Optimal. Leaf size=233 \[ -\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{11 i}{64 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{64 \sqrt{2} a^{7/2} d}+\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}} \]
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Rubi [A] time = 0.142192, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3487, 51, 63, 206} \[ -\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{11 i}{64 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{11 i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{64 \sqrt{2} a^{7/2} d}+\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 3487
Rule 51
Rule 63
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx &=-\frac{\left (i a^3\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x)^2 (a+x)^{11/2}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}-\frac{\left (11 i a^2\right ) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{11/2}} \, dx,x,i a \tan (c+d x)\right )}{4 d}\\ &=\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}-\frac{(11 i a) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{8 d}\\ &=\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}-\frac{(11 i) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{7/2}} \, dx,x,i a \tan (c+d x)\right )}{16 d}\\ &=\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}-\frac{(11 i) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{32 a d}\\ &=\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}-\frac{(11 i) \operatorname{Subst}\left (\int \frac{1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{64 a^2 d}\\ &=\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{11 i}{64 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{(11 i) \operatorname{Subst}\left (\int \frac{1}{(a-x) \sqrt{a+x}} \, dx,x,i a \tan (c+d x)\right )}{128 a^3 d}\\ &=\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{11 i}{64 a^3 d \sqrt{a+i a \tan (c+d x)}}-\frac{(11 i) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{64 a^3 d}\\ &=-\frac{11 i \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{64 \sqrt{2} a^{7/2} d}+\frac{11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac{i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac{11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac{11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}+\frac{11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac{11 i}{64 a^3 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 1.51676, size = 176, normalized size = 0.76 \[ -\frac{i e^{-11 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{5/2} \sec ^3(c+d x) \left (\sqrt{1+e^{2 i (c+d x)}} \left (-460 e^{2 i (c+d x)}-1338 e^{4 i (c+d x)}-2416 e^{6 i (c+d x)}-4618 e^{8 i (c+d x)}+315 e^{10 i (c+d x)}-70\right )+3465 e^{9 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{161280 a^3 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.365, size = 422, normalized size = 1.8 \begin{align*}{\frac{1}{80640\,{a}^{4}d}\sqrt{{\frac{a \left ( i\sin \left ( dx+c \right ) +\cos \left ( dx+c \right ) \right ) }{\cos \left ( dx+c \right ) }}} \left ( 71680\,i \left ( \cos \left ( dx+c \right ) \right ) ^{10}+71680\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{9}-43520\,i \left ( \cos \left ( dx+c \right ) \right ) ^{8}-7680\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}+512\,i \left ( \cos \left ( dx+c \right ) \right ) ^{6}+5632\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}\sin \left ( dx+c \right ) +3465\,i\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\arctan \left ({\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i-\sin \left ( dx+c \right ) \right ) }{2\,\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) \cos \left ( dx+c \right ) \sqrt{2}+1056\,i \left ( \cos \left ( dx+c \right ) \right ) ^{4}+3465\,i\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sqrt{2}\arctan \left ({\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i-\sin \left ( dx+c \right ) \right ) }{2\,\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) +3465\,\sqrt{2}\arctan \left ( 1/2\,{\frac{\sqrt{2} \left ( i\cos \left ( dx+c \right ) -i-\sin \left ( dx+c \right ) \right ) }{\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}}}} \right ) \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{\cos \left ( dx+c \right ) +1}}}\sin \left ( dx+c \right ) +7392\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +4620\,i \left ( \cos \left ( dx+c \right ) \right ) ^{2}+13860\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.74869, size = 1010, normalized size = 4.33 \begin{align*} \frac{{\left (-3465 i \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} d^{2}}} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left ({\left (2 \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3465 i \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} d^{2}}} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (-{\left (2 \, \sqrt{\frac{1}{2}} a^{4} d \sqrt{\frac{1}{a^{7} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt{2} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}{\left (-315 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 4303 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 7034 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 3754 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1798 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 530 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 70 i\right )} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-10 i \, d x - 10 i \, c\right )}}{40320 \, a^{4} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{7}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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