Optimal. Leaf size=155 \[ \frac{B+i A}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{-B+i A}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{B+i A}{6 a d (a+i a \tan (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.129972, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3526, 3479, 3480, 206} \[ \frac{B+i A}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{-B+i A}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{B+i A}{6 a d (a+i a \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3526
Rule 3479
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{(a+i a \tan (c+d x))^{5/2}} \, dx &=\frac{i A-B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{(A-i B) \int \frac{1}{(a+i a \tan (c+d x))^{3/2}} \, dx}{2 a}\\ &=\frac{i A-B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i A+B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac{(A-i B) \int \frac{1}{\sqrt{a+i a \tan (c+d x)}} \, dx}{4 a^2}\\ &=\frac{i A-B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i A+B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i A+B}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}+\frac{(A-i B) \int \sqrt{a+i a \tan (c+d x)} \, dx}{8 a^3}\\ &=\frac{i A-B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i A+B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i A+B}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}-\frac{(i A+B) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\sqrt{a+i a \tan (c+d x)}\right )}{4 a^2 d}\\ &=-\frac{(i A+B) \tanh ^{-1}\left (\frac{\sqrt{a+i a \tan (c+d x)}}{\sqrt{2} \sqrt{a}}\right )}{4 \sqrt{2} a^{5/2} d}+\frac{i A-B}{5 d (a+i a \tan (c+d x))^{5/2}}+\frac{i A+B}{6 a d (a+i a \tan (c+d x))^{3/2}}+\frac{i A+B}{4 a^2 d \sqrt{a+i a \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 2.46152, size = 176, normalized size = 1.14 \[ -\frac{e^{-6 i (c+d x)} \left (1+e^{2 i (c+d x)}\right )^{3/2} \sec ^2(c+d x) \left (\sqrt{1+e^{2 i (c+d x)}} \left (B \left (e^{2 i (c+d x)}-17 e^{4 i (c+d x)}+3\right )-i A \left (11 e^{2 i (c+d x)}+23 e^{4 i (c+d x)}+3\right )\right )+15 (B+i A) e^{5 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{240 a^2 d \sqrt{a+i a \tan (c+d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.023, size = 123, normalized size = 0.8 \begin{align*}{\frac{2\,i}{d} \left ( -{\frac{1}{5} \left ( -{\frac{A}{2}}-{\frac{i}{2}}B \right ) \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}-{\frac{-A+iB}{12\,a} \left ( a+ia\tan \left ( dx+c \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{-A+iB}{8\,{a}^{2}}{\frac{1}{\sqrt{a+ia\tan \left ( dx+c \right ) }}}}-{\frac{ \left ( A-iB \right ) \sqrt{2}}{16}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a+ia\tan \left ( dx+c \right ) }{\frac{1}{\sqrt{a}}}} \right ){a}^{-{\frac{5}{2}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.98948, size = 1095, normalized size = 7.06 \begin{align*} -\frac{{\left (15 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (\frac{{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt{2}{\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A + B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - 15 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (-\frac{{\left (2 \, \sqrt{\frac{1}{2}} a^{3} d \sqrt{-\frac{A^{2} - 2 i \, A B - B^{2}}{a^{5} d^{2}}} e^{\left (2 i \, d x + 2 i \, c\right )} - \sqrt{2}{\left ({\left (i \, A + B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A + B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}}{i \, A + B}\right ) - \sqrt{2}{\left ({\left (23 i \, A + 17 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (34 i \, A + 16 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (14 i \, A - 4 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, A - 3 \, B\right )} \sqrt{\frac{a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-6 i \, d x - 6 i \, c\right )}}{120 \, a^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]