Optimal. Leaf size=317 \[ \frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (\cot (c+d x)+i)}-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+23 i) A-(7+2 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((25+21 i) A-(9-5 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.68, antiderivative size = 317, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {3581, 3595, 3528, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (\cot (c+d x)+i)}-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} a^2 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+23 i) A-(7+2 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} a^2 d}+\frac {((25+21 i) A-(9-5 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3528
Rule 3534
Rule 3581
Rule 3595
Rubi steps
\begin {align*} \int \frac {\cot ^{\frac {3}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx &=\int \frac {\cot ^{\frac {5}{2}}(c+d x) (B+A \cot (c+d x))}{(i a+a \cot (c+d x))^2} \, dx\\ &=\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \frac {\cot ^{\frac {3}{2}}(c+d x) \left (-\frac {5}{2} a (i A-B)+\frac {1}{2} a (9 A+i B) \cot (c+d x)\right )}{i a+a \cot (c+d x)} \, dx}{4 a^2}\\ &=\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \sqrt {\cot (c+d x)} \left (-\frac {3}{2} a^2 (7 i A-3 B)+\frac {5}{2} a^2 (5 A+i B) \cot (c+d x)\right ) \, dx}{8 a^4}\\ &=-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {\int \frac {-\frac {5}{2} a^2 (5 A+i B)-\frac {3}{2} a^2 (7 i A-3 B) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx}{8 a^4}\\ &=-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {\operatorname {Subst}\left (\int \frac {\frac {5}{2} a^2 (5 A+i B)+\frac {3}{2} a^2 (7 i A-3 B) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{4 a^4 d}\\ &=-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {((25+21 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d}+\frac {((25-21 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{16 a^2 d}\\ &=-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}+\frac {((25+21 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}+\frac {((25+21 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 a^2 d}-\frac {((25-21 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 \sqrt {2} a^2 d}-\frac {((25-21 i) A+(9+5 i) B) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{32 \sqrt {2} a^2 d}\\ &=-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}-\frac {((25-21 i) A+(9+5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((25-21 i) A+(9+5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((25+21 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((25+21 i) A-(9-5 i) B) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}\\ &=-\frac {((25+21 i) A-(9-5 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {((25+21 i) A-(9-5 i) B) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {5 (5 A+i B) \sqrt {\cot (c+d x)}}{8 a^2 d}+\frac {(7 A+3 i B) \cot ^{\frac {3}{2}}(c+d x)}{8 a^2 d (i+\cot (c+d x))}+\frac {(A+i B) \cot ^{\frac {5}{2}}(c+d x)}{4 d (i a+a \cot (c+d x))^2}-\frac {((25-21 i) A+(9+5 i) B) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((25-21 i) A+(9+5 i) B) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt {2} a^2 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.90, size = 256, normalized size = 0.81 \[ \frac {\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left (\cot (c+d x) (-2 \cos (2 d x)+2 i \sin (2 d x)) ((-7 B+43 i A) \sin (2 (c+d x))+(41 A+5 i B) \cos (2 (c+d x))-9 A-5 i B)+(-\sin (2 c)+i \cos (2 c)) \sqrt {\sin (2 (c+d x))} \csc (c+d x) \left (((21-25 i) A+(5+9 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((23+2 i) A+(2+7 i) B) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt {\cot (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.58, size = 671, normalized size = 2.12 \[ \frac {{\left (2 \, a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left ({\left (8 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - 8 i \, a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} - 8 \, {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, {\left (i \, A + B\right )}}\right ) - 2 \, a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left ({\left (-8 i \, a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + 8 i \, a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} - 8 \, {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{4 \, {\left (i \, A + B\right )}}\right ) + a^{2} d \sqrt {\frac {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} + 23 i \, A - 7 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - a^{2} d \sqrt {\frac {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} - a^{2} d\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}} \sqrt {\frac {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} - 23 i \, A + 7 \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left (6 \, {\left (7 \, A + i \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (9 \, A + 5 i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} - A - i \, B\right )} \sqrt {\frac {i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} - 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{32 \, a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{\frac {3}{2}}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 4.40, size = 2507, normalized size = 7.91 \[ \text {Expression too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {cot}\left (c+d\,x\right )}^{3/2}\,\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \frac {\int \frac {A \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx + \int \frac {B \tan {\left (c + d x \right )} \cot ^{\frac {3}{2}}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________