Optimal. Leaf size=180 \[ -\frac{\left (2 a^2 A+3 a c C-A c^2\right ) \tanh ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{5/2}}-\frac{\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right )^2 (a+c \sinh (d+e x))}-\frac{(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac{B}{2 c e (a+c \sinh (d+e x))^2} \]
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Rubi [A] time = 0.269667, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258, Rules used = {4376, 2754, 12, 2660, 618, 204, 2668, 32} \[ -\frac{\left (2 a^2 A+3 a c C-A c^2\right ) \tanh ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{a^2+c^2}}\right )}{e \left (a^2+c^2\right )^{5/2}}-\frac{\left (a^2 (-C)+3 a A c+2 c^2 C\right ) \cosh (d+e x)}{2 e \left (a^2+c^2\right )^2 (a+c \sinh (d+e x))}-\frac{(A c-a C) \cosh (d+e x)}{2 e \left (a^2+c^2\right ) (a+c \sinh (d+e x))^2}-\frac{B}{2 c e (a+c \sinh (d+e x))^2} \]
Antiderivative was successfully verified.
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Rule 4376
Rule 2754
Rule 12
Rule 2660
Rule 618
Rule 204
Rule 2668
Rule 32
Rubi steps
\begin{align*} \int \frac{A+B \cosh (d+e x)+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx &=B \int \frac{\cosh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx+\int \frac{A+C \sinh (d+e x)}{(a+c \sinh (d+e x))^3} \, dx\\ &=-\frac{(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac{\int \frac{-2 (a A+c C)+(A c-a C) \sinh (d+e x)}{(a+c \sinh (d+e x))^2} \, dx}{2 \left (a^2+c^2\right )}+\frac{B \operatorname{Subst}\left (\int \frac{1}{(a+x)^3} \, dx,x,c \sinh (d+e x)\right )}{c e}\\ &=-\frac{B}{2 c e (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac{\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}+\frac{\int \frac{2 a^2 A-A c^2+3 a c C}{a+c \sinh (d+e x)} \, dx}{2 \left (a^2+c^2\right )^2}\\ &=-\frac{B}{2 c e (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac{\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}+\frac{\left (2 a^2 A-A c^2+3 a c C\right ) \int \frac{1}{a+c \sinh (d+e x)} \, dx}{2 \left (a^2+c^2\right )^2}\\ &=-\frac{B}{2 c e (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac{\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}-\frac{\left (i \left (2 a^2 A-A c^2+3 a c C\right )\right ) \operatorname{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 )^2 e}\\ &=-\frac{B}{2 c e (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac{\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}+\frac{\left (2 i \left (2 a^2 A-A c^2+3 a c C\right )\right ) \operatorname{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 )^2 e}\\ &=-\frac{\left (2 a^2 A-A c^2+3 a c C\right ) \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 )^{5/2} e}-\frac{B}{2 c e (a+c \sinh (d+e x))^2}-\frac{(A c-a C) \cosh (d+e x)}{2 \left (a^2+c^2\right ) e (a+c \sinh (d+e x))^2}-\frac{\left (3 a A c-a^2 C+2 c^2 C\right ) \cosh (d+e x)}{2 \left (a^2+c^2\right )^2 e (a+c \sinh (d+e x))}\\ \end{align*}
