Integrand size = 36, antiderivative size = 210 \[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {a} (9 i A+14 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {\sqrt {2} \sqrt {a} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]
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Time = 0.95 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3679, 3681, 3561, 212, 3680, 65, 214} \[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {\sqrt {a} (14 B+9 i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {\sqrt {2} \sqrt {a} (B+i A) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}-\frac {(6 B+i A) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}+\frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \]
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Rule 65
Rule 212
Rule 214
Rule 3561
Rule 3679
Rule 3680
Rule 3681
Rubi steps \begin{align*} \text {integral}& = -\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)} \left (\frac {1}{2} a (i A+6 B)-\frac {5}{2} a A \tan (c+d x)\right ) \, dx}{3 a} \\ & = -\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{4} a^2 (7 A-2 i B)-\frac {3}{4} a^2 (i A+6 B) \tan (c+d x)\right ) \, dx}{6 a^2} \\ & = \frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+\frac {\int \cot (c+d x) \sqrt {a+i a \tan (c+d x)} \left (-\frac {3}{8} a^3 (9 i A+14 B)+\frac {3}{8} a^3 (7 A-2 i B) \tan (c+d x)\right ) \, dx}{6 a^3} \\ & = \frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}+(A-i B) \int \sqrt {a+i a \tan (c+d x)} \, dx-\frac {(9 i A+14 B) \int \cot (c+d x) (a-i a \tan (c+d x)) \sqrt {a+i a \tan (c+d x)} \, dx}{16 a} \\ & = \frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {(2 a (i A+B)) \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{d}-\frac {(a (9 i A+14 B)) \text {Subst}\left (\int \frac {1}{x \sqrt {a+i a x}} \, dx,x,\tan (c+d x)\right )}{16 d} \\ & = -\frac {\sqrt {2} \sqrt {a} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d}-\frac {(9 A-14 i B) \text {Subst}\left (\int \frac {1}{i-\frac {i x^2}{a}} \, dx,x,\sqrt {a+i a \tan (c+d x)}\right )}{8 d} \\ & = \frac {\sqrt {a} (9 i A+14 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )}{8 d}-\frac {\sqrt {2} \sqrt {a} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{d}+\frac {(7 A-2 i B) \cot (c+d x) \sqrt {a+i a \tan (c+d x)}}{8 d}-\frac {(i A+6 B) \cot ^2(c+d x) \sqrt {a+i a \tan (c+d x)}}{12 d}-\frac {A \cot ^3(c+d x) \sqrt {a+i a \tan (c+d x)}}{3 d} \\ \end{align*}
Time = 4.70 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.75 \[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {-3 \sqrt {a} (9 i A+14 B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {a}}\right )+24 \sqrt {2} \sqrt {a} (i A+B) \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )+\cot (c+d x) \left (-21 A+6 i B+2 (i A+6 B) \cot (c+d x)+8 A \cot ^2(c+d x)\right ) \sqrt {a+i a \tan (c+d x)}}{24 d} \]
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Time = 0.29 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {2 i a^{4} \left (-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {7}{2}}}-\frac {-\frac {i \left (\left (-\frac {i B}{8}+\frac {7 A}{16}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}-\frac {5 A a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{6}+\left (\frac {9}{16} A \,a^{2}+\frac {1}{8} i B \,a^{2}\right ) \sqrt {a +i a \tan \left (d x +c \right )}\right )}{a^{3} \tan \left (d x +c \right )^{3}}-\frac {\left (-14 i B +9 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a^{3}}\right )}{d}\) | \(171\) |
default | \(\frac {2 i a^{4} \left (-\frac {\left (-i B +A \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{2 a^{\frac {7}{2}}}-\frac {-\frac {i \left (\left (-\frac {i B}{8}+\frac {7 A}{16}\right ) \left (a +i a \tan \left (d x +c \right )\right )^{\frac {5}{2}}-\frac {5 A a \left (a +i a \tan \left (d x +c \right )\right )^{\frac {3}{2}}}{6}+\left (\frac {9}{16} A \,a^{2}+\frac {1}{8} i B \,a^{2}\right ) \sqrt {a +i a \tan \left (d x +c \right )}\right )}{a^{3} \tan \left (d x +c \right )^{3}}-\frac {\left (-14 i B +9 A \right ) \operatorname {arctanh}\left (\frac {\sqrt {a +i a \tan \left (d x +c \right )}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a^{3}}\right )}{d}\) | \(171\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 823 vs. \(2 (163) = 326\).
