Optimal. Leaf size=219 \[ \frac{\left (8 a^2 A-12 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(4 a B+5 A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.963541, 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 = {3605, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ \frac{\left (8 a^2 A-12 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(4 a B+5 A b) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3605
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \cot ^3(c+d x) (a+b \tan (c+d x))^{3/2} (A+B \tan (c+d x)) \, dx &=-\frac{a 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 (5 A b+4 a B)-2 \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)-\frac{1}{2} b (3 a A-4 b B) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx\\ &=-\frac{(5 A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a 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} a \left (8 a^2 A-3 A b^2-12 a b B\right )+2 a \left (2 a A b+a^2 B-b^2 B\right ) \tan (c+d x)+\frac{1}{4} a b (5 A b+4 a B) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 a}\\ &=-\frac{(5 A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{\int \frac{2 a \left (2 a A b+a^2 B-b^2 B\right )-2 a \left (a^2 A-A b^2-2 a b B\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 a}-\frac{1}{8} \left (8 a^2 A-3 A b^2-12 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\\ &=-\frac{(5 A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{1}{2} \left ((a+i b)^2 (i A-B)\right ) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx-\frac{1}{2} \left ((a-i b)^2 (i A+B)\right ) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx-\frac{\left (8 a^2 A-3 A b^2-12 a b B\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{8 d}\\ &=-\frac{(5 A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}+\frac{\left ((a-i b)^2 (A-i B)\right ) \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 ((a+i b)^2 (A+i B)\right ) \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-3 A b^2-12 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 b d}\\ &=\frac{\left (8 a^2 A-3 A b^2-12 a b B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{(5 A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}-\frac{\left ((a+i b)^2 (i A-B)\right ) \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 ((a-i b)^2 (i A+B)\right ) \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-3 A b^2-12 a b B\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{4 \sqrt{a} d}-\frac{(a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d}-\frac{(a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d}-\frac{(5 A b+4 a B) \cot (c+d x) \sqrt{a+b \tan (c+d x)}}{4 d}-\frac{a A \cot ^2(c+d x) \sqrt{a+b \tan (c+d x)}}{2 d}\\ \end{align*}
Mathematica [A] time = 2.42674, size = 195, normalized size = 0.89 \[ \frac{\left (8 a^2 A-12 a b B-3 A b^2\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )-\sqrt{a} \left (4 (a-i b)^{3/2} (A-i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )+4 (a+i b)^{3/2} (A+i B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )+\cot (c+d x) \sqrt{a+b \tan (c+d x)} (2 a A \cot (c+d x)+4 a B+5 A b)\right )}{4 \sqrt{a} d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 2.227, size = 102706, normalized size = 469. \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(-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]
________________________________________________________________________________________
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]