Optimal. Leaf size=582 \[ \frac {(c+d x)^2}{2 a^2 d}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}-\frac {b^3 d \text {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {2 b d \text {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^3 d \text {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {2 b d \text {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \]
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Rubi [A]
time = 0.73, antiderivative size = 582, normalized size of antiderivative = 1.00, number of steps
used = 21, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3405,
3402, 2296, 2221, 2317, 2438, 2747, 31} \begin {gather*} \frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f \sqrt {b^2-a^2}}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{\sqrt {b^2-a^2}+b}\right )}{a^2 f \sqrt {b^2-a^2}}+\frac {b^2 (c+d x) \sin (e+f x)}{a f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {2 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f^2 \sqrt {b^2-a^2}}-\frac {2 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 f^2 \sqrt {b^2-a^2}}+\frac {b^2 d \log (a \cos (e+f x)+b)}{a^2 f^2 \left (a^2-b^2\right )}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{\sqrt {b^2-a^2}+b}\right )}{a^2 f \left (b^2-a^2\right )^{3/2}}-\frac {b^3 d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac {b^3 d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 f^2 \left (b^2-a^2\right )^{3/2}}+\frac {(c+d x)^2}{2 a^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 2221
Rule 2296
Rule 2317
Rule 2438
Rule 2747
Rule 3402
Rule 3405
Rule 4276
Rubi steps
\begin {align*} \int \frac {c+d x}{(a+b \sec (e+f x))^2} \, dx &=\int \left (\frac {c+d x}{a^2}+\frac {b^2 (c+d x)}{a^2 (b+a \cos (e+f x))^2}-\frac {2 b (c+d x)}{a^2 (b+a \cos (e+f x))}\right ) \, dx\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {(2 b) \int \frac {c+d x}{b+a \cos (e+f x)} \, dx}{a^2}+\frac {b^2 \int \frac {c+d x}{(b+a \cos (e+f x))^2} \, dx}{a^2}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2}-\frac {b^3 \int \frac {c+d x}{b+a \cos (e+f x)} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {\left (b^2 d\right ) \int \frac {\sin (e+f x)}{b+a \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) f}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)}{a+2 b e^{i (e+f x)}+a e^{2 i (e+f x)}} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}+\frac {(4 b) \int \frac {e^{i (e+f x)} (c+d x)}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}+\frac {\left (b^2 d\right ) \text {Subst}\left (\int \frac {1}{b+x} \, dx,x,a \cos (e+f x)\right )}{a^2 \left (a^2-b^2\right ) f^2}\\ &=\frac {(c+d x)^2}{2 a^2 d}+\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)}{2 b-2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac {\left (2 b^3\right ) \int \frac {e^{i (e+f x)} (c+d x)}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \left (-a^2+b^2\right )^{3/2}}-\frac {(2 i b d) \int \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f}+\frac {(2 i b d) \int \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \sqrt {-a^2+b^2} f}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {(2 b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {(2 b d) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {\left (i b^3 d\right ) \int \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b-2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {\left (i b^3 d\right ) \int \log \left (1+\frac {2 a e^{i (e+f x)}}{2 b+2 \sqrt {-a^2+b^2}}\right ) \, dx}{a^2 \left (-a^2+b^2\right )^{3/2} f}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}-\frac {2 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (b^3 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b-2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {\left (b^3 d\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {2 a x}{2 b+2 \sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}\\ &=\frac {(c+d x)^2}{2 a^2 d}-\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}+\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {i b^3 (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f}-\frac {2 i b (c+d x) \log \left (1+\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f}+\frac {b^2 d \log (b+a \cos (e+f x))}{a^2 \left (a^2-b^2\right ) f^2}-\frac {b^3 d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}+\frac {2 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^3 d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \left (-a^2+b^2\right )^{3/2} f^2}-\frac {2 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^2}+\frac {b^2 (c+d x) \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 9.80, size = 1037, normalized size = 1.78 \begin {gather*} \frac {(e+f x) (-2 d e+2 c f+d (e+f x)) (b+a \cos (e+f x))^2 \sec ^2(e+f x)}{2 a^2 f^2 (a+b \sec (e+f x))^2}+\frac {(b+a \cos (e+f x)) \sec ^2(e+f x) \left (b^2 d e \sin (e+f x)-b^2 c f \sin (e+f x)-b^2 d (e+f x) \sin (e+f x)\right )}{a (-a+b) (a+b) f^2 (a+b \sec (e+f x))^2}+\frac {b \cos ^2\left (\frac {1}{2} (e+f x)\right ) (b+a \cos (e+f x)) \left (-\frac {2 \left (2 a^2-b^2\right ) (d e-c f) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {-a-b}}\right )}{\sqrt {-a-b} \sqrt {a-b}}-b d \log \left (\sec ^2\left (\frac {1}{2} (e+f x)\right )\right )+b d \log \left (-\left ((b+a \cos (e+f x)) \sec ^2\left (\frac {1}{2} (e+f x)\right )\right )\right )-\frac {i \left (2 a^2-b^2\right ) d \left (\log \left (1+i \tan \left (\frac {1}{2} (e+f x)\right )\right ) \log \left (\frac {i \left (\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )-\log \left (1-i \tan \left (\frac {1}{2} (e+f x)\right )\right ) \log \left (\frac {\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{i \sqrt {a-b}+\sqrt {a+b}}\right )+\log \left (1-i \tan \left (\frac {1}{2} (e+f x)\right )\right ) \log \left (\frac {i \left (\sqrt {a+b}+\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )-\log \left (1+i \tan \left (\frac {1}{2} (e+f x)\right )\right ) \log \left (\frac {\sqrt {a+b}+\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{i \sqrt {a-b}+\sqrt {a+b}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1-i \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a-b}-i \sqrt {a+b}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1-i \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )-\text {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1+i \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a-b}-i \sqrt {a+b}}\right )+\text {PolyLog}\left (2,\frac {\sqrt {a-b} \left (1+i \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a-b}+i \sqrt {a+b}}\right )\right )}{\sqrt {a-b} \sqrt {a+b}}\right ) \sec ^2(e+f x) \left (\left (2 a^2-b^2\right ) (c f+d f x)+a b d \sin (e+f x)\right ) \left (\sqrt {a+b}-\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )\right ) \left (\sqrt {a+b}+\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )\right )}{a^2 \left (a^2-b^2\right ) f^2 (a+b \sec (e+f x))^2 \left (-\left (\left (2 a^2-b^2\right ) \left (d e-c f-i d \log \left (1-i \tan \left (\frac {1}{2} (e+f x)\right )\right )+i d \log \left (1+i \tan \left (\frac {1}{2} (e+f x)\right )\right )\right )\right )+a b d \sin (e+f x)\right )} \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. 1288 vs. \(2 (528 ) = 1056\).
