Optimal. Leaf size=386 \[ \frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (a^2-b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i \left (a^2-b^2\right ) f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i \left (a^2-b^2\right ) f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i b f \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2} \]
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Rubi [A]
time = 0.51, antiderivative size = 386, normalized size of antiderivative = 1.00, number of steps
used = 28, number of rules used = 15, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.469, Rules used =
{4639, 4493, 3377, 2718, 4495, 3855, 4489, 2715, 8, 3798, 2221, 2317, 2438, 4621, 4615}
\begin {gather*} \frac {i f \left (a^2-b^2\right ) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d^2}+\frac {i f \left (a^2-b^2\right ) \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b d^2}+\frac {i b f \text {PolyLog}\left (2,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a^2 b d}-\frac {\left (a^2-b^2\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a^2 b d}+\frac {i \left (a^2-b^2\right ) (e+f x)^2}{2 a^2 b f}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i b (e+f x)^2}{2 a^2 f}-\frac {f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2221
Rule 2317
Rule 2438
Rule 2715
Rule 2718
Rule 3377
Rule 3798
Rule 3855
Rule 4489
Rule 4493
Rule 4495
Rule 4615
Rule 4621
Rule 4639
Rubi steps
\begin {align*} \int \frac {(e+f x) \cos (c+d x) \cot ^2(c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x) \cos (c+d x) \cot ^2(c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cos ^2(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=-\frac {\int (e+f x) \cos (c+d x) \, dx}{a}+\frac {\int (e+f x) \cot (c+d x) \csc (c+d x) \, dx}{a}-\frac {b \int (e+f x) \cos ^2(c+d x) \cot (c+d x) \, dx}{a^2}+\frac {b^2 \int \frac {(e+f x) \cos ^3(c+d x)}{a+b \sin (c+d x)} \, dx}{a^2}\\ &=-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {(e+f x) \sin (c+d x)}{a d}+\frac {\int (e+f x) \cos (c+d x) \, dx}{a}-\frac {b \int (e+f x) \cot (c+d x) \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {(e+f x) \cos (c+d x)}{a+b \sin (c+d x)} \, dx+\frac {f \int \csc (c+d x) \, dx}{a d}+\frac {f \int \sin (c+d x) \, dx}{a d}\\ &=\frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac {f \cos (c+d x)}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}+\frac {(2 i b) \int \frac {e^{2 i (c+d x)} (e+f x)}{1-e^{2 i (c+d x)}} \, dx}{a^2}-\left (1-\frac {b^2}{a^2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{a-\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx-\left (1-\frac {b^2}{a^2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{a+\sqrt {a^2-b^2}-i b e^{i (c+d x)}} \, dx-\frac {f \int \sin (c+d x) \, dx}{a d}\\ &=\frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {(b f) \int \log \left (1-e^{2 i (c+d x)}\right ) \, dx}{a^2 d}+\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right ) \, dx}{b d}+\frac {\left (\left (1-\frac {b^2}{a^2}\right ) f\right ) \int \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right ) \, dx}{b d}\\ &=\frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}-\frac {(i b f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i (c+d x)}\right )}{2 a^2 d^2}-\frac {\left (i \left (1-\frac {b^2}{a^2}\right ) f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a-\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b d^2}-\frac {\left (i \left (1-\frac {b^2}{a^2}\right ) f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i b x}{a+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{b d^2}\\ &=\frac {i b (e+f x)^2}{2 a^2 f}+\frac {i \left (1-\frac {b^2}{a^2}\right ) (e+f x)^2}{2 b f}-\frac {f \tanh ^{-1}(\cos (c+d x))}{a d^2}-\frac {(e+f x) \csc (c+d x)}{a d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d}-\frac {\left (1-\frac {b^2}{a^2}\right ) (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d}-\frac {b (e+f x) \log \left (1-e^{2 i (c+d x)}\right )}{a^2 d}+\frac {i \left (1-\frac {b^2}{a^2}\right ) f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i \left (1-\frac {b^2}{a^2}\right ) f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{b d^2}+\frac {i b f \text {Li}_2\left (e^{2 i (c+d x)}\right )}{2 a^2 d^2}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(2314\) vs. \(2(386)=772\).
