Optimal. Leaf size=117 \[ -\frac {\sqrt {b} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 d}-\frac {(5 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d}+\frac {x \left (3 a^2+12 a b+8 b^2\right )}{8 a^3}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 a d} \]
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Rubi [A] time = 0.20, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4132, 470, 527, 522, 206, 208} \[ \frac {x \left (3 a^2+12 a b+8 b^2\right )}{8 a^3}-\frac {\sqrt {b} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 d}-\frac {(5 a+4 b) \sinh (c+d x) \cosh (c+d x)}{8 a^2 d}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 a d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 470
Rule 522
Rule 527
Rule 4132
Rubi steps
\begin {align*} \int \frac {\sinh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^3 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d}-\frac {\operatorname {Subst}\left (\int \frac {a+b+(4 a+3 b) x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{4 a d}\\ &=-\frac {(5 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d}-\frac {\operatorname {Subst}\left (\int \frac {-(a+b) (3 a+4 b)-b (5 a+4 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 a^2 d}\\ &=-\frac {(5 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d}-\frac {\left (b (a+b)^2\right ) \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a^3 d}+\frac {\left (3 a^2+12 a b+8 b^2\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 a^3 d}\\ &=\frac {\left (3 a^2+12 a b+8 b^2\right ) x}{8 a^3}-\frac {\sqrt {b} (a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^3 d}-\frac {(5 a+4 b) \cosh (c+d x) \sinh (c+d x)}{8 a^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 a d}\\ \end {align*}
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Mathematica [B] time = 2.48, size = 294, normalized size = 2.51 \[ -\frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\sqrt {b} \left (3 a^3+34 a^2 b+64 a b^2+32 b^3\right ) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )-\sqrt {b (\cosh (c)-\sinh (c))^4} \left (\sqrt {b} \sqrt {a+b} \left (a^2 \sinh (4 (c+d x))-2 a^2 c+12 a^2 d x-8 a (a+b) \sinh (2 (c+d x))+48 a b d x+32 b^2 d x\right )+a^2 (3 a+2 b) \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )\right )\right )}{64 a^3 \sqrt {b} d \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.59, size = 1681, normalized size = 14.37 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 2.88, size = 220, normalized size = 1.88 \[ \frac {\frac {8 \, {\left (3 \, a^{2} + 12 \, a b + 8 \, b^{2}\right )} {\left (d x + c\right )}}{a^{3}} + \frac {a e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a e^{\left (2 \, d x + 2 \, c\right )} - 8 \, b e^{\left (2 \, d x + 2 \, c\right )}}{a^{2}} - \frac {{\left (18 \, a^{2} e^{\left (4 \, d x + 4 \, c\right )} + 72 \, a b e^{\left (4 \, d x + 4 \, c\right )} + 48 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 8 \, a b e^{\left (2 \, d x + 2 \, c\right )} + a^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{a^{3}} - \frac {64 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{3}}}{64 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.41, size = 708, normalized size = 6.05 \[ \frac {1}{4 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{8 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {b}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3}{8 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {b}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 d a}-\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b}{2 d \,a^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b^{2}}{d \,a^{3}}-\frac {1}{4 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}+\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{8 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {b}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {3}{8 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {b}{2 d \,a^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 d a}+\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b}{2 d \,a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b^{2}}{d \,a^{3}}+\frac {\sqrt {b}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \sqrt {a +b}}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{d \,a^{2} \sqrt {a +b}}+\frac {b^{\frac {5}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,a^{3} \sqrt {a +b}}-\frac {\sqrt {b}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \sqrt {a +b}}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{d \,a^{2} \sqrt {a +b}}-\frac {b^{\frac {5}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,a^{3} \sqrt {a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 526, normalized size = 4.50 \[ \frac {3 \, b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a d} + \frac {3 \, {\left (d x + c\right )}}{8 \, a d} - \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, a^{2} d} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} + \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{2} d} - \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{2} d} - \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} {\left (d x + c\right )}}{2 \, a^{3} d} + \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, a^{2} d} + \frac {{\left (a^{2} b + 8 \, a b^{2} + 8 \, b^{3}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{16 \, \sqrt {{\left (a + b\right )} b} a^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.43, size = 328, normalized size = 2.80 \[ \frac {x\,\left (3\,a^2+12\,a\,b+8\,b^2\right )}{8\,a^3}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,a\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,a\,d}+\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,\left (a+b\right )}{8\,a^2\,d}-\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a+b\right )}{8\,a^2\,d}+\frac {\sqrt {b}\,\ln \left (\frac {4\,b\,{\left (a+b\right )}^3\,\left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}-\frac {8\,b^{3/2}\,{\left (a+b\right )}^{7/2}\,\left (a+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}\right )\,{\left (a+b\right )}^{3/2}}{2\,a^3\,d}-\frac {\sqrt {b}\,\ln \left (\frac {8\,b^{3/2}\,{\left (a+b\right )}^{7/2}\,\left (a+2\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}+\frac {4\,b\,{\left (a+b\right )}^3\,\left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^8}\right )\,{\left (a+b\right )}^{3/2}}{2\,a^3\,d} \]
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 \[ \int \frac {\sinh ^{4}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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