Optimal. Leaf size=95 \[ \frac {d x}{a}+\frac {(b d-2 a e) \tanh ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {b^2-4 a c}}\right )}{a \sqrt {b^2-4 a c} i}-\frac {d \log \left (a+b e^{h+i x}+c e^{2 h+2 i x}\right )}{2 a i} \]
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Rubi [A]
time = 0.10, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.162, Rules used = {2320, 814, 648,
632, 212, 642} \begin {gather*} \frac {(b d-2 a e) \tanh ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {b^2-4 a c}}\right )}{a i \sqrt {b^2-4 a c}}-\frac {d \log \left (a+b e^{h+i x}+c e^{2 h+2 i x}\right )}{2 a i}+\frac {d x}{a} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 632
Rule 642
Rule 648
Rule 814
Rule 2320
Rubi steps
\begin {align*} \int \frac {d+e e^{h+575 x}}{a+b e^{h+575 x}+c e^{2 h+1150 x}} \, dx &=\frac {1}{575} \text {Subst}\left (\int \frac {d+e x}{x \left (a+b x+c x^2\right )} \, dx,x,e^{h+575 x}\right )\\ &=\frac {1}{575} \text {Subst}\left (\int \left (\frac {d}{a x}+\frac {-b d+a e-c d x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,e^{h+575 x}\right )\\ &=\frac {d x}{a}+\frac {\text {Subst}\left (\int \frac {-b d+a e-c d x}{a+b x+c x^2} \, dx,x,e^{h+575 x}\right )}{575 a}\\ &=\frac {d x}{a}-\frac {d \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,e^{h+575 x}\right )}{1150 a}-\frac {(b d-2 a e) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,e^{h+575 x}\right )}{1150 a}\\ &=\frac {d x}{a}-\frac {d \log \left (a+b e^{h+575 x}+c e^{2 h+1150 x}\right )}{1150 a}+\frac {(b d-2 a e) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c e^{h+575 x}\right )}{575 a}\\ &=\frac {d x}{a}+\frac {(b d-2 a e) \tanh ^{-1}\left (\frac {b+2 c e^{h+575 x}}{\sqrt {b^2-4 a c}}\right )}{575 a \sqrt {b^2-4 a c}}-\frac {d \log \left (a+b e^{h+575 x}+c e^{2 h+1150 x}\right )}{1150 a}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 100, normalized size = 1.05 \begin {gather*} \frac {\frac {(-2 b d+4 a e) \tan ^{-1}\left (\frac {b+2 c e^{h+i x}}{\sqrt {-b^2+4 a c}}\right )}{\sqrt {-b^2+4 a c}}+d \left (2 \log \left (e^{h+i x}\right )-\log \left (a+e^{h+i x} \left (b+c e^{h+i x}\right )\right )\right )}{2 a i} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(182\) vs.
\(2(86)=172\).
time = 0.09, size = 183, normalized size = 1.93
method | result | size |
default | \(-\frac {d \ln \left (a +b \,{\mathrm e}^{i x} {\mathrm e}^{h}+c \,{\mathrm e}^{2 i x} {\mathrm e}^{2 h}\right )}{2 i a}-\frac {d \,{\mathrm e}^{h} b \arctan \left (\frac {{\mathrm e}^{h} b +2 \,{\mathrm e}^{2 h} {\mathrm e}^{i x} c}{\sqrt {4 a c \,{\mathrm e}^{2 h}-{\mathrm e}^{2 h} b^{2}}}\right )}{i a \sqrt {4 a c \,{\mathrm e}^{2 h}-{\mathrm e}^{2 h} b^{2}}}+\frac {d \ln \left ({\mathrm e}^{i x}\right )}{i a}+\frac {2 e \,{\mathrm e}^{h} \arctan \left (\frac {{\mathrm e}^{h} b +2 \,{\mathrm e}^{2 h} {\mathrm e}^{i x} c}{\sqrt {4 a c \,{\mathrm e}^{2 h}-{\mathrm e}^{2 h} b^{2}}}\right )}{i \sqrt {4 a c \,{\mathrm e}^{2 h}-{\mathrm e}^{2 h} b^{2}}}\) | \(183\) |
risch | \(\frac {4 a c d \,i^{2} x}{4 a^{2} c \,i^{2}-a \,b^{2} i^{2}}-\frac {b^{2} d \,i^{2} x}{4 a^{2} c \,i^{2}-a \,b^{2} i^{2}}+\frac {4 a c d h i}{4 a^{2} c \,i^{2}-a \,b^{2} i^{2}}-\frac {b^{2} d h i}{4 a^{2} c \,i^{2}-a \,b^{2} i^{2}}-\frac {2 \ln \left ({\mathrm e}^{i x +h}+\frac {2 b a e -b^{2} d +\sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 c \left (2 a e -b d \right )}\right ) c d}{\left (4 c a -b^{2}\right ) i}+\frac {\ln \left ({\mathrm e}^{i x +h}+\frac {2 b a e -b^{2} d +\sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 c \left (2 a e -b d \right )}\right ) b^{2} d}{2 a \left (4 c a -b^{2}\right ) i}+\frac {\ln \left ({\mathrm e}^{i x +h}+\frac {2 b a e -b^{2} d +\sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 c \left (2 a e -b d \right )}\right ) \sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 a \left (4 c a -b^{2}\right ) i}-\frac {2 \ln \left ({\mathrm e}^{i x +h}-\frac {-2 b a e +b^{2} d +\sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 c \left (2 a e -b d \right )}\right ) c d}{\left (4 c a -b^{2}\right ) i}+\frac {\ln \left ({\mathrm e}^{i x +h}-\frac {-2 b a e +b^{2} d +\sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 c \left (2 a e -b d \right )}\right ) b^{2} d}{2 a \left (4 c a -b^{2}\right ) i}-\frac {\ln \left ({\mathrm e}^{i x +h}-\frac {-2 b a e +b^{2} d +\sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 c \left (2 a e -b d \right )}\right ) \sqrt {-16 a^{3} c \,e^{2}+4 a^{2} b^{2} e^{2}+16 a^{2} b c d e -4 a \,b^{3} d e -4 a \,b^{2} c \,d^{2}+b^{4} d^{2}}}{2 a \left (4 c a -b^{2}\right ) i}\) | \(915\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 327 vs. \(2 (81) = 162\).
time = 0.39, size = 327, normalized size = 3.44 \begin {gather*} \frac {2 \, d x - {\left (a \sqrt {-\frac {b^{2} d^{2} - 4 \, a b d e + 4 \, a^{2} e^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} - i \, d\right )} \log \left (\frac {b^{2} d e - 2 \, a b e^{2} - {\left (i \, a b^{2} - 4 i \, a^{2} c\right )} \sqrt {-\frac {b^{2} d^{2} - 4 \, a b d e + 4 \, a^{2} e^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} e + 2 \, {\left (b c d - 2 \, a c e\right )} e^{\left (h + i \, x + 1\right )}}{2 \, {\left (b c d - 2 \, a c e\right )}}\right ) + {\left (a \sqrt {-\frac {b^{2} d^{2} - 4 \, a b d e + 4 \, a^{2} e^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} + i \, d\right )} \log \left (\frac {b^{2} d e - 2 \, a b e^{2} - {\left (-i \, a b^{2} + 4 i \, a^{2} c\right )} \sqrt {-\frac {b^{2} d^{2} - 4 \, a b d e + 4 \, a^{2} e^{2}}{a^{2} b^{2} - 4 \, a^{3} c}} e + 2 \, {\left (b c d - 2 \, a c e\right )} e^{\left (h + i \, x + 1\right )}}{2 \, {\left (b c d - 2 \, a c e\right )}}\right )}{2 \, a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.55, size = 116, normalized size = 1.22 \begin {gather*} \operatorname {RootSum} {\left (z^{2} \cdot \left (4 a^{2} c i^{2} - a b^{2} i^{2}\right ) + z \left (4 a c d i - b^{2} d i\right ) + a e^{2} - b d e + c d^{2}, \left ( i \mapsto i \log {\left (e^{h + i x} + \frac {4 i a^{2} c i - i a b^{2} i + a b e + 2 a c d - b^{2} d}{2 a c e - b c d} \right )} \right )\right )} + \frac {d x}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.61, size = 98, normalized size = 1.03 \begin {gather*} \frac {i \, {\left (b d e^{h} - 2 \, a e^{\left (h + 1\right )}\right )} \arctan \left (\frac {2 \, c e^{\left (h + i \, x\right )} + b}{\sqrt {-b^{2} + 4 \, a c}}\right ) e^{\left (-h\right )}}{\sqrt {-b^{2} + 4 \, a c} a} + \frac {i \, d \log \left (c e^{\left (2 \, h + 2 i \, x\right )} + b e^{\left (h + i \, x\right )} + a\right )}{2 \, a} - \frac {i \, d \log \left (e^{\left (i \, x\right )}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.78, size = 91, normalized size = 0.96 \begin {gather*} \frac {d\,x}{a}-\frac {d\,\ln \left (a+b\,{\mathrm {e}}^{i\,x}\,{\mathrm {e}}^h+c\,{\mathrm {e}}^{2\,h}\,{\mathrm {e}}^{2\,i\,x}\right )}{2\,a\,i}+\frac {\mathrm {atan}\left (\frac {b+2\,c\,{\mathrm {e}}^{i\,x}\,{\mathrm {e}}^h}{\sqrt {4\,a\,c-b^2}}\right )\,\left (2\,a\,e-b\,d\right )}{a\,i\,\sqrt {4\,a\,c-b^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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