Optimal. Leaf size=491 \[ -\frac{a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{d \left (a^2-b^2\right )^3}+\frac{a^{2/3} \sqrt [3]{b} \left (7 a^2 b^2+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )+a^4+b^4\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 d \left (a^2-b^2\right )^3}-\frac{a^{2/3} \sqrt [3]{b} \left (7 a^2 b^2+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )+a^4+b^4\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}-\frac{a^{2/3} \sqrt [3]{b} \left (3 a^{4/3} b^{2/3}+a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d \left (a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right )^3}-\frac{5 a-b}{16 d (a+b)^2 (1-\tanh (c+d x))}+\frac{5 a+b}{16 d (a-b)^2 (\tanh (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\tanh (c+d x))^2}-\frac{1}{16 d (a-b) (\tanh (c+d x)+1)^2}-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 d (a+b)^3}+\frac{3 a (a+5 b) \log (\tanh (c+d x)+1)}{16 d (a-b)^3} \]
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Rubi [A] time = 0.891549, antiderivative size = 491, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 10, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.435, Rules used = {3663, 6725, 1871, 1860, 31, 634, 617, 204, 628, 260} \[ -\frac{a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{d \left (a^2-b^2\right )^3}+\frac{a^{2/3} \sqrt [3]{b} \left (7 a^2 b^2+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )+a^4+b^4\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 d \left (a^2-b^2\right )^3}-\frac{a^{2/3} \sqrt [3]{b} \left (7 a^2 b^2+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )+a^4+b^4\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 d \left (a^2-b^2\right )^3}-\frac{a^{2/3} \sqrt [3]{b} \left (3 a^{4/3} b^{2/3}+a^2-b^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} d \left (a^{2/3} b^{2/3}+a^{4/3}+b^{4/3}\right )^3}-\frac{5 a-b}{16 d (a+b)^2 (1-\tanh (c+d x))}+\frac{5 a+b}{16 d (a-b)^2 (\tanh (c+d x)+1)}+\frac{1}{16 d (a+b) (1-\tanh (c+d x))^2}-\frac{1}{16 d (a-b) (\tanh (c+d x)+1)^2}-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 d (a+b)^3}+\frac{3 a (a+5 b) \log (\tanh (c+d x)+1)}{16 d (a-b)^3} \]
Antiderivative was successfully verified.
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Rule 3663
Rule 6725
Rule 1871
Rule 1860
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{\sinh ^4(c+d x)}{a+b \tanh ^3(c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{\left (1-x^2\right )^3 \left (a+b x^3\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{1}{8 (a+b) (-1+x)^3}+\frac{-5 a+b}{16 (a+b)^2 (-1+x)^2}-\frac{3 a (a-5 b)}{16 (a+b)^3 (-1+x)}+\frac{1}{8 (a-b) (1+x)^3}+\frac{-5 a-b}{16 (a-b)^2 (1+x)^2}+\frac{3 a (a+5 b)}{16 (a-b)^3 (1+x)}+\frac{a b \left (-3 a b \left (2 a^2+b^2\right )+\left (a^4+7 a^2 b^2+b^4\right ) x-3 a b \left (a^2+2 b^2\right ) x^2\right )}{\left (a^2-b^2\right )^3 \left (a+b x^3\right )}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac{3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}+\frac{1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac{5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac{1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac{5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac{(a b) \operatorname{Subst}\left (\int \frac{-3 a b \left (2 a^2+b^2\right )+\left (a^4+7 a^2 b^2+b^4\right ) x-3 a b \left (a^2+2 b^2\right ) x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac{3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}+\frac{1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac{5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac{1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac{5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac{(a b) \operatorname{Subst}\left (\int \frac{-3 a b \left (2 a^2+b^2\right )+\left (a^4+7 a^2 b^2+b^4\right ) x}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^3 d}-\frac{\left (3 a^2 b^2 \left (a^2+2 b^2\right )\right ) \operatorname{Subst}\left (\int \frac{x^2}{a+b x^3} \, dx,x,\tanh (c+d x)\right )}{\left (a^2-b^2\right )^3 d}\\ &=-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac{3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac{a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac{1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac{5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac{1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac{5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac{\left (\sqrt [3]{a} b^{2/3}\right ) \operatorname{Subst}\left (\int \frac{\sqrt [3]{a} \left (-6 a b^{4/3} \left (2 a^2+b^2\right )+\sqrt [3]{a} \left (a^4+7 a^2 b^2+b^4\right )\right )+\sqrt [3]{b} \left (3 a b^{4/3} \left (2 a^2+b^2\right )+\sqrt [3]{a} \left (a^4+7 a^2 b^2+b^4\right )\right ) x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{\left (a^{2/3} b^{2/3} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx,x,\tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}\\ &=-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac{3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac{a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}-\frac{a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac{1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac{5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac{1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac{5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac{\left (a b^{2/3} \left (a^2+3 a^{4/3} b^{2/3}-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{2 \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}+\frac{\left (a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right )\right ) \operatorname{Subst}\left (\int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx,x,\tanh (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}\\ &=-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac{3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac{a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac{a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}-\frac{a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac{1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac{5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac{1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac{5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}+\frac{\left (a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{4/3} b^{2/3}-b^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}\right )}{\left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}\\ &=-\frac{a^{2/3} \sqrt [3]{b} \left (a^2+3 a^{4/3} b^{2/3}-b^2\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} \tanh (c+d x)}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{\sqrt{3} \left (a^{4/3}+a^{2/3} b^{2/3}+b^{4/3}\right )^3 d}-\frac{3 a (a-5 b) \log (1-\tanh (c+d x))}{16 (a+b)^3 d}+\frac{3 a (a+5 b) \log (1+\tanh (c+d x))}{16 (a-b)^3 d}-\frac{a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} \tanh (c+d x)\right )}{3 \left (a^2-b^2\right )^3 d}+\frac{a^{2/3} \sqrt [3]{b} \left (a^4+7 a^2 b^2+b^4+3 a^{2/3} b^{4/3} \left (2 a^2+b^2\right )\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} \tanh (c+d x)+b^{2/3} \tanh ^2(c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}-\frac{a^2 b \left (a^2+2 b^2\right ) \log \left (a+b \tanh ^3(c+d x)\right )}{\left (a^2-b^2\right )^3 d}+\frac{1}{16 (a+b) d (1-\tanh (c+d x))^2}-\frac{5 a-b}{16 (a+b)^2 d (1-\tanh (c+d x))}-\frac{1}{16 (a-b) d (1+\tanh (c+d x))^2}+\frac{5 a+b}{16 (a-b)^2 d (1+\tanh (c+d x))}\\ \end{align*}
Mathematica [C] time = 4.37695, size = 645, normalized size = 1.31 \[ \frac{3 \left (-8 a \left (a^2 b+a^3+2 a b^2+2 b^3\right ) \sinh (2 (c+d x))+a (a-b) \left (12 \left (a^2-6 a b+5 b^2\right ) (c+d x)+(a+b)^2 \sinh (4 (c+d x))\right )+4 b \left (5 a^2 b+5 a^3+a b^2+b^3\right ) \cosh (2 (c+d x))+b (-(a-b)) (a+b)^2 \cosh (4 (c+d x))\right )-32 a b \text{RootSum}\left [\text{$\#$1}^3 a+3 \text{$\#$1}^2 a+\text{$\#$1}^3 b-3 \text{$\#$1}^2 b+3 \text{$\#$1} a+3 \text{$\#$1} b+a-b\& ,\frac{-10 \text{$\#$1}^2 a^2 b \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+20 \text{$\#$1}^2 a^2 b c+20 \text{$\#$1}^2 a^2 b d x+5 \text{$\#$1}^2 a^3 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-10 \text{$\#$1}^2 a^3 c-10 \text{$\#$1}^2 a^3 d x+10 \text{$\#$1}^2 a b^2 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-20 \text{$\#$1}^2 a b^2 c-20 \text{$\#$1}^2 a b^2 d x-2 \text{$\#$1}^2 b^3 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+4 \text{$\#$1}^2 b^3 c+4 \text{$\#$1}^2 b^3 d x-2 \text{$\#$1} a^2 b \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+4 \text{$\#$1} a^2 b c+4 \text{$\#$1} a^2 b d x+3 a^3 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+4 \text{$\#$1} a^3 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-8 \text{$\#$1} a^3 c-8 \text{$\#$1} a^3 d x+6 a b^2 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-4 \text{$\#$1} a b^2 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )+8 \text{$\#$1} a b^2 c+8 \text{$\#$1} a b^2 d x+2 \text{$\#$1} b^3 \log \left (e^{2 (c+d x)}-\text{$\#$1}\right )-4 \text{$\#$1} b^3 c-4 \text{$\#$1} b^3 d x-6 a^3 c-6 a^3 d x-12 a b^2 c-12 a b^2 d x}{\text{$\#$1}^2 a-\text{$\#$1}^2 b+2 \text{$\#$1} a+2 \text{$\#$1} b+a-b}\& \right ]}{96 d (a-b)^2 (a+b)^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.128, size = 603, normalized size = 1.2 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \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 [A] time = 2.33021, size = 489, normalized size = 1. \begin{align*} \frac{\frac{24 \,{\left (a^{2} + 5 \, a b\right )} d x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 90 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 4 \, a b e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} - 2 \, a b + b^{2}\right )} e^{\left (-4 \, d x\right )}}{a^{3} e^{\left (4 \, c\right )} - 3 \, a^{2} b e^{\left (4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, c\right )} - b^{3} e^{\left (4 \, c\right )}} - \frac{64 \,{\left (a^{4} b + 2 \, a^{2} b^{3}\right )} \log \left ({\left | a e^{\left (6 \, d x + 6 \, c\right )} + b e^{\left (6 \, d x + 6 \, c\right )} + 3 \, a e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a e^{\left (2 \, d x + 2 \, c\right )} + 3 \, b e^{\left (2 \, d x + 2 \, c\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac{a e^{\left (4 \, d x + 24 \, c\right )} + b e^{\left (4 \, d x + 24 \, c\right )} - 8 \, a e^{\left (2 \, d x + 22 \, c\right )} + 4 \, b e^{\left (2 \, d x + 22 \, c\right )}}{a^{2} e^{\left (20 \, c\right )} + 2 \, a b e^{\left (20 \, c\right )} + b^{2} e^{\left (20 \, c\right )}}}{64 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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