Optimal. Leaf size=850 \[ \frac {3 e^3 \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)} b^4}{2 a^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {3 e^3 \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)} b^4}{2 a^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac {3 e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac {e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac {7 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} b^2}{a^4 d \sqrt {\sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) e^3 \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)} b^2}{a^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) e^3 \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)} b^2}{a^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac {2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{a^{9/2} d}-\frac {2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{a^{9/2} d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {6 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}} \]
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Rubi [A] time = 2.13, antiderivative size = 850, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {3872, 2912, 2635, 2640, 2639, 2693, 2867, 2701, 2807, 2805, 329, 298, 205, 208, 2695} \[ \frac {3 e^3 \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)} b^4}{2 a^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {3 e^3 \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)} b^4}{2 a^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {3 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac {3 e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b^3}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac {e (e \sin (c+d x))^{3/2} b^2}{a^3 d (b+a \cos (c+d x))}-\frac {7 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)} b^2}{a^4 d \sqrt {\sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) e^3 \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)} b^2}{a^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 \left (a^2-b^2\right ) e^3 \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)} b^2}{a^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2} b}{3 a^3 d}+\frac {2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{a^{9/2} d}-\frac {2 \left (a^2-b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right ) b}{a^{9/2} d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {6 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 205
Rule 208
Rule 298
Rule 329
Rule 2635
Rule 2639
Rule 2640
Rule 2693
Rule 2695
Rule 2701
Rule 2805
Rule 2807
Rule 2867
Rule 2912
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{5/2}}{(a+b \sec (c+d x))^2} \, dx &=\int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{5/2}}{(-b-a \cos (c+d x))^2} \, dx\\ &=\int \left (\frac {(e \sin (c+d x))^{5/2}}{a^2}+\frac {b^2 (e \sin (c+d x))^{5/2}}{a^2 (b+a \cos (c+d x))^2}-\frac {2 b (e \sin (c+d x))^{5/2}}{a^2 (b+a \cos (c+d x))}\right ) \, dx\\ &=\frac {\int (e \sin (c+d x))^{5/2} \, dx}{a^2}-\frac {(2 b) \int \frac {(e \sin (c+d x))^{5/2}}{b+a \cos (c+d x)} \, dx}{a^2}+\frac {b^2 \int \frac {(e \sin (c+d x))^{5/2}}{(b+a \cos (c+d x))^2} \, dx}{a^2}\\ &=\frac {4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (3 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 a^2}+\frac {\left (2 b e^2\right ) \int \frac {(-a-b \cos (c+d x)) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^3}-\frac {\left (3 b^2 e^2\right ) \int \frac {\cos (c+d x) \sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^3}\\ &=\frac {4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (3 b^2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{2 a^4}-\frac {\left (2 b^2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{a^4}+\frac {\left (3 b^3 e^2\right ) \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{2 a^4}-\frac {\left (2 b \left (a^2-b^2\right ) e^2\right ) \int \frac {\sqrt {e \sin (c+d x)}}{b+a \cos (c+d x)} \, dx}{a^4}+\frac {\left (3 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {\sin (c+d x)}}\\ &=\frac {6 e^2 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^2 d \sqrt {\sin (c+d x)}}+\frac {4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (3 b^4 e^3\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^5}+\frac {\left (3 b^4 e^3\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^5}+\frac {\left (b^2 \left (a^2-b^2\right ) e^3\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}-a \sin (c+d x)\right )} \, dx}{a^5}-\frac {\left (b^2 \left (a^2-b^2\right ) e^3\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {a^2-b^2}+a \sin (c+d x)\right )} \, dx}{a^5}-\frac {\left (3 b^3 e^3\right ) \operatorname {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^3 d}+\frac {\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname {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^3 d}-\frac {\left (3 b^2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{2 a^4 \sqrt {\sin (c+d x)}}-\frac {\left (2 b^2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{a^4 \sqrt {\sin (c+d x)}}\\ &=\frac {6 e^2 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^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^4 d \sqrt {\sin (c+d x)}}+\frac {4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}-\frac {\left (3 b^3 e^3\right ) \operatorname {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^3 d}+\frac {\left (4 b \left (a^2-b^2\right ) e^3\right ) \operatorname {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^3 d}-\frac {\left (3 b^4 e^3 \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^5 \sqrt {e \sin (c+d x)}}+\frac {\left (3 b^4 e^3 \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^5 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \left (a^2-b^2\right ) e^3 \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^5 \sqrt {e \sin (c+d x)}}-\frac {\left (b^2 \left (a^2-b^2\right ) e^3 \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^5 \sqrt {e \sin (c+d x)}}\\ &=\frac {3 b^4 e^3 \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^5 \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 ) e^3 \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^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {3 b^4 e^3 \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^5 \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 ) e^3 \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^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {6 e^2 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^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^4 d \sqrt {\sin (c+d x)}}+\frac {4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}+\frac {\left (3 b^3 e^3\right ) \operatorname {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^4 d}-\frac {\left (3 b^3 e^3\right ) \operatorname {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^4 d}-\frac {\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^4 d}+\frac {\left (2 b \left (a^2-b^2\right ) e^3\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^4 d}\\ &=-\frac {3 b^3 e^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}+\frac {2 b \left (a^2-b^2\right )^{3/4} e^{5/2} \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{9/2} d}+\frac {3 b^3 e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{2 a^{9/2} \sqrt [4]{a^2-b^2} d}-\frac {2 b \left (a^2-b^2\right )^{3/4} e^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{9/2} d}+\frac {3 b^4 e^3 \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^5 \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 ) e^3 \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^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {3 b^4 e^3 \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^5 \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 ) e^3 \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^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {6 e^2 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^2 d \sqrt {\sin (c+d x)}}-\frac {7 b^2 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{a^4 d \sqrt {\sin (c+d x)}}+\frac {4 b e (e \sin (c+d x))^{3/2}}{3 a^3 d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}+\frac {b^2 e (e \sin (c+d x))^{3/2}}{a^3 d (b+a \cos (c+d x))}\\ \end {align*}
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Mathematica [C] time = 15.15, size = 886, normalized size = 1.04 \[ \frac {(b+a \cos (c+d x))^2 \csc ^2(c+d x) \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left (\frac {\sin (c+d x) b^2}{a^3 (b+a \cos (c+d x))}+\frac {4 \sin (c+d x) b}{3 a^3}-\frac {\sin (2 (c+d x))}{5 a^2}\right )}{d (a+b \sec (c+d x))^2}-\frac {(b+a \cos (c+d x))^2 \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left (\frac {\left (35 b^2-6 a^2\right ) \left (8 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) a^{5/2}+3 \sqrt {2} b \left (b^2-a^2\right )^{3/4} \left (2 \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}\right )-2 \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{b^2-a^2}}+1\right )-\log \left (a \sin (c+d x)-\sqrt {2} \sqrt {a} \sqrt [4]{b^2-a^2} \sqrt {\sin (c+d x)}+\sqrt {b^2-a^2}\right )+\log \left (a \sin (c+d x)+\sqrt {2} \sqrt {a} \sqrt [4]{b^2-a^2} \sqrt {\sin (c+d x)}+\sqrt {b^2-a^2}\right )\right )\right ) \left (\sqrt {1-\sin ^2(c+d x)} a+b\right ) \cos ^2(c+d x)}{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 {28 a b \left (\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 (b^2-a^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \tan ^{-1}\left (1-\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \tan ^{-1}\left (\frac {(1+i) \sqrt {a} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}+1\right )-\log \left (i a \sin (c+d x)-(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+\sqrt {a^2-b^2}\right )+\log \left (i a \sin (c+d x)+(1+i) \sqrt {a} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+\sqrt {a^2-b^2}\right )\right )}{\sqrt {a} \sqrt [4]{a^2-b^2}}\right ) \left (\sqrt {1-\sin ^2(c+d x)} a+b\right ) \cos (c+d x)}{(b+a \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}\right )}{10 a^3 d (a+b \sec (c+d x))^2 \sin ^{\frac {5}{2}}(c+d x)} \]
Warning: Unable to verify antiderivative.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 16.08, size = 2540, normalized size = 2.99 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}}{{\left (b+a\,\cos \left (c+d\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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