Optimal. Leaf size=113 \[ -\frac {2 (a A+c C) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{3/2} e}-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))} \]
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Rubi [A]
time = 0.14, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4461, 2833, 12,
2739, 632, 210, 2747, 32} \begin {gather*} -\frac {2 (a A+c C) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{3/2}}-\frac {(A c-a C) \cosh (d+e x)}{e \left (a^2+c^2\right ) (a+c \sinh (d+e x))}-\frac {B}{c e (a+c \sinh (d+e x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 210
Rule 632
Rule 2739
Rule 2747
Rule 2833
Rule 4461
Rubi steps
\begin {align*} \int \frac {A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx &=B \int \frac {\cosh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx+\int \frac {A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx\\ &=-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}-\frac {\int \frac {-a A-c C}{a+c \sinh (d+e x)} \, dx}{a^2+c^2}+\frac {B \text {Subst}\left (\int \frac {1}{(a+x)^2} \, dx,x,c \sinh (d+e x)\right )}{c e}\\ &=-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}+\frac {(a A+c C) \int \frac {1}{a+c \sinh (d+e x)} \, dx}{a^2+c^2}\\ &=-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}-\frac {(2 i (a A+c C)) \text {Subst}\left (\int \frac {1}{a-2 i c x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right ) e}\\ &=-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}+\frac {(4 i (a A+c C)) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+c^2\right )-x^2} \, dx,x,-2 i c+2 a \tan \left (\frac {1}{2} (i d+i e x)\right )\right )}{\left (a^2+c^2\right ) e}\\ &=-\frac {2 (a A+c C) \tanh ^{-1}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a^2+c^2}}\right )}{\left (a^2+c^2\right )^{3/2} e}-\frac {B}{c e (a+c \sinh (d+e x))}-\frac {(A c-a C) \cosh (d+e x)}{\left (a^2+c^2\right ) e (a+c \sinh (d+e x))}\\ \end {align*}
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Mathematica [A]
time = 0.38, size = 113, normalized size = 1.00 \begin {gather*} \frac {\frac {2 (a A+c C) \text {ArcTan}\left (\frac {c-a \tanh \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2-c^2}}\right )}{\sqrt {-a^2-c^2}}-\frac {B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)}{c (a+c \sinh (d+e x))}}{\left (a^2+c^2\right ) e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 6.00, size = 151, normalized size = 1.34
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (A \,c^{2}-B \,a^{2}-B \,c^{2}-C a c \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{2}+c^{2}\right )}-\frac {A c -C a}{a^{2}+c^{2}}\right )}{a \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a}+\frac {2 \left (A a +C c \right ) \arctanh \left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}}}}{e}\) | \(151\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (A \,c^{2}-B \,a^{2}-B \,c^{2}-C a c \right ) \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )}{a \left (a^{2}+c^{2}\right )}-\frac {A c -C a}{a^{2}+c^{2}}\right )}{a \left (\tanh ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-2 c \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-a}+\frac {2 \left (A a +C c \right ) \arctanh \left (\frac {2 a \tanh \left (\frac {e x}{2}+\frac {d}{2}\right )-2 c}{2 \sqrt {a^{2}+c^{2}}}\right )}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}}}}{e}\) | \(151\) |
risch | \(\frac {2 A a c \,{\mathrm e}^{e x +d}-2 B \,a^{2} {\mathrm e}^{e x +d}-2 B \,c^{2} {\mathrm e}^{e x +d}-2 C \,a^{2} {\mathrm e}^{e x +d}-2 A \,c^{2}+2 C a c}{c e \left (a^{2}+c^{2}\right ) \left (c \,{\mathrm e}^{2 e x +2 d}+2 a \,{\mathrm e}^{e x +d}-c \right )}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a -a^{4}-2 a^{2} c^{2}-c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) A a}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}+\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a -a^{4}-2 a^{2} c^{2}-c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) C c}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a +a^{4}+2 a^{2} c^{2}+c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) A a}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}-\frac {\ln \left ({\mathrm e}^{e x +d}+\frac {\left (a^{2}+c^{2}\right )^{\frac {3}{2}} a +a^{4}+2 a^{2} c^{2}+c^{4}}{c \left (a^{2}+c^{2}\right )^{\frac {3}{2}}}\right ) C c}{\left (a^{2}+c^{2}\right )^{\frac {3}{2}} e}\) | \(360\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 346 vs.
\(2 (110) = 220\).
time = 0.51, size = 346, normalized size = 3.06 \begin {gather*} {\left (\frac {a e^{\left (-1\right )} \log \left (\frac {c e^{\left (-x e - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-x e - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{2} + c^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (a e^{\left (-x e - d\right )} + c\right )} e^{\left (-1\right )}}{a^{2} c + c^{3} + 2 \, {\left (a^{3} + a c^{2}\right )} e^{\left (-x e - d\right )} - {\left (a^{2} c + c^{3}\right )} e^{\left (-2 \, x e - 2 \, d\right )}}\right )} A + {\left (\frac {c e^{\left (-1\right )} \log \left (\frac {c e^{\left (-x e - d\right )} - a - \sqrt {a^{2} + c^{2}}}{c e^{\left (-x e - d\right )} - a + \sqrt {a^{2} + c^{2}}}\right )}{{\left (a^{2} + c^{2}\right )}^{\frac {3}{2}}} + \frac {2 \, {\left (a^{2} e^{\left (-x e - d\right )} + a c\right )} e^{\left (-1\right )}}{a^{2} c^{2} + c^{4} + 2 \, {\left (a^{3} c + a c^{3}\right )} e^{\left (-x e - d\right )} - {\left (a^{2} c^{2} + c^{4}\right )} e^{\left (-2 \, x e - 2 \, d\right )}}\right )} C - \frac {2 \, B e^{\left (-x e - d - 1\right )}}{2 \, a c e^{\left (-x e - d\right )} - c^{2} e^{\left (-2 \, x e - 2 \, d\right )} + c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 837 vs.