time = 0.72, size = 1289, normalized size = 2.21
method | result | size |
risch | \(\frac {d \,x^{2}}{2 a^{2}}+\frac {c x}{a^{2}}+\frac {i b^{3} d \ln \left (\frac {a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}\right ) e}{a^{2} \left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}+\frac {2 i b^{2} \left (d x +c \right ) \left (b \,{\mathrm e}^{i \left (f x +e \right )}+a \right )}{a^{2} \left (a^{2}-b^{2}\right ) f \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{i \left (f x +e \right )}+a \right )}-\frac {2 b^{2} d \ln \left ({\mathrm e}^{i \left (f x +e \right )}\right )}{a^{2} \left (a^{2}-b^{2}\right ) f^{2}}+\frac {b^{2} d \ln \left (a \,{\mathrm e}^{2 i \left (f x +e \right )}+2 b \,{\mathrm e}^{i \left (f x +e \right )}+a \right )}{a^{2} \left (a^{2}-b^{2}\right ) f^{2}}-\frac {4 i b d e \arctan \left (\frac {2 a \,{\mathrm e}^{i \left (f x +e \right )}+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}} f^{2}}-\frac {i b^{3} d \ln \left (\frac {-a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}-b}{-b +\sqrt {-a^{2}+b^{2}}}\right ) x}{a^{2} \left (a^{2}-b^{2}\right ) f \sqrt {-a^{2}+b^{2}}}-\frac {2 i b d \ln \left (\frac {a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}\right ) e}{\left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}+\frac {2 i b d \ln \left (\frac {-a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}-b}{-b +\sqrt {-a^{2}+b^{2}}}\right ) x}{\left (a^{2}-b^{2}\right ) f \sqrt {-a^{2}+b^{2}}}-\frac {i b^{3} d \ln \left (\frac {-a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}-b}{-b +\sqrt {-a^{2}+b^{2}}}\right ) e}{a^{2} \left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 i b^{3} c \arctan \left (\frac {2 a \,{\mathrm e}^{i \left (f x +e \right )}+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}} f}-\frac {2 i b d \ln \left (\frac {a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}\right ) x}{\left (a^{2}-b^{2}\right ) f \sqrt {-a^{2}+b^{2}}}+\frac {2 b d \dilog \left (\frac {-a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}-b}{-b +\sqrt {-a^{2}+b^{2}}}\right )}{\left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}-\frac {2 b d \dilog \left (\frac {a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}\right )}{\left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}+\frac {2 i b^{3} d e \arctan \left (\frac {2 a \,{\mathrm e}^{i \left (f x +e \right )}+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{2} \left (a^{2}-b^{2}\right )^{\frac {3}{2}} f^{2}}+\frac {2 i b d \ln \left (\frac {-a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}-b}{-b +\sqrt {-a^{2}+b^{2}}}\right ) e}{\left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}+\frac {4 i b c \arctan \left (\frac {2 a \,{\mathrm e}^{i \left (f x +e \right )}+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}} f}+\frac {i b^{3} d \ln \left (\frac {a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}\right ) x}{a^{2} \left (a^{2}-b^{2}\right ) f \sqrt {-a^{2}+b^{2}}}-\frac {b^{3} d \dilog \left (\frac {-a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}-b}{-b +\sqrt {-a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}+\frac {b^{3} d \dilog \left (\frac {a \,{\mathrm e}^{i \left (f x +e \right )}+\sqrt {-a^{2}+b^{2}}+b}{b +\sqrt {-a^{2}+b^{2}}}\right )}{a^{2} \left (a^{2}-b^{2}\right ) f^{2} \sqrt {-a^{2}+b^{2}}}\) | \(1289\) |
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. 2155 vs. \(2 (531) = 1062\).
time = 3.61, size = 2155, normalized size = 3.70 \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 {c + d x}{\left (a + b \sec {\left (e + f x \right )}\right )^{2}}\, 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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