Optimal. Leaf size=321 \[ -\frac {715 i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2048 \sqrt {2} a^{3/2} d}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac {715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}+\frac {715 i}{2048 a d \sqrt {a+i a \tan (c+d x)}} \]
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Rubi [A] time = 0.18, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3487, 51, 63, 206} \[ -\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}-\frac {715 i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2048 \sqrt {2} a^{3/2} d}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac {715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}+\frac {715 i}{2048 a d \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 206
Rule 3487
Rubi steps
\begin {align*} \int \frac {\cos ^6(c+d x)}{(a+i a \tan (c+d x))^{3/2}} \, dx &=-\frac {\left (i a^7\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 (a+x)^{11/2}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {\left (5 i a^6\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 (a+x)^{11/2}} \, dx,x,i a \tan (c+d x)\right )}{4 d}\\ &=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {\left (65 i a^5\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 (a+x)^{11/2}} \, dx,x,i a \tan (c+d x)\right )}{32 d}\\ &=-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}-\frac {\left (715 i a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{11/2}} \, dx,x,i a \tan (c+d x)\right )}{128 d}\\ &=\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}-\frac {\left (715 i a^3\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{9/2}} \, dx,x,i a \tan (c+d x)\right )}{256 d}\\ &=\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}-\frac {\left (715 i a^2\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{7/2}} \, dx,x,i a \tan (c+d x)\right )}{512 d}\\ &=\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}-\frac {(715 i a) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,i a \tan (c+d x)\right )}{1024 d}\\ &=\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac {715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}-\frac {(715 i) \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,i a \tan (c+d x)\right )}{2048 d}\\ &=\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac {715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}+\frac {715 i}{2048 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(715 i) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{4096 a d}\\ &=\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac {715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}+\frac {715 i}{2048 a d \sqrt {a+i a \tan (c+d x)}}-\frac {(715 i) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{2048 a d}\\ &=-\frac {715 i \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{2048 \sqrt {2} a^{3/2} d}+\frac {715 i a^3}{1152 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3 (a+i a \tan (c+d x))^{9/2}}-\frac {5 i a^5}{16 d (a-i a \tan (c+d x))^2 (a+i a \tan (c+d x))^{9/2}}-\frac {65 i a^4}{64 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {715 i a^2}{1792 d (a+i a \tan (c+d x))^{7/2}}+\frac {143 i a}{512 d (a+i a \tan (c+d x))^{5/2}}+\frac {715 i}{3072 d (a+i a \tan (c+d x))^{3/2}}+\frac {715 i}{2048 a d \sqrt {a+i a \tan (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 2.09, size = 203, normalized size = 0.63 \[ \frac {i e^{-8 i (c+d x)} \left (1136 e^{2 i (c+d x)}+5440 e^{4 i (c+d x)}+17344 e^{6 i (c+d x)}+57632 e^{8 i (c+d x)}+33301 e^{10 i (c+d x)}-13209 e^{12 i (c+d x)}-1974 e^{14 i (c+d x)}-168 e^{16 i (c+d x)}-45045 e^{9 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \sinh ^{-1}\left (e^{i (c+d x)}\right )+112\right )}{129024 a d \left (1+e^{2 i (c+d x)}\right ) \sqrt {a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.63, size = 338, normalized size = 1.05 \[ \frac {{\left (-45045 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 45045 i \, \sqrt {\frac {1}{2}} a^{2} d \sqrt {\frac {1}{a^{3} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{3} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-168 i \, e^{\left (16 i \, d x + 16 i \, c\right )} - 1974 i \, e^{\left (14 i \, d x + 14 i \, c\right )} - 13209 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 33301 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 57632 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 17344 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 5440 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 1136 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 112 i\right )}\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{258048 \, a^{2} d} \]
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 \[ \int \frac {\cos \left (d x + c\right )^{6}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.42, size = 422, normalized size = 1.31 \[ \frac {\sqrt {\frac {a \left (i \sin \left (d x +c \right )+\cos \left (d x +c \right )\right )}{\cos \left (d x +c \right )}}\, \left (114688 i \left (\cos ^{10}\left (d x +c \right )\right )+114688 \sin \left (d x +c \right ) \left (\cos ^{9}\left (d x +c \right )\right )+4096 i \left (\cos ^{8}\left (d x +c \right )\right )+61440 \sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )+6656 i \left (\cos ^{6}\left (d x +c \right )\right )+73216 \left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )+13728 i \left (\cos ^{4}\left (d x +c \right )\right )+45045 i \cos \left (d x +c \right ) \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+45045 i \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+96096 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+45045 \arctan \left (\frac {\left (i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )\right ) \sqrt {2}}{2 \sin \left (d x +c \right ) \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}}\right ) \sin \left (d x +c \right ) \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}+60060 i \left (\cos ^{2}\left (d x +c \right )\right )+180180 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{516096 d \,a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.81, size = 261, normalized size = 0.81 \[ \frac {i \, {\left (\frac {45045 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{\sqrt {a}} + \frac {4 \, {\left (45045 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{7} - 240240 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{6} a + 396396 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} a^{2} - 164736 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a^{3} - 36608 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{4} - 19968 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{5} - 15360 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{6} - 14336 \, a^{7}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {15}{2}} - 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {13}{2}} a + 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{2} - 8 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{3}}\right )}}{516096 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^6}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\cos ^{6}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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