Optimal. Leaf size=219 \[ \frac{\left (8 a^2 A-4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{3/2} d}-\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(4 a B+A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.861336, antiderivative size = 219, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3608, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{\left (8 a^2 A-4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{3/2} d}-\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(4 a B+A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3608
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \cot ^3(c+d x) \sqrt{a+b \tan (c+d x)} (A+B \tan (c+d x)) \, dx &=-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{1}{2} \int \frac{\cot ^2(c+d x) \left (\frac{1}{2} (-A b-4 a B)+2 (a A-b B) \tan (c+d x)+\frac{3}{2} A b \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{(A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{\int \frac{\cot (c+d x) \left (\frac{1}{4} \left (-8 a^2 A-A b^2+4 a b B\right )-2 a (A b+a B) \tan (c+d x)-\frac{1}{4} b (A b+4 a B) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 a}\\ &=-\frac{(A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{\int \frac{-2 a (A b+a B)+2 a (a A-b B) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 a}+\frac{\left (-8 a^2 A-A b^2+4 a b B\right ) \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{8 a}\\ &=-\frac{(A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{1}{2} ((i a-b) (A+i B)) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx-\frac{(2 a (A b+a B)+2 i a (a A-b B)) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{4 a}-\frac{\left (8 a^2 A+A b^2-4 a b B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{8 a d}\\ &=-\frac{(A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{((a-i b) (A-i B)) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 d}+\frac{((a+i b) (A+i B)) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 d}-\frac{\left (8 a^2 A+A b^2-4 a b B\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{4 a b d}\\ &=\frac{\left (8 a^2 A+A b^2-4 a b B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{3/2} d}-\frac{(A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{((i a+b) (A-i B)) \operatorname{Subst}\left (\int \frac{1}{-1-\frac{i a}{b}+\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{b d}-\frac{((i a-b) (A+i B)) \operatorname{Subst}\left (\int \frac{1}{-1+\frac{i a}{b}-\frac{i x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{b d}\\ &=\frac{\left (8 a^2 A+A b^2-4 a b B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 a^{3/2} d}-\frac{\sqrt{a-i b} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{\sqrt{a+i b} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 a d}-\frac{A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}\\ \end{align*}
Mathematica [A] time = 4.6523, size = 271, normalized size = 1.24 \[ \frac{\frac{\left (8 a^2 A-4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\frac{4 \left (-a A b+a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-\sqrt{-b^2}}}\right )}{\sqrt{a-\sqrt{-b^2}}}-\frac{4 \left (a A b+a \sqrt{-b^2} B+A \sqrt{-b^2} b+b^2 (-B)\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+\sqrt{-b^2}}}\right )}{\sqrt{a+\sqrt{-b^2}}}-\frac{b \cot (c+d x) \sqrt{a+b \tan (c+d x)} (2 a A \cot (c+d x)+4 a B+A b)}{a}}{b}}{4 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.935, size = 81276, normalized size = 371.1 \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: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (A + B \tan{\left (c + d x \right )}\right ) \sqrt{a + b \tan{\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]