Optimal. Leaf size=181 \[ -\frac {5 i a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}-\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {5 i a^5 \sqrt {a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac {5 i a^4 \sqrt {a+i a \tan (c+d x)}}{64 d (a-i a \tan (c+d x))} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3487, 51, 63, 206} \[ -\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {5 i a^5 \sqrt {a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac {5 i a^4 \sqrt {a+i a \tan (c+d x)}}{64 d (a-i a \tan (c+d x))}-\frac {5 i a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 206
Rule 3487
Rubi steps
\begin {align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x))^{7/2} \, dx &=-\frac {\left (i a^7\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^4 \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{d}\\ &=-\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {\left (5 i a^6\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^3 \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{12 d}\\ &=-\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {5 i a^5 \sqrt {a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac {\left (5 i a^5\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x)^2 \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{32 d}\\ &=-\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {5 i a^5 \sqrt {a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac {5 i a^4 \sqrt {a+i a \tan (c+d x)}}{64 d (a-i a \tan (c+d x))}-\frac {\left (5 i a^4\right ) \operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,i a \tan (c+d x)\right )}{128 d}\\ &=-\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {5 i a^5 \sqrt {a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac {5 i a^4 \sqrt {a+i a \tan (c+d x)}}{64 d (a-i a \tan (c+d x))}-\frac {\left (5 i a^4\right ) \operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{64 d}\\ &=-\frac {5 i a^{7/2} \tanh ^{-1}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} d}-\frac {i a^6 \sqrt {a+i a \tan (c+d x)}}{6 d (a-i a \tan (c+d x))^3}-\frac {5 i a^5 \sqrt {a+i a \tan (c+d x)}}{48 d (a-i a \tan (c+d x))^2}-\frac {5 i a^4 \sqrt {a+i a \tan (c+d x)}}{64 d (a-i a \tan (c+d x))}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 1.11, size = 129, normalized size = 0.71 \[ -\frac {i a^3 e^{-i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \left (26 e^{2 i (c+d x)}+8 e^{4 i (c+d x)}+33\right )+15 \sinh ^{-1}\left (e^{i (c+d x)}\right )\right ) \sqrt {a+i a \tan (c+d x)}}{384 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.89, size = 276, normalized size = 1.52 \[ \frac {15 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d \log \left (\frac {{\left (128 \, a^{4} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (128 i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + 128 i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, a^{3}}\right ) - 15 \, \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} d \log \left (\frac {{\left (128 \, a^{4} e^{\left (i \, d x + i \, c\right )} + \sqrt {2} \sqrt {\frac {1}{2}} \sqrt {-\frac {a^{7}}{d^{2}}} {\left (-128 i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - 128 i \, d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-i \, d x - i \, c\right )}}{32 \, a^{3}}\right ) + \sqrt {2} {\left (-8 i \, a^{3} e^{\left (7 i \, d x + 7 i \, c\right )} - 34 i \, a^{3} e^{\left (5 i \, d x + 5 i \, c\right )} - 59 i \, a^{3} e^{\left (3 i \, d x + 3 i \, c\right )} - 33 i \, a^{3} e^{\left (i \, d x + i \, c\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{384 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 1.63, size = 1088, normalized size = 6.01 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.60, size = 176, normalized size = 0.97 \[ \frac {i \, {\left (15 \, \sqrt {2} a^{\frac {9}{2}} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right ) + \frac {4 \, {\left (15 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a^{5} - 80 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} a^{6} + 132 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} a^{7}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} - 6 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a + 12 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{2} - 8 \, a^{3}}\right )}}{768 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^6\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________