Optimal. Leaf size=1523 \[ -\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \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)^3 \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)^3 \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)^3 \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 {6 i b^2 d^2 (c+d x) \text {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \text {PolyLog}\left (2,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {6 b^2 d^3 \text {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \text {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {6 i b^3 d^2 (c+d x) \text {PolyLog}\left (3,-\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^3}+\frac {12 i b d^2 (c+d x) \text {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 i b^3 d^2 (c+d x) \text {PolyLog}\left (3,-\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^3}-\frac {12 i b d^2 (c+d x) \text {PolyLog}\left (3,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 b^3 d^3 \text {PolyLog}\left (4,-\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^4}-\frac {12 b d^3 \text {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}-\frac {6 b^3 d^3 \text {PolyLog}\left (4,-\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^4}+\frac {12 b d^3 \text {PolyLog}\left (4,-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.05, antiderivative size = 1523, normalized size of antiderivative = 1.00, number of steps
used = 36, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4276, 3405,
3402, 2296, 2221, 2611, 6744, 2320, 6724, 4618} \begin {gather*} \frac {(c+d x)^4}{4 a^2 d}+\frac {2 i b \log \left (\frac {e^{i (e+f x)} a}{b-\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \sqrt {b^2-a^2} f}-\frac {i b^3 \log \left (\frac {e^{i (e+f x)} a}{b-\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \left (b^2-a^2\right )^{3/2} f}-\frac {2 i b \log \left (\frac {e^{i (e+f x)} a}{b+\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \sqrt {b^2-a^2} f}+\frac {i b^3 \log \left (\frac {e^{i (e+f x)} a}{b+\sqrt {b^2-a^2}}+1\right ) (c+d x)^3}{a^2 \left (b^2-a^2\right )^{3/2} f}+\frac {b^2 \sin (e+f x) (c+d x)^3}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {3 b^2 d \log \left (\frac {e^{i (e+f x)} a}{b-i \sqrt {a^2-b^2}}+1\right ) (c+d x)^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d \log \left (\frac {e^{i (e+f x)} a}{b+i \sqrt {a^2-b^2}}+1\right ) (c+d x)^2}{a^2 \left (a^2-b^2\right ) f^2}+\frac {6 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \sqrt {b^2-a^2} f^2}-\frac {3 b^3 d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac {6 b d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \sqrt {b^2-a^2} f^2}+\frac {3 b^3 d \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)^2}{a^2 \left (b^2-a^2\right )^{3/2} f^2}-\frac {6 i b^2 d^2 \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right ) (c+d x)}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right ) (c+d x)}{a^2 \left (a^2-b^2\right ) f^3}+\frac {12 i b d^2 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \sqrt {b^2-a^2} f^3}-\frac {6 i b^3 d^2 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \left (b^2-a^2\right )^{3/2} f^3}-\frac {12 i b d^2 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \sqrt {b^2-a^2} f^3}+\frac {6 i b^3 d^2 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right ) (c+d x)}{a^2 \left (b^2-a^2\right )^{3/2} f^3}+\frac {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {12 b d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} f^4}+\frac {6 b^3 d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} f^4}+\frac {12 b d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \sqrt {b^2-a^2} f^4}-\frac {6 b^3 d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {b^2-a^2}}\right )}{a^2 \left (b^2-a^2\right )^{3/2} f^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2221
Rule 2296
Rule 2320
Rule 2611
Rule 3402
Rule 3405
Rule 4276
Rule 4618
Rule 6724
Rule 6744
Rubi steps
\begin {align*} \int \frac {(c+d x)^3}{(a+b \sec (e+f x))^2} \, dx &=\int \left (\frac {(c+d x)^3}{a^2}+\frac {b^2 (c+d x)^3}{a^2 (b+a \cos (e+f x))^2}-\frac {2 b (c+d x)^3}{a^2 (b+a \cos (e+f x))}\right ) \, dx\\ &=\frac {(c+d x)^4}{4 a^2 d}-\frac {(2 b) \int \frac {(c+d x)^3}{b+a \cos (e+f x)} \, dx}{a^2}+\frac {b^2 \int \frac {(c+d x)^3}{(b+a \cos (e+f x))^2} \, dx}{a^2}\\ &=\frac {(c+d x)^4}{4 a^2 d}+\frac {b^2 (c+d x)^3 \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)^3}{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)^3}{b+a \cos (e+f x)} \, dx}{a^2 \left (a^2-b^2\right )}-\frac {\left (3 b^2 d\right ) \int \frac {(c+d x)^2 \sin (e+f x)}{b+a \cos (e+f x)} \, dx}{a \left (a^2-b^2\right ) f}\\ &=-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {b^2 (c+d x)^3 \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)^3}{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)^3}{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)^3}{2 b+2 \sqrt {-a^2+b^2}+2 a e^{i (e+f x)}} \, dx}{a \sqrt {-a^2+b^2}}-\frac {\left (3 b^2 d\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{i b-\sqrt {a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f}-\frac {\left (3 b^2 d\right ) \int \frac {e^{i (e+f x)} (c+d x)^2}{i b+\sqrt {a^2-b^2}+i a e^{i (e+f x)}} \, dx}{a \left (a^2-b^2\right ) f}\\ &=-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {2 i b (c+d x)^3 \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)^3 \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 (c+d x)^3 \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)^3}{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)^3}{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 (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {i a e^{i (e+f x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac {\left (6 b^2 d^2\right ) \int (c+d x) \log \left (1+\frac {i a e^{i (e+f x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^2}-\frac {(6 i b d) \int (c+d x)^2 \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 {(6 i b d) \int (c+d x)^2 \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 {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \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)^3 \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)^3 \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)^3 \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 {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}+\frac {6 b d (c+d x)^2 \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 {6 b d (c+d x)^2 \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)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (6 i b^2 