Optimal. Leaf size=210 \[ -\frac {35 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}+\frac {35 i a^3}{128 d \sqrt {a+i a \tan (c+d x)}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A]
time = 0.10, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {3568, 44, 53,
65, 212} \begin {gather*} -\frac {35 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}+\frac {35 i a^3}{128 d \sqrt {a+i a \tan (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 44
Rule 53
Rule 65
Rule 212
Rule 3568
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{5/2} \, dx &=-\frac {\left (i a^7\right ) \text {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {\left (7 i a^6\right ) \text {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{12 d}\\ &=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {\left (35 i a^5\right ) \text {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{96 d}\\ &=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}-\frac {\left (35 i a^4\right ) \text {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{128 d}\\ &=\frac {35 i a^3}{128 d \sqrt {a+i a \tan (c+d x)}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}-\frac {\left (35 i a^3\right ) \text {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{256 d}\\ &=\frac {35 i a^3}{128 d \sqrt {a+i a \tan (c+d x)}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}-\frac {\left (35 i a^3\right ) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{128 d}\\ &=-\frac {35 i a^{5/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{128 \sqrt {2} d}+\frac {35 i a^3}{128 d \sqrt {a+i a \tan (c+d x)}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 \sqrt {a+i a \tan (c+d x)}}-\frac {7 i a^5}{48 d (a-i a \tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}}-\frac {35 i a^4}{192 d (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 1.01, size = 142, normalized size = 0.68 \begin {gather*} -\frac {i a^2 e^{-2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (\sqrt {1+e^{2 i (c+d x)}} \left (-48+87 e^{2 i (c+d x)}+38 e^{4 i (c+d x)}+8 e^{6 i (c+d x)}\right )+105 e^{i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{768 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. 1087 vs. \(2 (170 ) = 340\).
time = 1.07, size = 1088, normalized size = 5.18
method | result | size |
default | \(\text {Expression too large to display}\) | \(1088\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 194, normalized size = 0.92 \begin {gather*} \frac {i \, {\left (105 \, \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 (105 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{4} - 560 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{5} + 924 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{6} - 384 \, a^{7}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}} - 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a + 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{2} - 8 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{3}}\right )}}{1536 \, a d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 309, normalized size = 1.47 \begin {gather*} -\frac {{\left (105 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} \log \left (\frac {4 \, {\left (a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + 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 )}}{a^{2}}\right ) - 105 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} d e^{\left (i \, d x + i \, c\right )} \log \left (\frac {4 \, {\left (a^{3} e^{\left (i \, d x + i \, c\right )} - \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{5}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - 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 )}}{a^{2}}\right ) - \sqrt {2} {\left (-8 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 46 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 125 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} - 39 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 48 i \, a^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{768 \, d} \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(-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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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