Optimal. Leaf size=719 \[ \frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}} \]
[Out]
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Rubi [A]
time = 0.58, antiderivative size = 719, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {3596, 3579,
3582, 3580, 3576, 303, 1176, 631, 210, 1179, 642} \begin {gather*} -\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}+\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac {5 i a^{3/2} e^{7/2} \sec (c+d x) \log \left (\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))+a\right )}{16 \sqrt {2} d \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {5 i a \cos ^2(c+d x)}{8 d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}}+\frac {i a}{3 d \sqrt {a+i a \tan (c+d x)} (e \cos (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 303
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3576
Rule 3579
Rule 3580
Rule 3582
Rule 3596
Rubi steps
\begin {align*} \int \frac {\sqrt {a+i a \tan (c+d x)}}{(e \cos (c+d x))^{7/2}} \, dx &=\frac {\int (e \sec (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)} \, dx}{(e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {(5 a) \int \frac {(e \sec (c+d x))^{7/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{6 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 e^2\right ) \int (e \sec (c+d x))^{3/2} \sqrt {a+i a \tan (c+d x)} \, dx}{8 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 a e^2\right ) \int \frac {(e \sec (c+d x))^{3/2}}{\sqrt {a+i a \tan (c+d x)}} \, dx}{16 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 a e^3 \sec (c+d x)\right ) \int \sqrt {e \sec (c+d x)} \sqrt {a-i a \tan (c+d x)} \, dx}{16 (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 i a^2 e^5 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{4 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}-\frac {\left (5 i a^2 e^4 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {a-e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^2 e^4 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {a+e x^2}{a^2+e^2 x^4} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{8 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 i a^2 e^3 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^2 e^3 \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}+x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}+2 x}{-\frac {a}{e}-\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {\frac {\sqrt {2} \sqrt {a}}{\sqrt {e}}-2 x}{-\frac {a}{e}+\frac {\sqrt {2} \sqrt {a} x}{\sqrt {e}}-x^2} \, dx,x,\frac {\sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}\right )}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}+\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {\left (5 i a^{3/2} e^{7/2} \sec (c+d x)\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right )}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}\\ &=\frac {i a}{3 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a \cos ^2(c+d x)}{8 d (e \cos (c+d x))^{7/2} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {a} \sqrt {e \sec (c+d x)}}\right ) \sec (c+d x)}{8 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}+\frac {5 i a^{3/2} e^{7/2} \log \left (a-\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i a^{3/2} e^{7/2} \log \left (a+\frac {\sqrt {2} \sqrt {a} \sqrt {e} \sqrt {a-i a \tan (c+d x)}}{\sqrt {e \sec (c+d x)}}+\cos (c+d x) (a-i a \tan (c+d x))\right ) \sec (c+d x)}{16 \sqrt {2} d (e \cos (c+d x))^{7/2} (e \sec (c+d x))^{7/2} \sqrt {a-i a \tan (c+d x)} \sqrt {a+i a \tan (c+d x)}}-\frac {5 i \cos ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d (e \cos (c+d x))^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 2.96, size = 305, normalized size = 0.42 \begin {gather*} \frac {\sqrt {\cos (c+d x)} \left (-\frac {40}{3} i \cos ^{\frac {3}{2}}(c+d x)+\frac {5}{8} i e^{-\frac {7}{2} i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^3 \sqrt {e^{-i (c+d x)} \left (1+e^{2 i (c+d x)}\right )} \left (2 \text {ArcTan}\left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}\right )+\log \left (1-\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )-\log \left (1+\sqrt {2} e^{\frac {1}{2} i (c+d x)}+e^{i (c+d x)}\right )\right )+\frac {32}{3} \sqrt {\cos (c+d x)} (i \cos (c+d x)+\sin (c+d x))+20 \cos ^{\frac {5}{2}}(c+d x) (i \cos (c+d x)+\sin (c+d x))\right ) \sqrt {a+i a \tan (c+d x)}}{32 d (e \cos (c+d x))^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.10, size = 417, normalized size = 0.58
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \cos \left (d x +c \right ) \left (-1+\cos \left (d x +c \right )\right )^{4} \left (30 i \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-15 i \left (\cos ^{3}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right )}{2}\right )+15 i \left (\cos ^{3}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )+20 i \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-30 \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}+15 \left (\cos ^{3}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1-\sin \left (d x +c \right )\right )}{2}\right )+15 \left (\cos ^{3}\left (d x +c \right )\right ) \arctanh \left (\frac {\sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \left (\cos \left (d x +c \right )+1+\sin \left (d x +c \right )\right )}{2}\right )+16 i \sin \left (d x +c \right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}-10 \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}+4 \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\, \cos \left (d x +c \right )-16 \sqrt {\frac {1}{1+\cos \left (d x +c \right )}}\right )}{48 d \sin \left (d x +c \right )^{7} \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )-1\right ) \left (\frac {1}{1+\cos \left (d x +c \right )}\right )^{\frac {7}{2}} \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}\) | \(417\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than
twice the leaf count of optimal. 2638 vs. \(2 (469) = 938\).
time = 0.85, size = 2638, normalized size = 3.67 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 600, normalized size = 0.83 \begin {gather*} \frac {\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-5 i \, e^{\left (5 i \, d x + 5 i \, c\right )} + 42 i \, e^{\left (3 i \, d x + 3 i \, c\right )} + 15 i \, e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - 6 \, {\left (d e^{\frac {7}{2}} + d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} \sqrt {\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + \frac {8}{5} \, d \sqrt {\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} e^{\frac {7}{2}}\right ) + 6 \, {\left (d e^{\frac {7}{2}} + d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} \sqrt {\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - \frac {8}{5} \, d \sqrt {\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} e^{\frac {7}{2}}\right ) + 6 \, {\left (d e^{\frac {7}{2}} + d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} \sqrt {-\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} + \frac {8}{5} \, d \sqrt {-\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} e^{\frac {7}{2}}\right ) - 6 \, {\left (d e^{\frac {7}{2}} + d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )} \sqrt {-\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} \log \left (\sqrt {2} \sqrt {\frac {1}{2}} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {e^{\left (2 i \, d x + 2 i \, c\right )} + 1} e^{\left (-\frac {1}{2} i \, d x - \frac {1}{2} i \, c\right )} - \frac {8}{5} \, d \sqrt {-\frac {25 i \, a e^{\left (-7\right )}}{64 \, d^{2}}} e^{\frac {7}{2}}\right )}{12 \, {\left (d e^{\frac {7}{2}} + d e^{\left (6 i \, d x + 6 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c + \frac {7}{2}\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c + \frac {7}{2}\right )}\right )}} \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 {\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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