Optimal. Leaf size=316 \[ \frac{(B+2 i A) \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{(2 i A+(1-i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}-\frac{(2 i A+(1-i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}-\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (a \cot (c+d x)+i a)^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.761368, antiderivative size = 316, normalized size of antiderivative = 1., 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, 3596, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac{(B+2 i A) \sqrt{\cot (c+d x)}}{8 d \left (a^3 \cot (c+d x)+i a^3\right )}+\frac{(2 i A+(1-i) B) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}-\frac{(2 i A+(1-i) B) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{32 \sqrt{2} a^3 d}-\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) (B+(1+i) A) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )}{\sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (a \cot (c+d x)+i a)^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (a \cot (c+d x)+i a)^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3581
Rule 3595
Rule 3596
Rule 3534
Rule 1168
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{A+B \tan (c+d x)}{\sqrt{\cot (c+d x)} (a+i a \tan (c+d x))^3} \, dx &=\int \frac{\cot ^{\frac{3}{2}}(c+d x) (B+A \cot (c+d x))}{(i a+a \cot (c+d x))^3} \, dx\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{\int \frac{\sqrt{\cot (c+d x)} \left (-\frac{3}{2} a (i A-B)+\frac{3}{2} a (3 A-i B) \cot (c+d x)\right )}{(i a+a \cot (c+d x))^2} \, dx}{6 a^2}\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{\int \frac{-3 i a^2 A+3 a^2 (3 A-2 i B) \cot (c+d x)}{\sqrt{\cot (c+d x)} (i a+a \cot (c+d x))} \, dx}{24 a^4}\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{(2 i A+B) \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\int \frac{-3 i a^3 B-3 a^3 (2 i A+B) \cot (c+d x)}{\sqrt{\cot (c+d x)}} \, dx}{48 a^6}\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{(2 i A+B) \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\operatorname{Subst}\left (\int \frac{3 i a^3 B+3 a^3 (2 i A+B) x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{24 a^6 d}\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{(2 i A+B) \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\left (\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1+x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}-\frac{(2 i A+(1-i) B) \operatorname{Subst}\left (\int \frac{1-x^2}{1+x^4} \, dx,x,\sqrt{\cot (c+d x)}\right )}{16 a^3 d}\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{(2 i A+B) \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{\left (\left (\frac{1}{32}+\frac{i}{32}\right ) ((1+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1-\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+\frac{\left (\left (\frac{1}{32}+\frac{i}{32}\right ) ((1+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{1+\sqrt{2} x+x^2} \, dx,x,\sqrt{\cot (c+d x)}\right )}{a^3 d}+\frac{(2 i A+(1-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^3 d}+\frac{(2 i A+(1-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^3 d}\\ &=\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{(2 i A+B) \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{(2 i A+(1-i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}-\frac{(2 i A+(1-i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}+-\frac{\left (\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+i) A+B)\right ) \operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}\\ &=-\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+i) A+B) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{\left (\frac{1}{16}+\frac{i}{16}\right ) ((1+i) A+B) \tan ^{-1}\left (1+\sqrt{2} \sqrt{\cot (c+d x)}\right )}{\sqrt{2} a^3 d}+\frac{(A+i B) \cot ^{\frac{3}{2}}(c+d x)}{6 d (i a+a \cot (c+d x))^3}+\frac{A \sqrt{\cot (c+d x)}}{4 a d (i a+a \cot (c+d x))^2}+\frac{(2 i A+B) \sqrt{\cot (c+d x)}}{8 d \left (i a^3+a^3 \cot (c+d x)\right )}+\frac{(2 i A+(1-i) B) \log \left (1-\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}-\frac{(2 i A+(1-i) B) \log \left (1+\sqrt{2} \sqrt{\cot (c+d x)}+\cot (c+d x)\right )}{32 \sqrt{2} a^3 d}\\ \end{align*}
Mathematica [A] time = 3.78045, size = 272, normalized size = 0.86 \[ \frac{e^{-4 i (c+d x)} \sqrt{\cot (c+d x)} \sec (c+d x) (\cos (3 (c+d x))-i \sin (3 (c+d x))) \left (\left (-2 e^{2 i (c+d x)}+e^{4 i (c+d x)}+2 e^{6 i (c+d x)}-1\right ) \left (A e^{2 i (c+d x)}+A-2 i B e^{2 i (c+d x)}+i B\right )-6 (A-i B) e^{6 i (c+d x)} \sqrt{-1+e^{2 i (c+d x)}} \sqrt{1+e^{2 i (c+d x)}} \tanh ^{-1}\left (\sqrt{\frac{-1+e^{2 i (c+d x)}}{1+e^{2 i (c+d x)}}}\right )-3 A e^{6 i (c+d x)} \sqrt{-1+e^{4 i (c+d x)}} \tan ^{-1}\left (\sqrt{-1+e^{4 i (c+d x)}}\right )\right )}{96 a^3 d} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.602, size = 5075, normalized size = 16.1 \begin{align*} \text{output too large to display} \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: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 1.56556, size = 1706, normalized size = 5.4 \begin{align*} \text{result too large to display} \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 )}^{3} \sqrt{\cot \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]