Optimal. Leaf size=740 \[ -\frac {e^{3/2} \left (a^2-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}+\frac {e^{3/2} \left (a^2-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a b^2 d}-\frac {e^{3/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}+\frac {e^{3/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 \sqrt {2} e^2 \sqrt {a^2-b^2} \sqrt {\sin (c+d x)} \Pi \left (\frac {b}{a-\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}\right )\right |-1\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} e^2 \sqrt {a^2-b^2} \sqrt {\sin (c+d x)} \Pi \left (\frac {b}{a+\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}\right )\right |-1\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {a e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {a e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} b^2 d}+\frac {a e^{3/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}-\frac {a e^{3/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}+\frac {e^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{b d \sqrt {e \tan (c+d x)}} \]
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Rubi [A] time = 1.01, 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 = {3891, 3884, 3476, 329, 211, 1165, 628, 1162, 617, 204, 2614, 2573, 2641, 3892, 2733, 2729, 2907, 1213, 537} \[ -\frac {e^{3/2} \left (a^2-b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} a b^2 d}+\frac {e^{3/2} \left (a^2-b^2\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} a b^2 d}-\frac {e^{3/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}+\frac {e^{3/2} \left (a^2-b^2\right ) \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} a b^2 d}-\frac {2 \sqrt {2} e^2 \sqrt {a^2-b^2} \sqrt {\sin (c+d x)} \Pi \left (\frac {b}{a-\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}\right )\right |-1\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {2 \sqrt {2} e^2 \sqrt {a^2-b^2} \sqrt {\sin (c+d x)} \Pi \left (\frac {b}{a+\sqrt {a^2-b^2}};\left .\sin ^{-1}\left (\frac {\sqrt {-\cos (c+d x)}}{\sqrt {\sin (c+d x)+1}}\right )\right |-1\right )}{a b d \sqrt {-\cos (c+d x)} \sqrt {e \tan (c+d x)}}+\frac {a e^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} b^2 d}-\frac {a e^{3/2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {e \tan (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} b^2 d}+\frac {a e^{3/2} \log \left (\sqrt {e} \tan (c+d x)-\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}-\frac {a e^{3/2} \log \left (\sqrt {e} \tan (c+d x)+\sqrt {2} \sqrt {e \tan (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} b^2 d}+\frac {e^2 \sqrt {\sin (2 c+2 d x)} \sec (c+d x) F\left (\left .c+d x-\frac {\pi }{4}\right |2\right )}{b d \sqrt {e \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 211
Rule 329
Rule 537
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1213
Rule 2573
Rule 2614
Rule 2641
Rule 2729
Rule 2733
Rule 2907
Rule 3476
Rule 3884
Rule 3891
Rule 3892
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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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] time = 6.91, size = 202, normalized size = 0.27 \[ \frac {2 e \sqrt {1-\tan \left (\frac {1}{2} (c+d x)\right )} \cot \left (\frac {1}{2} (c+d x)\right ) \sqrt {e \tan (c+d x)} \sqrt {\frac {-\sin (c+d x)+\cos (c+d x)-1}{\cos (c+d x)+1}} \left (-\Pi \left (-\frac {\sqrt {a-b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )-\Pi \left (\frac {\sqrt {a-b}}{\sqrt {a+b}};\left .\sin ^{-1}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )+\Pi \left (-i;\left .\sin ^{-1}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )+\Pi \left (i;\left .\sin ^{-1}\left (\sqrt {-\tan \left (\frac {1}{2} (c+d x)\right )}\right )\right |-1\right )\right )}{a d} \]
Antiderivative was successfully verified.
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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 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {3}{2}}}{b \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.64, size = 1801, normalized size = 2.43 \[ \text {result too large to display} \]
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 \[ \int \frac {\left (e \tan \left (d x + c\right )\right )^{\frac {3}{2}}}{b \sec \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\mathrm {tan}\left (c+d\,x\right )\right )}^{3/2}}{b+a\,\cos \left (c+d\,x\right )} \,d x \]
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 \[ \int \frac {\left (e \tan {\left (c + d x \right )}\right )^{\frac {3}{2}}}{a + b \sec {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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