d^3\right ) \int \text {Li}_2\left (-\frac {i a e^{i (e+f x)}}{i b-\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^3}+\frac {\left (6 i b^2 d^3\right ) \int \text {Li}_2\left (-\frac {i a e^{i (e+f x)}}{i b+\sqrt {a^2-b^2}}\right ) \, dx}{a^2 \left (a^2-b^2\right ) f^3}-\frac {\left (12 b d^2\right ) \int (c+d x) \text {Li}_2\left (-\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^2}+\frac {\left (12 b d^2\right ) \int (c+d x) \text {Li}_2\left (-\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^2}+\frac {\left (3 i b^3 d\right ) \int (c+d x)^2 \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 (3 i b^3 d\right ) \int (c+d x)^2 \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 {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \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)^3 \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)^3 \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)^3 \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 {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}-\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (6 b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {i a x}{-i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {\left (6 b^2 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (-\frac {i a x}{i b+\sqrt {a^2-b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {\left (12 i b d^3\right ) \int \text {Li}_3\left (-\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^3}+\frac {\left (12 i b d^3\right ) \int \text {Li}_3\left (-\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^3}+\frac {\left (6 b^3 d^2\right ) \int (c+d x) \text {Li}_2\left (-\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^2}-\frac {\left (6 b^3 d^2\right ) \int (c+d x) \text {Li}_2\left (-\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^2}\\ &=-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \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)^3 \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)^3 \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)^3 \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 {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {6 i b^3 d^2 (c+d x) \text {Li}_3\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^3}+\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 i b^3 d^2 (c+d x) \text {Li}_3\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^3}-\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}-\frac {\left (12 b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {a x}{-b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {\left (12 b d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {a x}{b+\sqrt {-a^2+b^2}}\right )}{x} \, dx,x,e^{i (e+f x)}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {\left (6 i b^3 d^3\right ) \int \text {Li}_3\left (-\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^3}-\frac {\left (6 i b^3 d^3\right ) \int \text {Li}_3\left (-\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^3}\\ &=-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \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)^3 \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)^3 \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)^3 \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 {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {6 i b^3 d^2 (c+d x) \text {Li}_3\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^3}+\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 i b^3 d^2 (c+d x) \text {Li}_3\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^3}-\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}-\frac {12 b d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {12 b d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {b^2 (c+d x)^3 \sin (e+f x)}{a \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {\left (6 b^3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {a x}{-b+\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^4}-\frac {\left (6 b^3 d^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (-\frac {a x}{b+\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^4}\\ &=-\frac {i b^2 (c+d x)^3}{a^2 \left (a^2-b^2\right ) f}+\frac {(c+d x)^4}{4 a^2 d}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}+\frac {3 b^2 d (c+d x)^2 \log \left (1+\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^2}-\frac {i b^3 (c+d x)^3 \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)^3 \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)^3 \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)^3 \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 {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {6 i b^2 d^2 (c+d x) \text {Li}_2\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^3}-\frac {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {3 b^3 d (c+d x)^2 \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 {6 b d (c+d x)^2 \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 {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}+\frac {6 b^2 d^3 \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+i \sqrt {a^2-b^2}}\right )}{a^2 \left (a^2-b^2\right ) f^4}-\frac {6 i b^3 d^2 (c+d x) \text {Li}_3\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^3}+\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 i b^3 d^2 (c+d x) \text {Li}_3\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^3}-\frac {12 i b d^2 (c+d x) \text {Li}_3\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^3}+\frac {6 b^3 d^3 \text {Li}_4\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^4}-\frac {12 b d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b-\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}-\frac {6 b^3 d^3 \text {Li}_4\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^4}+\frac {12 b d^3 \text {Li}_4\left (-\frac {a e^{i (e+f x)}}{b+\sqrt {-a^2+b^2}}\right )}{a^2 \sqrt {-a^2+b^2} f^4}+\frac {b^2 (c+d x)^3 \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 [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(9005\) vs. \(2(1523)=3046\).
time = 27.10, size = 9005, normalized size = 5.91 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [F]
time = 0.31, size = 0, normalized size = 0.00 \[\int \frac {\left (d x +c \right )^{3}}{\left (a +b \sec \left (f x +e \right )\right )^{2}}\, dx\]
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. 7103 vs. \(2 (1385) = 2770\).
time = 3.54, size = 7103, normalized size = 4.66 \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 (c + d x\right )^{3}}{\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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