Time = 0.28 (sec) , antiderivative size = 823, normalized size of antiderivative = 3.92 \[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\text {Too large to display} \]
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\[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int \sqrt {i a \left (\tan {\left (c + d x \right )} - i\right )} \left (A + B \tan {\left (c + d x \right )}\right ) \cot ^{4}{\left (c + d x \right )}\, dx \]
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Time = 0.31 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.19 \[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\frac {i \, a^{3} {\left (\frac {2 \, {\left (3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}} {\left (7 \, A - 2 i \, B\right )} - 40 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {3}{2}} A a + 3 \, \sqrt {i \, a \tan \left (d x + c\right ) + a} {\left (9 \, A + 2 i \, B\right )} a^{2}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} - 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} + 3 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} - a^{5}} + \frac {24 \, \sqrt {2} {\left (A - i \, B\right )} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}} - \frac {3 \, {\left (9 \, A - 14 i \, B\right )} \log \left (\frac {\sqrt {i \, a \tan \left (d x + c\right ) + a} - \sqrt {a}}{\sqrt {i \, a \tan \left (d x + c\right ) + a} + \sqrt {a}}\right )}{a^{\frac {5}{2}}}\right )}}{48 \, d} \]
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\[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=\int { {\left (B \tan \left (d x + c\right ) + A\right )} \sqrt {i \, a \tan \left (d x + c\right ) + a} \cot \left (d x + c\right )^{4} \,d x } \]
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Time = 8.45 (sec) , antiderivative size = 735, normalized size of antiderivative = 3.50 \[ \int \cot ^4(c+d x) \sqrt {a+i a \tan (c+d x)} (A+B \tan (c+d x)) \, dx=-\frac {\frac {\left (9\,A\,a^3+B\,a^3\,2{}\mathrm {i}\right )\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,1{}\mathrm {i}}{8\,d}+\frac {\left (7\,A\,a-B\,a\,2{}\mathrm {i}\right )\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{5/2}\,1{}\mathrm {i}}{8\,d}-\frac {A\,a^2\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,5{}\mathrm {i}}{3\,d}}{3\,a\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2-3\,a^2\,\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )-{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^3+a^3}-\frac {\mathrm {atan}\left (\frac {47\,\sqrt {32}\,A^3\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{8\,\left (47{}\mathrm {i}\,d\,A^3\,a^5+51\,d\,A^2\,B\,a^5+64{}\mathrm {i}\,d\,A\,B^2\,a^5+68\,d\,B^3\,a^5\right )}-\frac {\sqrt {32}\,B^3\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,17{}\mathrm {i}}{2\,\left (47{}\mathrm {i}\,d\,A^3\,a^5+51\,d\,A^2\,B\,a^5+64{}\mathrm {i}\,d\,A\,B^2\,a^5+68\,d\,B^3\,a^5\right )}+\frac {8\,\sqrt {32}\,A\,B^2\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{47{}\mathrm {i}\,d\,A^3\,a^5+51\,d\,A^2\,B\,a^5+64{}\mathrm {i}\,d\,A\,B^2\,a^5+68\,d\,B^3\,a^5}-\frac {\sqrt {32}\,A^2\,B\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,51{}\mathrm {i}}{8\,\left (47{}\mathrm {i}\,d\,A^3\,a^5+51\,d\,A^2\,B\,a^5+64{}\mathrm {i}\,d\,A\,B^2\,a^5+68\,d\,B^3\,a^5\right )}\right )\,\left (B+A\,1{}\mathrm {i}\right )\,\sqrt {\frac {a}{32}}\,8{}\mathrm {i}}{d}+\frac {\sqrt {a}\,\mathrm {atan}\left (\frac {423\,A^3\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{8\,\left (\frac {423{}\mathrm {i}\,d\,A^3\,a^5}{8}+\frac {347\,d\,A^2\,B\,a^5}{4}+\frac {139{}\mathrm {i}\,d\,A\,B^2\,a^5}{2}+119\,d\,B^3\,a^5\right )}-\frac {B^3\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,119{}\mathrm {i}}{\frac {423{}\mathrm {i}\,d\,A^3\,a^5}{8}+\frac {347\,d\,A^2\,B\,a^5}{4}+\frac {139{}\mathrm {i}\,d\,A\,B^2\,a^5}{2}+119\,d\,B^3\,a^5}+\frac {139\,A\,B^2\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}}{2\,\left (\frac {423{}\mathrm {i}\,d\,A^3\,a^5}{8}+\frac {347\,d\,A^2\,B\,a^5}{4}+\frac {139{}\mathrm {i}\,d\,A\,B^2\,a^5}{2}+119\,d\,B^3\,a^5\right )}-\frac {A^2\,B\,a^{9/2}\,d\,\sqrt {a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}}\,347{}\mathrm {i}}{4\,\left (\frac {423{}\mathrm {i}\,d\,A^3\,a^5}{8}+\frac {347\,d\,A^2\,B\,a^5}{4}+\frac {139{}\mathrm {i}\,d\,A\,B^2\,a^5}{2}+119\,d\,B^3\,a^5\right )}\right )\,\left (14\,B+A\,9{}\mathrm {i}\right )\,1{}\mathrm {i}}{8\,d} \]
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