Optimal. Leaf size=201 \[ \frac{b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac{\sqrt{b} \left (35 a^2 b+35 a^3+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{16 a^{7/2} d (a+b)^4}+\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b \tanh (c+d x)}{6 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^3}+\frac{x}{(a+b)^4} \]
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Rubi [A] time = 0.277135, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {3661, 414, 527, 522, 206, 205} \[ \frac{b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 d (a+b)^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac{\sqrt{b} \left (35 a^2 b+35 a^3+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{16 a^{7/2} d (a+b)^4}+\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 d (a+b)^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b \tanh (c+d x)}{6 a d (a+b) \left (a+b \tanh ^2(c+d x)\right )^3}+\frac{x}{(a+b)^4} \]
Antiderivative was successfully verified.
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Rule 3661
Rule 414
Rule 527
Rule 522
Rule 206
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+b \tanh ^2(c+d x)\right )^4} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \left (a+b x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}-\frac{\operatorname{Subst}\left (\int \frac{b-6 (a+b)+5 b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 a (a+b) d}\\ &=\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}+\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{3 \left (8 a^2+11 a b+5 b^2\right )-3 b (11 a+5 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{24 a^2 (a+b)^2 d}\\ &=\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}+\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}-\frac{\operatorname{Subst}\left (\int \frac{-3 \left (16 a^3+19 a^2 b+16 a b^2+5 b^3\right )+3 b \left (19 a^2+16 a b+5 b^2\right ) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{48 a^3 (a+b)^3 d}\\ &=\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}+\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^4 d}+\frac{\left (b \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{16 a^3 (a+b)^4 d}\\ &=\frac{x}{(a+b)^4}+\frac{\sqrt{b} \left (35 a^3+35 a^2 b+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{16 a^{7/2} (a+b)^4 d}+\frac{b \tanh (c+d x)}{6 a (a+b) d \left (a+b \tanh ^2(c+d x)\right )^3}+\frac{b (11 a+5 b) \tanh (c+d x)}{24 a^2 (a+b)^2 d \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{b \left (19 a^2+16 a b+5 b^2\right ) \tanh (c+d x)}{16 a^3 (a+b)^3 d \left (a+b \tanh ^2(c+d x)\right )}\\ \end{align*}
Mathematica [A] time = 0.546873, size = 203, normalized size = 1.01 \[ \frac{\frac{3 b \left (19 a^2+16 a b+5 b^2\right ) (a+b) \tanh (c+d x)}{a^3 \left (a+b \tanh ^2(c+d x)\right )}+\frac{3 \sqrt{b} \left (35 a^2 b+35 a^3+21 a b^2+5 b^3\right ) \tan ^{-1}\left (\frac{\sqrt{b} \tanh (c+d x)}{\sqrt{a}}\right )}{a^{7/2}}+\frac{2 b (11 a+5 b) (a+b)^2 \tanh (c+d x)}{a^2 \left (a+b \tanh ^2(c+d x)\right )^2}+\frac{8 b (a+b)^3 \tanh (c+d x)}{a \left (a+b \tanh ^2(c+d x)\right )^3}-24 \log (1-\tanh (c+d x))+24 \log (\tanh (c+d x)+1)}{48 d (a+b)^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.034, size = 608, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.31529, size = 1030, normalized size = 5.12 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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