Optimal. Leaf size=137 \[ -\frac {3 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}-\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac {3 i a^3 \sqrt {a+i a \tan (c+d x)}}{16 d (a-i a \tan (c+d x))} \]
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Rubi [A] time = 0.09, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3487, 51, 63, 206} \[ -\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac {3 i a^3 \sqrt {a+i a \tan (c+d x)}}{16 d (a-i a \tan (c+d x))}-\frac {3 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 3487
Rubi steps
\begin {align*} \int \cos ^4(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac {\left (i a^5\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac {\left (3 i a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{8 d}\\ &=-\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac {3 i a^3 \sqrt {a+i a \tan (c+d x)}}{16 d (a-i a \tan (c+d x))}-\frac {\left (3 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{32 d}\\ &=-\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac {3 i a^3 \sqrt {a+i a \tan (c+d x)}}{16 d (a-i a \tan (c+d x))}-\frac {\left (3 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{16 d}\\ &=-\frac {3 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{16 \sqrt {2} d}-\frac {i a^4 \sqrt {a+i a \tan (c+d x)}}{4 d (a-i a \tan (c+d x))^2}-\frac {3 i a^3 \sqrt {a+i a \tan (c+d x)}}{16 d (a-i a \tan (c+d x))}\\ \end {align*}
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Mathematica [A] time = 0.84, size = 116, normalized size = 0.85 \[ -\frac {i a^2 e^{-i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (5+2 e^{2 i (c+d x)}\right )+3 \sinh ^{-1}\left (e^{i (c+d x)}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{32 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 262, normalized size = 1.91 \[ \frac {3 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d \log \left (\frac {{\left (32 \, a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (32 i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + 32 i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{8 \, a^{2}}\right ) - 3 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d \log \left (\frac {{\left (32 \, a^{3} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (-32 i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - 32 i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{8 \, a^{2}}\right ) + \sqrt {2} {\left (-2 i \, a^{2} e^{\left (5 i \, d x + 5 i \, c\right )} - 7 i \, a^{2} e^{\left (3 i \, d x + 3 i \, c\right )} - 5 i \, a^{2} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{32 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [B] time = 1.34, size = 744, normalized size = 5.43 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.62, size = 140, normalized size = 1.02 \[ \frac {i \, {\left (3 \, \sqrt {2} a^{\frac {7}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{4} - 10 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} - 4 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a + 4 \, a^{2}}\right )}}{64 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^4\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]
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.
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