+d x) E\left (\arcsin \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 d}+\frac {2 (a-b) \sqrt {a+b} \left (8 a^2+9 b^2\right ) \coth (c+d x) E\left (\arcsin \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}}}{15 b^4 d}-\frac {4 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \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 d}+\frac {2 \sqrt {a+b} \left (8 a^2-2 a b+9 b^2\right ) \coth (c+d x) \operatorname {EllipticF}\left (\arcsin \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}}}{15 b^3 d}+\frac {2 \sqrt {a+b} \coth (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \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 d}-\frac {8 a \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{15 b^2 d}+\frac {2 \text {sech}(c+d x) \sqrt {a+b \text {sech}(c+d x)} \tanh (c+d x)}{5 b d} \\ \end{align*}
Timed out. \[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\text {\$Aborted} \]
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\[\int \frac {\tanh \left (d x +c \right )^{4}}{\sqrt {a +b \,\operatorname {sech}\left (d x +c \right )}}d x\]
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\[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{4}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {\tanh ^{4}{\left (c + d x \right )}}{\sqrt {a + b \operatorname {sech}{\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{4}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
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\[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int { \frac {\tanh \left (d x + c\right )^{4}}{\sqrt {b \operatorname {sech}\left (d x + c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {\tanh ^4(c+d x)}{\sqrt {a+b \text {sech}(c+d x)}} \, dx=\int \frac {{\mathrm {tanh}\left (c+d\,x\right )}^4}{\sqrt {a+\frac {b}{\mathrm {cosh}\left (c+d\,x\right )}}} \,d x \]
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