Integrand size = 21, antiderivative size = 166 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 b \sqrt {\tan (c+d x)}}{d} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=-\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} d}+\frac {(a+b) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2} d}+\frac {2 b \sqrt {\tan (c+d x)}}{d} \]
[In]
[Out]
Rule 210
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 1182
Rule 3609
Rule 3615
Rubi steps \begin{align*} \text {integral}& = \frac {2 b \sqrt {\tan (c+d x)}}{d}+\int \frac {-b+a \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {2 b \sqrt {\tan (c+d x)}}{d}+\frac {2 \text {Subst}\left (\int \frac {-b+a x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {2 b \sqrt {\tan (c+d x)}}{d}+\frac {(a-b) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d}-\frac {(a+b) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = \frac {2 b \sqrt {\tan (c+d x)}}{d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d}+\frac {(a+b) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{2 \sqrt {2} d} \\ & = \frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 b \sqrt {\tan (c+d x)}}{d}+\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a-b) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d} \\ & = -\frac {(a-b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a+b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a+b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 b \sqrt {\tan (c+d x)}}{d} \\ \end{align*}
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.48 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=\frac {\sqrt [4]{-1} (i a+b) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )-(-1)^{3/4} (a+i b) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 b \sqrt {\tan (c+d x)}}{d} \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.14
method | result | size |
derivativedivides | \(\frac {2 b \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(189\) |
default | \(\frac {2 b \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(189\) |
parts | \(\frac {a \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}+\frac {b \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(192\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (136) = 272\).
Time = 0.25 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.51 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=-\frac {d \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} \log \left ({\left (a d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - {\left (a^{2} b - b^{3}\right )} d\right )} \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - d \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} \log \left (-{\left (a d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - {\left (a^{2} b - b^{3}\right )} d\right )} \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - d \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} \log \left ({\left (a d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + {\left (a^{2} b - b^{3}\right )} d\right )} \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) + d \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} \log \left (-{\left (a d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + {\left (a^{2} b - b^{3}\right )} d\right )} \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - 4 \, b \sqrt {\tan \left (d x + c\right )}}{2 \, d} \]
[In]
[Out]
\[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \sqrt {\tan {\left (c + d x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.36 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.81 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=\frac {2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (a - b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} {\left (a + b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} {\left (a + b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + 8 \, b \sqrt {\tan \left (d x + c\right )}}{4 \, d} \]
[In]
[Out]
\[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \sqrt {\tan \left (d x + c\right )} \,d x } \]
[In]
[Out]
Time = 5.81 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.92 \[ \int \sqrt {\tan (c+d x)} (a+b \tan (c+d x)) \, dx=\frac {2\,b\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {\sqrt {2}\,a\,\left (\ln \left (\sqrt {2}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}-\mathrm {tan}\left (c+d\,x\right )-1\right )-\ln \left (\mathrm {tan}\left (c+d\,x\right )+\sqrt {2}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}+1\right )\right )}{4\,d}+\frac {\sqrt {2}\,a\,\left (\mathrm {atan}\left (\sqrt {2}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}-1\right )+\mathrm {atan}\left (\sqrt {2}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}+1\right )\right )}{2\,d}+\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d}+\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d} \]
[In]
[Out]