Optimal. Leaf size=421 \[ \frac {2 b \left (18 a^2 B+21 a A b-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right )}{d \sqrt {\cot (c+d x)}}+\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a^2 b (A+B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 b^2 (11 a B+7 A b)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b B (a \cot (c+d x)+b)^2}{7 d \cot ^{\frac {7}{2}}(c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.78, antiderivative size = 421, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3581, 3605, 3635, 3628, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {2 b \left (18 a^2 B+21 a A b-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b+a^3 B-3 a b^2 B-A b^3\right )}{d \sqrt {\cot (c+d x)}}+\frac {\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)-\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)+a^3 (A+B)-3 a b^2 (A+B)-b^3 (A-B)\right ) \log \left (\cot (c+d x)+\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (-3 a^2 b (A+B)+a^3 (A-B)-3 a b^2 (A-B)+b^3 (A+B)\right ) \tan ^{-1}\left (\sqrt {2} \sqrt {\cot (c+d x)}+1\right )}{\sqrt {2} d}+\frac {2 b^2 (11 a B+7 A b)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b B (a \cot (c+d x)+b)^2}{7 d \cot ^{\frac {7}{2}}(c+d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rule 3529
Rule 3534
Rule 3581
Rule 3605
Rule 3628
Rule 3635
Rubi steps
\begin {align*} \int \frac {(a+b \tan (c+d x))^3 (A+B \tan (c+d x))}{\sqrt {\cot (c+d x)}} \, dx &=\int \frac {(b+a \cot (c+d x))^3 (B+A \cot (c+d x))}{\cot ^{\frac {9}{2}}(c+d x)} \, dx\\ &=\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {2}{7} \int \frac {(b+a \cot (c+d x)) \left (-\frac {1}{2} b (7 A b+11 a B)-\frac {7}{2} \left (2 a A b+a^2 B-b^2 B\right ) \cot (c+d x)-\frac {1}{2} a (7 a A-3 b B) \cot ^2(c+d x)\right )}{\cot ^{\frac {7}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {\frac {1}{2} b \left (21 a A b+18 a^2 B-7 b^2 B\right )+\frac {7}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)+\frac {1}{2} a^2 (7 a A-3 b B) \cot ^2(c+d x)}{\cot ^{\frac {5}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {\frac {7}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )+\frac {7}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right ) \cot (c+d x)}{\cot ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {2}{7} \int \frac {\frac {7}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )-\frac {7}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) \cot (c+d x)}{\sqrt {\cot (c+d x)}} \, dx\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {4 \operatorname {Subst}\left (\int \frac {-\frac {7}{2} \left (a^3 A-3 a A b^2-3 a^2 b B+b^3 B\right )+\frac {7}{2} \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right ) x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{7 d}\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\cot (c+d x)}\right )}{d}\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 \sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \operatorname {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\cot (c+d x)}\right )}{2 d}\\ &=\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (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} d}+\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (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} d}\\ &=\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}-\frac {\left (a^3 (A-B)-3 a b^2 (A-B)-3 a^2 b (A+B)+b^3 (A+B)\right ) \tan ^{-1}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )}{\sqrt {2} d}+\frac {2 b^2 (7 A b+11 a B)}{35 d \cot ^{\frac {5}{2}}(c+d x)}+\frac {2 b \left (21 a A b+18 a^2 B-7 b^2 B\right )}{21 d \cot ^{\frac {3}{2}}(c+d x)}+\frac {2 \left (3 a^2 A b-A b^3+a^3 B-3 a b^2 B\right )}{d \sqrt {\cot (c+d x)}}+\frac {2 b B (b+a \cot (c+d x))^2}{7 d \cot ^{\frac {7}{2}}(c+d x)}+\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}-\frac {\left (3 a^2 b (A-B)-b^3 (A-B)+a^3 (A+B)-3 a b^2 (A+B)\right ) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )}{2 \sqrt {2} d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 2.49, size = 327, normalized size = 0.78 \[ \frac {2 \sqrt {\tan (c+d x)} \sqrt {\cot (c+d x)} \left (\frac {1}{3} b \left (3 a^2 B+3 a A b-b^2 B\right ) \tan ^{\frac {3}{2}}(c+d x)-\frac {\left (a^3 (A-B)-3 a^2 b (A+B)+3 a b^2 (B-A)+b^3 (A+B)\right ) \left (\tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )-\tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{2 \sqrt {2}}+\left (a^3 B+3 a^2 A b-3 a b^2 B-A b^3\right ) \sqrt {\tan (c+d x)}+\frac {\left (a^3 (A+B)+3 a^2 b (A-B)-3 a b^2 (A+B)+b^3 (B-A)\right ) \left (\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )}{4 \sqrt {2}}+\frac {1}{5} b^2 (3 a B+A b) \tan ^{\frac {5}{2}}(c+d x)+\frac {1}{7} b^3 B \tan ^{\frac {7}{2}}(c+d x)\right )}{d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \tan \left (d x + c\right ) + A\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{3}}{\sqrt {\cot \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 2.10, size = 5111, normalized size = 12.14 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.95, size = 370, normalized size = 0.88 \[ \frac {8 \, {\left (15 \, B b^{3} + \frac {21 \, {\left (3 \, B a b^{2} + A b^{3}\right )}}{\tan \left (d x + c\right )} + \frac {35 \, {\left (3 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )}}{\tan \left (d x + c\right )^{2}} + \frac {105 \, {\left (B a^{3} + 3 \, A a^{2} b - 3 \, B a b^{2} - A b^{3}\right )}}{\tan \left (d x + c\right )^{3}}\right )} \tan \left (d x + c\right )^{\frac {7}{2}} - 210 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 210 \, \sqrt {2} {\left ({\left (A - B\right )} a^{3} - 3 \, {\left (A + B\right )} a^{2} b - 3 \, {\left (A - B\right )} a b^{2} + {\left (A + B\right )} b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) - 105 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + 105 \, \sqrt {2} {\left ({\left (A + B\right )} a^{3} + 3 \, {\left (A - B\right )} a^{2} b - 3 \, {\left (A + B\right )} a b^{2} - {\left (A - B\right )} b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )}{420 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (A+B\,\mathrm {tan}\left (c+d\,x\right )\right )\,{\left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )}^3}{\sqrt {\mathrm {cot}\left (c+d\,x\right )}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (A + B \tan {\left (c + d x \right )}\right ) \left (a + b \tan {\left (c + d x \right )}\right )^{3}}{\sqrt {\cot {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________