Optimal. Leaf size=524 \[ -\frac {e x}{2 b}+\frac {\left (a^2-b^2\right ) e x}{b^3}-\frac {f x^2}{4 b}+\frac {\left (a^2-b^2\right ) f x^2}{2 b^3}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}+\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a b^2 d}-\frac {f \cos ^2(c+d x)}{4 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {i f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {f \sin (c+d x)}{a d^2}-\frac {\left (a^2-b^2\right ) f \sin (c+d x)}{a b^2 d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d} \]
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Rubi [A]
time = 0.59, antiderivative size = 524, normalized size of antiderivative = 1.00, number of steps
used = 31, number of rules used = 14, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used =
{4639, 4493, 4490, 2713, 3377, 2717, 4268, 2317, 2438, 4621, 3391, 3404, 2296, 2221}
\begin {gather*} \frac {f \left (a^2-b^2\right )^{3/2} \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {f \left (a^2-b^2\right )^{3/2} \text {PolyLog}\left (2,\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a b^3 d^2}+\frac {i f \text {PolyLog}\left (2,-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {PolyLog}\left (2,e^{i (c+d x)}\right )}{a d^2}-\frac {f \left (a^2-b^2\right ) \sin (c+d x)}{a b^2 d^2}+\frac {\left (a^2-b^2\right ) (e+f x) \cos (c+d x)}{a b^2 d}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{\sqrt {a^2-b^2}+a}\right )}{a b^3 d}+\frac {e x \left (a^2-b^2\right )}{b^3}+\frac {f x^2 \left (a^2-b^2\right )}{2 b^3}-\frac {f \sin (c+d x)}{a d^2}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}-\frac {f \cos ^2(c+d x)}{4 b d^2}-\frac {(e+f x) \sin (c+d x) \cos (c+d x)}{2 b d}-\frac {e x}{2 b}-\frac {f x^2}{4 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2713
Rule 2717
Rule 3377
Rule 3391
Rule 3404
Rule 4268
Rule 4490
Rule 4493
Rule 4621
Rule 4639
Rubi steps
\begin {align*} \int \frac {(e+f x) \cos ^3(c+d x) \cot (c+d x)}{a+b \sin (c+d x)} \, dx &=\frac {\int (e+f x) \cos ^3(c+d x) \cot (c+d x) \, dx}{a}-\frac {b \int \frac {(e+f x) \cos ^4(c+d x)}{a+b \sin (c+d x)} \, dx}{a}\\ &=\frac {\int (e+f x) \cos (c+d x) \cot (c+d x) \, dx}{a}-\frac {\int (e+f x) \cos ^2(c+d x) \, dx}{b}+\left (\frac {a}{b}-\frac {b}{a}\right ) \int \frac {(e+f x) \cos ^2(c+d x)}{a+b \sin (c+d x)} \, dx\\ &=-\frac {f \cos ^2(c+d x)}{4 b d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\int (e+f x) \csc (c+d x) \, dx}{a}-\frac {\int (e+f x) \sin (c+d x) \, dx}{a}+\left (\frac {1}{a}-\frac {a}{b^2}\right ) \int (e+f x) \sin (c+d x) \, dx-\frac {\int (e+f x) \, dx}{2 b}+\frac {\left (a^2-b^2\right ) \int (e+f x) \, dx}{b^3}-\frac {\left (a^2-b^2\right )^2 \int \frac {e+f x}{a+b \sin (c+d x)} \, dx}{a b^3}\\ &=-\frac {e x}{2 b}+\frac {\left (a^2-b^2\right ) e x}{b^3}-\frac {f x^2}{4 b}+\frac {\left (a^2-b^2\right ) f x^2}{2 b^3}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x) \cos (c+d x)}{d}-\frac {f \cos ^2(c+d x)}{4 b d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (2 \left (a^2-b^2\right )^2\right ) \int \frac {e^{i (c+d x)} (e+f x)}{i b+2 a e^{i (c+d x)}-i b e^{2 i (c+d x)}} \, dx}{a b^3}-\frac {f \int \cos (c+d x) \, dx}{a d}-\frac {f \int \log \left (1-e^{i (c+d x)}\right ) \, dx}{a d}+\frac {f \int \log \left (1+e^{i (c+d x)}\right ) \, dx}{a d}+\frac {\left (\left (\frac {1}{a}-\frac {a}{b^2}\right ) f\right ) \int \cos (c+d x) \, dx}{d}\\ &=-\frac {e x}{2 b}+\frac {\left (a^2-b^2\right ) e x}{b^3}-\frac {f x^2}{4 b}+\frac {\left (a^2-b^2\right ) f x^2}{2 b^3}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x) \cos (c+d x)}{d}-\frac {f \cos ^2(c+d x)}{4 b d^2}-\frac {f \sin (c+d x)}{a d^2}+\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}+\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a-2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a b^2}-\frac {\left (2 i \left (a^2-b^2\right )^{3/2}\right ) \int \frac {e^{i (c+d x)} (e+f x)}{2 a+2 \sqrt {a^2-b^2}-2 i b e^{i (c+d x)}} \, dx}{a b^2}+\frac {(i f) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}-\frac {(i f) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i (c+d x)}\right )}{a d^2}\\ &=-\frac {e x}{2 b}+\frac {\left (a^2-b^2\right ) e x}{b^3}-\frac {f x^2}{4 b}+\frac {\left (a^2-b^2\right ) f x^2}{2 b^3}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x) \cos (c+d x)}{d}-\frac {f \cos ^2(c+d x)}{4 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {i f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {f \sin (c+d x)}{a d^2}+\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (i \left (a^2-b^2\right )^{3/2} f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a-2 \sqrt {a^2-b^2}}\right ) \, dx}{a b^3 d}+\frac {\left (i \left (a^2-b^2\right )^{3/2} f\right ) \int \log \left (1-\frac {2 i b e^{i (c+d x)}}{2 a+2 \sqrt {a^2-b^2}}\right ) \, dx}{a b^3 d}\\ &=-\frac {e x}{2 b}+\frac {\left (a^2-b^2\right ) e x}{b^3}-\frac {f x^2}{4 b}+\frac {\left (a^2-b^2\right ) f x^2}{2 b^3}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x) \cos (c+d x)}{d}-\frac {f \cos ^2(c+d x)}{4 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {i f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}-\frac {f \sin (c+d x)}{a d^2}+\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}-\frac {\left (\left (a^2-b^2\right )^{3/2} f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a-2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a b^3 d^2}+\frac {\left (\left (a^2-b^2\right )^{3/2} f\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {2 i b x}{2 a+2 \sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (c+d x)}\right )}{a b^3 d^2}\\ &=-\frac {e x}{2 b}+\frac {\left (a^2-b^2\right ) e x}{b^3}-\frac {f x^2}{4 b}+\frac {\left (a^2-b^2\right ) f x^2}{2 b^3}-\frac {2 (e+f x) \tanh ^{-1}\left (e^{i (c+d x)}\right )}{a d}+\frac {(e+f x) \cos (c+d x)}{a d}-\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) (e+f x) \cos (c+d x)}{d}-\frac {f \cos ^2(c+d x)}{4 b d^2}+\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d}-\frac {i \left (a^2-b^2\right )^{3/2} (e+f x) \log \left (1-\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d}+\frac {i f \text {Li}_2\left (-e^{i (c+d x)}\right )}{a d^2}-\frac {i f \text {Li}_2\left (e^{i (c+d x)}\right )}{a d^2}+\frac {\left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a-\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {\left (a^2-b^2\right )^{3/2} f \text {Li}_2\left (\frac {i b e^{i (c+d x)}}{a+\sqrt {a^2-b^2}}\right )}{a b^3 d^2}-\frac {f \sin (c+d x)}{a d^2}+\frac {\left (\frac {1}{a}-\frac {a}{b^2}\right ) f \sin (c+d x)}{d^2}-\frac {(e+f x) \cos (c+d x) \sin (c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A]
time = 9.54, size = 934, normalized size = 1.78 \begin {gather*} -\frac {\left (-2 a^2+3 b^2\right ) (c+d x) (2 d e-2 c f+f (c+d x))}{4 b^3 d^2}+\frac {a (d e-c f+f (c+d x)) \cos (c+d x)}{b^2 d^2}-\frac {f \cos (2 (c+d x))}{8 b d^2}+\frac {e \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d}-\frac {c f \log \left (\tan \left (\frac {1}{2} (c+d x)\right )\right )}{a d^2}+\frac {f \left ((c+d x) \left (\log \left (1-e^{i (c+d x)}\right )-\log \left (1+e^{i (c+d x)}\right )\right )+i \left (\text {Li}_2\left (-e^{i (c+d x)}\right )-\text {Li}_2\left (e^{i (c+d x)}\right )\right )\right )}{a d^2}-\frac {\left (a^2-b^2\right )^2 (d e+d f x) \left (\frac {2 (d e-c f) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{-i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a+i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b+\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )}{a-i \left (b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}+\frac {i f \left (\log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (-\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a-b+\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a \left (i+\tan \left (\frac {1}{2} (c+d x)\right )\right )}{i a-b+\sqrt {-a^2+b^2}}\right )\right )}{\sqrt {-a^2+b^2}}-\frac {i f \left (\log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right ) \log \left (\frac {b-\sqrt {-a^2+b^2}+a \tan \left (\frac {1}{2} (c+d x)\right )}{i a+b-\sqrt {-a^2+b^2}}\right )+\text {Li}_2\left (\frac {a+i a \tan \left (\frac {1}{2} (c+d x)\right )}{a+i \left (-b+\sqrt {-a^2+b^2}\right )}\right )\right )}{\sqrt {-a^2+b^2}}\right )}{a b^3 d^2 \left (d e-c f+i f \log \left (1-i \tan \left (\frac {1}{2} (c+d x)\right )\right )-i f \log \left (1+i \tan \left (\frac {1}{2} (c+d x)\right )\right )\right )}-\frac {a f \sin (c+d x)}{b^2 d^2}-\frac {(d e-c f+f (c+d x)) \sin (2 (c+d x))}{4 b d^2} \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. 1876 vs. \(2 (476 ) = 952\).
time = 1.72, size = 1877, normalized size = 3.58
method | result | size |
risch | \(\text {Expression too large to display}\) | \(1877\) |
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. 1622 vs. \(2 (472) = 944\).
time = 0.68, size = 1622, normalized size = 3.10 \begin {gather*} \text {Too large to display} \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 ^{3}{\left (c + d x \right )} \cot {\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(-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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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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