Mathematica [A] time = 0.679457, size = 170, normalized size = 0.94 \[ \frac{-\frac{\left (a^2+c^2\right ) \left (B \left (a^2+c^2\right )+c (A c-a C) \cosh (d+e x)\right )}{c (a+c \sinh (d+e x))^2}+\frac{2 \left (2 a^2 A+3 a c C-A c^2\right ) \tan ^{-1}\left (\frac{c-a \tanh \left (\frac{1}{2} (d+e x)\right )}{\sqrt{-a^2-c^2}}\right )}{\sqrt{-a^2-c^2}}+\frac{\left (a^2 C-3 a A c-2 c^2 C\right ) \cosh (d+e x)}{a+c \sinh (d+e x)}}{2 e \left (a^2+c^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.116, size = 416, normalized size = 2.3 \begin{align*}{\frac{1}{e} \left ( -2\,{\frac{1}{ \left ( a \left ( \tanh \left ( 1/2\,ex+d/2 \right ) \right ) ^{2}-2\,c\tanh \left ( 1/2\,ex+d/2 \right ) -a \right ) ^{2}} \left ( -1/2\,{\frac{ \left ( 5\,A{a}^{2}{c}^{2}+2\,A{c}^{4}-2\,B{a}^{4}-4\,B{a}^{2}{c}^{2}-2\,B{c}^{4}-3\,C{a}^{3}c \right ) \left ( \tanh \left ( 1/2\,ex+d/2 \right ) \right ) ^{3}}{a \left ({a}^{4}+2\,{a}^{2}{c}^{2}+{c}^{4} \right ) }}-1/2\,{\frac{ \left ( 4\,A{a}^{4}c-7\,A{a}^{2}{c}^{3}-2\,A{c}^{5}+2\,B{a}^{4}c+4\,B{a}^{2}{c}^{3}+2\,B{c}^{5}-2\,C{a}^{5}+5\,C{a}^{3}{c}^{2}-2\,Ca{c}^{4} \right ) \left ( \tanh \left ( 1/2\,ex+d/2 \right ) \right ) ^{2}}{ \left ({a}^{4}+2\,{a}^{2}{c}^{2}+{c}^{4} \right ){a}^{2}}}+1/2\,{\frac{ \left ( 11\,A{a}^{2}{c}^{2}+2\,A{c}^{4}-2\,B{a}^{4}-4\,B{a}^{2}{c}^{2}-2\,B{c}^{4}-5\,C{a}^{3}c+4\,Ca{c}^{3} \right ) \tanh \left ( 1/2\,ex+d/2 \right ) }{a \left ({a}^{4}+2\,{a}^{2}{c}^{2}+{c}^{4} \right ) }}+1/2\,{\frac{4\,A{a}^{2}c+A{c}^{3}-2\,C{a}^{3}+Ca{c}^{2}}{{a}^{4}+2\,{a}^{2}{c}^{2}+{c}^{4}}} \right ) }+{\frac{2\,{a}^{2}A-A{c}^{2}+3\,Cac}{{a}^{4}+2\,{a}^{2}{c}^{2}+{c}^{4}}{\it Artanh} \left ({\frac{1}{2} \left ( 2\,a\tanh \left ( 1/2\,ex+d/2 \right ) -2\,c \right ){\frac{1}{\sqrt{{a}^{2}+{c}^{2}}}}} \right ){\frac{1}{\sqrt{{a}^{2}+{c}^{2}}}}} \right ) } \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 [B] time = 2.3097, size = 4178, normalized size = 23.21 \begin{align*} \text{result too large to display} \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 = 1.17361, size = 586, normalized size = 3.26 \begin{align*} -\frac{{\left (2 \, A a^{2} + 3 \, C a c - A c^{2}\right )} \log \left (\frac{{\left | -2 \, c e^{\left (x e + d\right )} - 2 \, a - 2 \, \sqrt{a^{2} + c^{2}} \right |}}{{\left | -2 \, c e^{\left (x e + d\right )} - 2 \, a + 2 \, \sqrt{a^{2} + c^{2}} \right |}}\right )}{2 \,{\left (a^{4} e + 2 \, a^{2} c^{2} e + c^{4} e\right )} \sqrt{a^{2} + c^{2}}} + \frac{2 \, A a^{2} c^{2} e^{\left (3 \, x e + 3 \, d\right )} + 3 \, C a c^{3} e^{\left (3 \, x e + 3 \, d\right )} - A c^{4} e^{\left (3 \, x e + 3 \, d\right )} - 2 \, B a^{4} e^{\left (2 \, x e + 2 \, d\right )} - 2 \, C a^{4} e^{\left (2 \, x e + 2 \, d\right )} + 6 \, A a^{3} c e^{\left (2 \, x e + 2 \, d\right )} - 4 \, B a^{2} c^{2} e^{\left (2 \, x e + 2 \, d\right )} + 5 \, C a^{2} c^{2} e^{\left (2 \, x e + 2 \, d\right )} - 3 \, A a c^{3} e^{\left (2 \, x e + 2 \, d\right )} - 2 \, B c^{4} e^{\left (2 \, x e + 2 \, d\right )} - 2 \, C c^{4} e^{\left (2 \, x e + 2 \, d\right )} + 4 \, C a^{3} c e^{\left (x e + d\right )} - 10 \, A a^{2} c^{2} e^{\left (x e + d\right )} - 5 \, C a c^{3} e^{\left (x e + d\right )} - A c^{4} e^{\left (x e + d\right )} - C a^{2} c^{2} + 3 \, A a c^{3} + 2 \, C c^{4}}{{\left (a^{4} c e + 2 \, a^{2} c^{3} e + c^{5} e\right )}{\left (c e^{\left (2 \, x e + 2 \, d\right )} + 2 \, a e^{\left (x e + d\right )} - c\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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