\(2 (110) = 220\).
time = 0.43, size = 837, normalized size = 7.41 \begin {gather*} \frac {2 \, C a^{3} c - 2 \, A a^{2} c^{2} + 2 \, C a c^{3} - 2 \, A c^{4} - {\left (A a c^{2} + C c^{3} - {\left (A a c^{2} + C c^{3}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - {\left (A a c^{2} + C c^{3}\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - 2 \, {\left (A a^{2} c + C a c^{2}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) - 2 \, {\left (A a^{2} c + C a c^{2} + {\left (A a c^{2} + C c^{3}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )\right )} \sqrt {a^{2} + c^{2}} \log \left (\frac {c^{2} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + c^{2} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + 2 \, a c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + 2 \, a^{2} + c^{2} + 2 \, {\left (c^{2} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + a c\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) - 2 \, \sqrt {a^{2} + c^{2}} {\left (c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + c \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + a\right )}}{c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + c \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + 2 \, a \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + 2 \, {\left (c \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + a\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) - c}\right ) - 2 \, {\left ({\left (B + C\right )} a^{4} - A a^{3} c + {\left (2 \, B + C\right )} a^{2} c^{2} - A a c^{3} + B c^{4}\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) - 2 \, {\left ({\left (B + C\right )} a^{4} - A a^{3} c + {\left (2 \, B + C\right )} a^{2} c^{2} - A a c^{3} + B c^{4}\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )}{{\left ({\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \cosh \left (1\right ) + {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} + {\left ({\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \cosh \left (1\right ) + {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \sinh \left (1\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )^{2} - {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \cosh \left (1\right ) + 2 \, {\left ({\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} \cosh \left (1\right ) + {\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) - {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \sinh \left (1\right ) + 2 \, {\left ({\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} \cosh \left (1\right ) + {\left ({\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \cosh \left (1\right ) + {\left (a^{4} c^{2} + 2 \, a^{2} c^{4} + c^{6}\right )} \sinh \left (1\right )\right )} \cosh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right ) + {\left (a^{5} c + 2 \, a^{3} c^{3} + a c^{5}\right )} \sinh \left (1\right )\right )} \sinh \left (x \cosh \left (1\right ) + x \sinh \left (1\right ) + d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 170, normalized size = 1.50 \begin {gather*} \frac {\frac {{\left (A a + C c\right )} \log \left (\frac {{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a - 2 \, \sqrt {a^{2} + c^{2}} \right |}}{{\left | 2 \, c e^{\left (e x + d\right )} + 2 \, a + 2 \, \sqrt {a^{2} + c^{2}} \right |}}\right )}{{\left (a^{2} + c^{2}\right )}^{\frac {3}{2}}} - \frac {2 \, {\left (B a^{2} e^{\left (e x + d\right )} + C a^{2} e^{\left (e x + d\right )} - A a c e^{\left (e x + d\right )} + B c^{2} e^{\left (e x + d\right )} - C a c + A c^{2}\right )}}{{\left (a^{2} c + c^{3}\right )} {\left (c e^{\left (2 \, e x + 2 \, d\right )} + 2 \, a e^{\left (e x + d\right )} - c\right )}}}{e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.20, size = 279, normalized size = 2.47 \begin {gather*} \frac {\ln \left (\frac {2\,\left (A\,a+C\,c\right )\,\left (c-a\,{\mathrm {e}}^{d+e\,x}\right )}{c\,{\left (a^2+c^2\right )}^{3/2}}-\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (A\,a+C\,c\right )}{c\,\left (a^2+c^2\right )}\right )\,\left (A\,a+C\,c\right )}{e\,{\left (a^2+c^2\right )}^{3/2}}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (A\,a+C\,c\right )}{c\,\left (a^2+c^2\right )}-\frac {2\,\left (A\,a+C\,c\right )\,\left (c-a\,{\mathrm {e}}^{d+e\,x}\right )}{c\,{\left (a^2+c^2\right )}^{3/2}}\right )\,\left (A\,a+C\,c\right )}{e\,{\left (a^2+c^2\right )}^{3/2}}-\frac {\frac {2\,\left (A\,c^3-C\,a\,c^2\right )}{c\,e\,\left (a^2\,c+c^3\right )}+\frac {2\,{\mathrm {e}}^{d+e\,x}\,\left (B\,c^4+B\,a^2\,c^2+C\,a^2\,c^2-A\,a\,c^3\right )}{c^2\,e\,\left (a^2\,c+c^3\right )}}{2\,a\,{\mathrm {e}}^{d+e\,x}-c+c\,{\mathrm {e}}^{2\,d+2\,e\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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