time = 13.63, size = 2314, normalized size = 5.99 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1731 vs. \(2 (351 ) = 702\).
time = 0.33, size = 1732, normalized size = 4.49
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1732\) |
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. 1432 vs. \(2 (349) = 698\).
time = 0.63, size = 1432, normalized size = 3.71 \begin {gather*} -\frac {2 \, a b d f x - i \, b^{2} f {\rm Li}_2\left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + i \, b^{2} f {\rm Li}_2\left (\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + i \, b^{2} f {\rm Li}_2\left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) - i \, b^{2} f {\rm Li}_2\left (-\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) \sin \left (d x + c\right ) + 2 \, a b d e - i \, {\left (a^{2} - b^{2}\right )} f {\rm Li}_2\left (\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) \sin \left (d x + c\right ) - i \, {\left (a^{2} - b^{2}\right )} f {\rm Li}_2\left (\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) \sin \left (d x + c\right ) + i \, {\left (a^{2} - b^{2}\right )} f {\rm Li}_2\left (\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) \sin \left (d x + c\right ) + i \, {\left (a^{2} - b^{2}\right )} f {\rm Li}_2\left (\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b} + 1\right ) \sin \left (d x + c\right ) - {\left ({\left (a^{2} - b^{2}\right )} c f - {\left (a^{2} - b^{2}\right )} d e\right )} \log \left (2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) \sin \left (d x + c\right ) - {\left ({\left (a^{2} - b^{2}\right )} c f - {\left (a^{2} - b^{2}\right )} d e\right )} \log \left (2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) \sin \left (d x + c\right ) - {\left ({\left (a^{2} - b^{2}\right )} c f - {\left (a^{2} - b^{2}\right )} d e\right )} \log \left (-2 \, b \cos \left (d x + c\right ) + 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} + 2 i \, a\right ) \sin \left (d x + c\right ) - {\left ({\left (a^{2} - b^{2}\right )} c f - {\left (a^{2} - b^{2}\right )} d e\right )} \log \left (-2 \, b \cos \left (d x + c\right ) - 2 i \, b \sin \left (d x + c\right ) + 2 \, b \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - 2 i \, a\right ) \sin \left (d x + c\right ) + {\left ({\left (a^{2} - b^{2}\right )} d f x + {\left (a^{2} - b^{2}\right )} c f\right )} \log \left (-\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) \sin \left (d x + c\right ) + {\left ({\left (a^{2} - b^{2}\right )} d f x + {\left (a^{2} - b^{2}\right )} c f\right )} \log \left (-\frac {i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) + i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) \sin \left (d x + c\right ) + {\left ({\left (a^{2} - b^{2}\right )} d f x + {\left (a^{2} - b^{2}\right )} c f\right )} \log \left (-\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) + {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) \sin \left (d x + c\right ) + {\left ({\left (a^{2} - b^{2}\right )} d f x + {\left (a^{2} - b^{2}\right )} c f\right )} \log \left (-\frac {-i \, a \cos \left (d x + c\right ) - a \sin \left (d x + c\right ) - {\left (b \cos \left (d x + c\right ) - i \, b \sin \left (d x + c\right )\right )} \sqrt {-\frac {a^{2} - b^{2}}{b^{2}}} - b}{b}\right ) \sin \left (d x + c\right ) + {\left (b^{2} d f x + b^{2} d e + a b f\right )} \log \left (\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (b^{2} d f x + b^{2} d e + a b f\right )} \log \left (\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (b^{2} d e - {\left (b^{2} c + a b\right )} f\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2} i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (b^{2} d e - {\left (b^{2} c + a b\right )} f\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) - \frac {1}{2} i \, \sin \left (d x + c\right ) + \frac {1}{2}\right ) \sin \left (d x + c\right ) + {\left (b^{2} d f x + b^{2} c f\right )} \log \left (-\cos \left (d x + c\right ) + i \, \sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + {\left (b^{2} d f x + b^{2} c f\right )} \log \left (-\cos \left (d x + c\right ) - i \, \sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right )}{2 \, a^{2} b d^{2} \sin \left (d x + c\right )} \end {gather*}
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 \begin {gather*} \int \frac {\left (e + f x\right ) \cos {\left (c + d x \right )} \cot ^{2}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F(-1)]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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