Optimal. Leaf size=1070 \[ -\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}+\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}-\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 2.10, antiderivative size = 1070, normalized size of antiderivative = 1.00, number of steps
used = 35, number of rules used = 16, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3957,
2991, 2715, 2721, 2719, 2772, 2944, 2946, 2780, 2886, 2884, 335, 304, 211, 214, 2774}
\begin {gather*} -\frac {2 b^2 \left (a^2-b^2\right )^2 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} e^5}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} e^5}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} e^5}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {\sin (c+d x)} e^5}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 b \left (a^2-b^2\right )^{7/4} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{a^{13/2} d}-\frac {7 b^3 \left (a^2-b^2\right )^{3/4} \text {ArcTan}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{2 a^{13/2} d}-\frac {2 b \left (a^2-b^2\right )^{7/4} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{a^{13/2} d}+\frac {7 b^3 \left (a^2-b^2\right )^{3/4} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) e^{9/2}}{2 a^{13/2} d}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} e^4}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} e^4}{5 a^6 d \sqrt {\sin (c+d x)}}+\frac {14 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} e^4}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {14 \cos (c+d x) (e \sin (c+d x))^{3/2} e^3}{45 a^2 d}-\frac {7 b^2 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}+\frac {4 b \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2} e^3}{15 a^5 d}-\frac {2 \cos (c+d x) (e \sin (c+d x))^{7/2} e}{9 a^2 d}+\frac {4 b (e \sin (c+d x))^{7/2} e}{7 a^3 d}+\frac {b^2 (e \sin (c+d x))^{7/2} e}{a^3 d (b+a \cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 214
Rule 304
Rule 335
Rule 2715
Rule 2719
Rule 2721
Rule 2772
Rule 2774
Rule 2780
Rule 2884
Rule 2886
Rule 2944
Rule 2946
Rule 2991
Rule 3957
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{9/2}}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{9/2}}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac {(e \sin (c+d x))^{9/2}}{a^2}+\frac {b^2 (e \sin (c+d x))^{9/2}}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b (e \sin (c+d x))^{9/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx\\ &=\frac {\int (e \sin (c+d x))^{9/2} \, dx}{a^2}-\frac {(2 b) \int \frac {(e \sin (c+d x))^{9/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac {b^2 \int \frac {(e \sin (c+d x))^{9/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2}\\ &=\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (7 e^2\right ) \int (e \sin (c+d x))^{5/2} \, dx}{9 a^2}+\frac {\left (2 b e^2\right ) \int \frac {(-a-b \cos (c+d x)) (e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac {\left (7 b^2 e^2\right ) \int \frac {\cos (c+d x) (e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{2 a^3}\\ &=-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (7 e^4\right ) \int \sqrt {e \sin (c+d x)} \, dx}{15 a^2}+\frac {\left (4 b e^4\right ) \int \frac {\left (-\frac {1}{2} a \left (5 a^2-2 b^2\right )-\frac {1}{2} b \left (8 a^2-5 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{5 a^5}-\frac {\left (7 b^2 e^4\right ) \int \frac {\left (-a b+\frac {1}{2} \left (3 a^2-5 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{5 a^5}\\ &=-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (7 b^2 \left (3 a^2-5 b^2\right ) e^4\right ) \int \sqrt {e \sin (c+d x)} \, dx}{10 a^6}-\frac {\left (2 b^2 \left (8 a^2-5 b^2\right ) e^4\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 a^6}+\frac {\left (7 b^3 \left (a^2-b^2\right ) e^4\right ) \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^6}-\frac {\left (2 b \left (a^2-b^2\right )^2 e^4\right ) \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^6}+\frac {\left (7 e^4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2 \sqrt {\sin (c+d x)}}\\ &=\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^7}+\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^7}+\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^7}-\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^7}-\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{2 a^5 d}+\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (-a^2+b^2\right ) e^2+a^2 x^2} \, dx,x,e \sin (c+d x)\right )}{a^5 d}-\frac {\left (7 b^2 \left (3 a^2-5 b^2\right ) e^4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{10 a^6 \sqrt {\sin (c+d x)}}-\frac {\left (2 b^2 \left (8 a^2-5 b^2\right ) e^4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a^6 \sqrt {\sin (c+d x)}}\\ &=\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}+\frac {\left (4 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {x^2}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^5 d}-\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{4 a^7 \sqrt {e \sin (c+d x)}}+\frac {\left (7 b^4 \left (a^2-b^2\right ) e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{4 a^7 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^7 \sqrt {e \sin (c+d x)}}-\frac {\left (b^2 \left (a^2-b^2\right )^2 e^5 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^7 \sqrt {e \sin (c+d x)}}\\ &=\frac {7 b^4 \left (a^2-b^2\right ) e^5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{2 a^6 d}-\frac {\left (7 b^3 \left (a^2-b^2\right ) e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{2 a^6 d}-\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^6 d}+\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^6 d}\\ &=-\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}+\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^3 \left (a^2-b^2\right )^{3/4} e^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{13/2} d}-\frac {2 b \left (a^2-b^2\right )^{7/4} e^{9/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{13/2} d}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \Pi \left (\frac {2 a}{a-\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {7 b^4 \left (a^2-b^2\right ) e^5 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{2 a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 b^2 \left (a^2-b^2\right )^2 e^5 \Pi \left (\frac {2 a}{a+\sqrt {a^2-b^2}};\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{a^7 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {14 e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 \left (3 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {4 b^2 \left (8 a^2-5 b^2\right ) e^4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^6 d \sqrt {\sin (c+d x)}}-\frac {14 e^3 \cos (c+d x) (e \sin (c+d x))^{3/2}}{45 a^2 d}-\frac {7 b^2 e^3 (5 b-3 a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e^3 \left (5 \left (a^2-b^2\right )+3 a b \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{15 a^5 d}+\frac {4 b e (e \sin (c+d x))^{7/2}}{7 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{7/2}}{9 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{7/2}}{a^3 d (b+a \cos (c+d x))}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in
optimal.
time = 24.25, size = 974, normalized size = 0.91 \begin {gather*} \frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{9/2} \left (\frac {\left (14 a^4-159 a^2 b^2+165 b^4\right ) \cos ^2(c+d x) \left (3 \sqrt {2} b \left (-a^2+b^2\right )^{3/4} \left (2 \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+\sqrt {2} \sqrt {a} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+a \sin (c+d x)\right )\right )+8 a^{5/2} F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)\right ) \left (b+a \sqrt {1-\sin ^2(c+d x)}\right )}{12 a^{3/2} \left (a^2-b^2\right ) (b+a \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (-46 a^3 b+66 a b^3\right ) \cos (c+d x) \left (\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \text {ArcTan}\left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \text {ArcTan}\left (1+\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+i a \sin (c+d x)\right )\right )}{\sqrt {a} \sqrt [4]{a^2-b^2}}+\frac {b F_1\left (\frac {3}{4};\frac {1}{2},1;\frac {7}{4};\sin ^2(c+d x),\frac {a^2 \sin ^2(c+d x)}{a^2-b^2}\right ) \sin ^{\frac {3}{2}}(c+d x)}{3 \left (-a^2+b^2\right )}\right ) \left (b+a \sqrt {1-\sin ^2(c+d x)}\right )}{(b+a \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{30 a^5 d (a+b \sec (c+d x))^2 \sin ^{\frac {9}{2}}(c+d x)}+\frac {(b+a \cos (c+d x))^2 \csc ^4(c+d x) \sec ^2(c+d x) (e \sin (c+d x))^{9/2} \left (-\frac {b \left (-37 a^2+56 b^2\right ) \sin (c+d x)}{21 a^5}+\frac {a^2 b^2 \sin (c+d x)-b^4 \sin (c+d x)}{a^5 (b+a \cos (c+d x))}-\frac {\left (19 a^2-54 b^2\right ) \sin (2 (c+d x))}{90 a^4}-\frac {b \sin (3 (c+d x))}{7 a^3}+\frac {\sin (4 (c+d x))}{36 a^2}\right )}{d (a+b \sec (c+d x))^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.86, size = 2081, normalized size = 1.94
method | result | size |
default | \(\text {Expression too large to display}\) | \(2081\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [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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Fricas [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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Sympy [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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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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{9/2}}{{\left (b+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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