Integrand size = 16, antiderivative size = 170 \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d} \]
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Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {3742, 427, 542, 537, 223, 212, 385, 209} \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d} \]
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Rule 209
Rule 212
Rule 223
Rule 385
Rule 427
Rule 537
Rule 542
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^{5/2}}{1+x^2} \, dx,x,\tan (c+d x)\right )}{d} \\ & = \frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^2} \left (a (4 a-b)+(7 a-4 b) b x^2\right )}{1+x^2} \, dx,x,\tan (c+d x)\right )}{4 d} \\ & = \frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {\text {Subst}\left (\int \frac {a \left (8 a^2-9 a b+4 b^2\right )+b \left (15 a^2-20 a b+8 b^2\right ) x^2}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (c+d x)\right )}{8 d} \\ & = \frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{\left (1+x^2\right ) \sqrt {a+b x^2}} \, dx,x,\tan (c+d x)\right )}{d}+\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\tan (c+d x)\right )}{8 d} \\ & = \frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d}+\frac {(a-b)^3 \text {Subst}\left (\int \frac {1}{1-(-a+b) x^2} \, dx,x,\frac {\tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {\left (b \left (15 a^2-20 a b+8 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d} \\ & = \frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d} \\ \end{align*}
Time = 1.23 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {-8 (a-b)^{5/2} \arctan \left (\frac {\sqrt {b}+\sqrt {b} \tan ^2(c+d x)-\tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{\sqrt {a-b}}\right )-\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \log \left (-\sqrt {b} \tan (c+d x)+\sqrt {a+b \tan ^2(c+d x)}\right )+b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)} \left (9 a-4 b+2 b \tan ^2(c+d x)\right )}{8 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(460\) vs. \(2(148)=296\).
Time = 0.16 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.71
method | result | size |
derivativedivides | \(\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )^{3} \sqrt {a +b \tan \left (d x +c \right )^{2}}}{4 d}+\frac {9 b a \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{8 d}+\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{8 d}-\frac {b^{2} \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{2 d}-\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{2 d}-\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}+\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\) | \(461\) |
default | \(\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )^{3} \sqrt {a +b \tan \left (d x +c \right )^{2}}}{4 d}+\frac {9 b a \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{8 d}+\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{8 d}-\frac {b^{2} \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{2 d}-\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{2 d}-\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}+\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\) | \(461\) |
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Time = 1.30 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.14 \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\left [\frac {{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {b} \log \left (2 \, b \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {b} \tan \left (d x + c\right ) + a\right ) + 8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-a + b} \tan \left (d x + c\right ) - a}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{16 \, d}, \frac {16 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a}}{\sqrt {a - b} \tan \left (d x + c\right )}\right ) + {\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {b} \log \left (2 \, b \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {b} \tan \left (d x + c\right ) + a\right ) + 2 \, {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{16 \, d}, -\frac {{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-b}}{b \tan \left (d x + c\right )}\right ) - 4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-a + b} \tan \left (d x + c\right ) - a}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{8 \, d}, \frac {8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a}}{\sqrt {a - b} \tan \left (d x + c\right )}\right ) - {\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-b}}{b \tan \left (d x + c\right )}\right ) + {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{8 \, d}\right ] \]
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\[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
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\[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \]
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Timed out. \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \]
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