Optimal. Leaf size=289 \[ -\frac{b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac{b \left (10 a^2 A b^2+a^4 A-6 a^3 b B-2 a b^3 B+5 A b^4\right )}{a^3 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}-\frac{(-B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 1.25245, antiderivative size = 289, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {3609, 3649, 3653, 3539, 3537, 63, 208, 3634} \[ -\frac{b \left (3 a^2 A-2 a b B+5 A b^2\right )}{3 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac{b \left (10 a^2 A b^2+a^4 A-6 a^3 b B-2 a b^3 B+5 A b^4\right )}{a^3 d \left (a^2+b^2\right )^2 \sqrt{a+b \tan (c+d x)}}+\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d}+\frac{(B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{d (a-i b)^{5/2}}-\frac{(-B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{d (a+i b)^{5/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3609
Rule 3649
Rule 3653
Rule 3539
Rule 3537
Rule 63
Rule 208
Rule 3634
Rubi steps
\begin{align*} \int \frac{\cot ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^{5/2}} \, dx &=-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{\int \frac{\cot (c+d x) \left (\frac{1}{2} (5 A b-2 a B)+a A \tan (c+d x)+\frac{5}{2} A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{2 \int \frac{\cot (c+d x) \left (\frac{3}{4} \left (a^2+b^2\right ) (5 A b-2 a B)+\frac{3}{2} a^2 (a A+b B) \tan (c+d x)+\frac{3}{4} b \left (3 a^2 A+5 A b^2-2 a b B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^{3/2}} \, dx}{3 a^2 \left (a^2+b^2\right )}\\ &=-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{4 \int \frac{\cot (c+d x) \left (\frac{3}{8} \left (a^2+b^2\right )^2 (5 A b-2 a B)+\frac{3}{4} a^3 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+\frac{3}{8} b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right ) \tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}\\ &=-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{4 \int \frac{\frac{3}{4} a^3 \left (a^2 A-A b^2+2 a b B\right )-\frac{3}{4} a^3 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{3 a^3 \left (a^2+b^2\right )^2}-\frac{(5 A b-2 a B) \int \frac{\cot (c+d x) \left (1+\tan ^2(c+d x)\right )}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 a^3}\\ &=-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{(A-i B) \int \frac{1+i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a-i b)^2}-\frac{(A+i B) \int \frac{1-i \tan (c+d x)}{\sqrt{a+b \tan (c+d x)}} \, dx}{2 (a+i b)^2}-\frac{(5 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\tan (c+d x)\right )}{2 a^3 d}\\ &=-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}+\frac{(i A-B) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a+i b x}} \, dx,x,-i \tan (c+d x)\right )}{2 (a+i b)^2 d}-\frac{(i A+B) \operatorname{Subst}\left (\int \frac{1}{(-1+x) \sqrt{a-i b x}} \, dx,x,i \tan (c+d x)\right )}{2 (a-i b)^2 d}-\frac{(5 A b-2 a B) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \tan (c+d x)}\right )}{a^3 b d}\\ &=\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d}-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}-\frac{(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 (i a+b)^2 d}-\frac{(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 )}{(i a-b)^2 b d}\\ &=\frac{(5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{a^{7/2} d}+\frac{(i A+B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{(a-i b)^{5/2} d}-\frac{(i A-B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{(a+i b)^{5/2} d}-\frac{b \left (3 a^2 A+5 A b^2-2 a b B\right )}{3 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}-\frac{A \cot (c+d x)}{a d (a+b \tan (c+d x))^{3/2}}-\frac{b \left (a^4 A+10 a^2 A b^2+5 A b^4-6 a^3 b B-2 a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d \sqrt{a+b \tan (c+d x)}}\\ \end{align*}
Mathematica [A] time = 4.87337, size = 306, normalized size = 1.06 \[ \frac{\frac{b \left (-3 a^2 A+2 a b B-5 A b^2\right )}{\left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}+\frac{3 \left (-\frac{b \left (10 a^2 A b^2+a^4 A-6 a^3 b B-2 a b^3 B+5 A b^4\right )}{\sqrt{a+b \tan (c+d x)}}+\frac{\left (a^2+b^2\right )^2 (5 A b-2 a B) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a}}\right )}{\sqrt{a}}+\frac{a^3 (a+i b)^2 (B+i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a-i b}}\right )}{\sqrt{a-i b}}+\frac{a^3 (a-i b)^2 (B-i A) \tanh ^{-1}\left (\frac{\sqrt{a+b \tan (c+d x)}}{\sqrt{a+i b}}\right )}{\sqrt{a+i b}}\right )}{a \left (a^2+b^2\right )^2}-\frac{3 a A \cot (c+d x)}{(a+b \tan (c+d x))^{3/2}}}{3 a^2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 9.231, size = 339349, normalized size = 1174.2 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \tan \left (d x + c\right ) + A\right )} \cot \left (d x + c\right )^{2}}{{\left (b \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]