Optimal. Leaf size=740 \[ \frac {a e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}+\frac {a e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {a e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 \sqrt {2} \sqrt {a^2-b^2} e^2 \Pi \left (\frac {b}{a-\sqrt {a^2-b^2}};\left .\text {ArcSin}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )\right |-1\right ) \sqrt {\sin (c+d x)}}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} \sqrt {a^2-b^2} e^2 \Pi \left (\frac {b}{a+\sqrt {a^2-b^2}};\left .\text {ArcSin}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )\right |-1\right ) \sqrt {\sin (c+d x)}}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {e^2 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{b d \sqrt {e \tan (c+d x)}} \]
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Rubi [A]
time = 0.67, antiderivative size = 740, normalized size of antiderivative = 1.00, number of steps
used = 35, number of rules used = 19, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3976, 3969,
3557, 335, 217, 1179, 642, 1176, 631, 210, 2694, 2653, 2720, 3977, 2812, 2808, 2986, 1227, 551}
\begin {gather*} -\frac {2 \sqrt {2} e^2 \sqrt {a^2-b^2} \sqrt {\sin (c+d x)} \Pi \left (\frac {b}{a-\sqrt {a^2-b^2}};\left .\text {ArcSin}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}\right )\right |-1\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} e^2 \sqrt {a^2-b^2} \sqrt {\sin (c+d x)} \Pi \left (\frac {b}{a+\sqrt {a^2-b^2}};\left .\text {ArcSin}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}\right )\right |-1\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}-\frac {e^{3/2} \left (a^2-b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}+\frac {e^{3/2} \left (a^2-b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a b^2 d}-\frac {e^{3/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}+\frac {e^{3/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}+\frac {a e^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {a e^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} b^2 d}+\frac {a e^{3/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}-\frac {a e^{3/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}+\frac {e^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{b d \sqrt {e \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 335
Rule 551
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1227
Rule 2653
Rule 2694
Rule 2720
Rule 2808
Rule 2812
Rule 2986
Rule 3557
Rule 3969
Rule 3976
Rule 3977
Rubi steps
\begin {align*} \int \frac {(e \tan (c+d x))^{3/2}}{a+b \sec (c+d x)} \, dx &=-\frac {e^2 \int \frac {a-b \sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{b^2}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \frac {1}{(a+b \sec (c+d x)) \sqrt {e \tan (c+d x)}} \, dx}{b^2}\\ &=-\frac {\left (a e^2\right ) \int \frac {1}{\sqrt {e \tan (c+d x)}} \, dx}{b^2}+\frac {e^2 \int \frac {\sec (c+d x)}{\sqrt {e \tan (c+d x)}} \, dx}{b}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \frac {1}{\sqrt {e \tan (c+d x)}} \, dx}{a b^2}-\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \frac {1}{(b+a \cos (c+d x)) \sqrt {e \tan (c+d x)}} \, dx}{a b}\\ &=-\frac {\left (a e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \tan (c+d x)\right )}{b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (e^2+x^2\right )} \, dx,x,e \tan (c+d x)\right )}{a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^2\right ) \int \frac {\sqrt {e \cot (c+d x)}}{b+a \cos (c+d x)} \, dx}{a b \sqrt {e \cot (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {\left (e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \sqrt {\sin (c+d x)}} \, dx}{b \sqrt {\cos (c+d x)} \sqrt {e \tan (c+d x)}}\\ &=-\frac {\left (2 a e^3\right ) \text {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{b^2 d}+\frac {\left (2 \left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {\sqrt {-\cos (c+d x)}}{(b+a \cos (c+d x)) \sqrt {\sin (c+d x)}} \, dx}{a b \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {\left (e^2 \sec (c+d x) \sqrt {\sin (2 c+2 d x)}\right ) \int \frac {1}{\sqrt {\sin (2 c+2 d x)}} \, dx}{b \sqrt {e \tan (c+d x)}}\\ &=\frac {e^2 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{b d \sqrt {e \tan (c+d x)}}-\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{b^2 d}-\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{a b^2 d}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) \left (1-\frac {a}{\sqrt {a^2-b^2}}\right ) e^2 \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-a+\sqrt {a^2-b^2}+b x^2\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) \left (1+\frac {a}{\sqrt {a^2-b^2}}\right ) e^2 \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\left (-a-\sqrt {a^2-b^2}+b x^2\right ) \sqrt {1-x^4}} \, dx,x,\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}\\ &=\frac {e^2 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{b d \sqrt {e \tan (c+d x)}}+\frac {\left (a e^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a e^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^{3/2}\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 b^2 d}-\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^2\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \tan (c+d x)}\right )}{2 a b^2 d}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) \left (1-\frac {a}{\sqrt {a^2-b^2}}\right ) e^2 \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (-a+\sqrt {a^2-b^2}+b x^2\right )} \, dx,x,\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}-\frac {\left (2 \sqrt {2} \left (a^2-b^2\right ) \left (1+\frac {a}{\sqrt {a^2-b^2}}\right ) e^2 \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+x^2} \left (-a-\sqrt {a^2-b^2}+b x^2\right )} \, dx,x,\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}\\ &=\frac {a e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {a e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 \sqrt {2} \sqrt {a^2-b^2} e^2 \Pi \left (\frac {b}{a-\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )\right |-1\right ) \sqrt {\sin (c+d x)}}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} \sqrt {a^2-b^2} e^2 \Pi \left (\frac {b}{a+\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )\right |-1\right ) \sqrt {\sin (c+d x)}}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {e^2 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{b d \sqrt {e \tan (c+d x)}}-\frac {\left (a e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (a e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (\left (a^2-b^2\right ) e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}\\ &=\frac {a e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}-\frac {a e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}+\frac {a e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}-\frac {\left (a^2-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {a e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} b^2 d}+\frac {\left (a^2-b^2\right ) e^{3/2} \log \left (\sqrt {e}+\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 \sqrt {2} \sqrt {a^2-b^2} e^2 \Pi \left (\frac {b}{a-\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )\right |-1\right ) \sqrt {\sin (c+d x)}}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} \sqrt {a^2-b^2} e^2 \Pi \left (\frac {b}{a+\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {1+\sin (c+d x)}}\right )\right |-1\right ) \sqrt {\sin (c+d x)}}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {e^2 F\left (\left .c-\frac {\pi }{4}+d x\right |2\right ) \sec (c+d x) \sqrt {\sin (2 c+2 d x)}}{b d \sqrt {e \tan (c+d x)}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 17.20, size = 202, normalized size = 0.27 \begin {gather*} \frac {2 e \cot \left (\frac {1}{2} (c+d x)\right ) \left (\Pi \left (-i;\left .\text {ArcSin}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )+\Pi \left (i;\left .\text {ArcSin}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )-\Pi \left (-\frac {\sqrt {a-b}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {a-b}}{\sqrt {a+b}};\left .\text {ArcSin}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )\right ) \sqrt {\frac {-1+\cos (c+d x)-\sin (c+d x)}{1+\cos (c+d x)}} \sqrt {1-\tan \left (\frac {1}{2} (c+d x)\right )} \sqrt {e \tan (c+d x)}}{a d} \end {gather*}
Antiderivative was successfully verified.
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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. 1829 vs. \(2 (636 ) = 1272\).
time = 0.24, size = 1830, normalized size = 2.47
method | result | size |
default | \(\text {Expression too large to display}\) | \(1830\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (e \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}{a + b \sec {\left (c + d x \right )}}\, 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]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{b+a\,\cos \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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