Integrand size = 37, antiderivative size = 213 \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=-\frac {2 A b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 C \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 A E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \]
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Time = 0.72 (sec) , antiderivative size = 213, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.297, Rules used = {4194, 3944, 2886, 2884, 3947, 3941, 2734, 2732, 3943, 2742, 2740} \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=-\frac {2 A b \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 A \sqrt {a+b \sec (c+d x)} E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right )}{a d \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}}}+\frac {2 C \sqrt {\sec (c+d x)} \sqrt {\frac {a \cos (c+d x)+b}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right )}{d \sqrt {a+b \sec (c+d x)}} \]
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Rule 2732
Rule 2734
Rule 2740
Rule 2742
Rule 2884
Rule 2886
Rule 3941
Rule 3943
Rule 3944
Rule 3947
Rule 4194
Rubi steps \begin{align*} \text {integral}& = A \int \frac {1}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx+C \int \frac {\sec ^{\frac {3}{2}}(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx \\ & = \frac {A \int \frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {\sec (c+d x)}} \, dx}{a}-\frac {(A b) \int \frac {\sqrt {\sec (c+d x)}}{\sqrt {a+b \sec (c+d x)}} \, dx}{a}+\frac {\left (C \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {b+a \cos (c+d x)}} \, dx}{\sqrt {a+b \sec (c+d x)}} \\ & = -\frac {\left (A b \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {b+a \cos (c+d x)}} \, dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {\left (C \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {\sec (c+d x)}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{\sqrt {a+b \sec (c+d x)}}+\frac {\left (A \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {b+a \cos (c+d x)} \, dx}{a \sqrt {b+a \cos (c+d x)} \sqrt {\sec (c+d x)}} \\ & = \frac {2 C \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}}-\frac {\left (A b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}\right ) \int \frac {1}{\sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}}} \, dx}{a \sqrt {a+b \sec (c+d x)}}+\frac {\left (A \sqrt {a+b \sec (c+d x)}\right ) \int \sqrt {\frac {b}{a+b}+\frac {a \cos (c+d x)}{a+b}} \, dx}{a \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ & = -\frac {2 A b \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{a d \sqrt {a+b \sec (c+d x)}}+\frac {2 C \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \operatorname {EllipticPi}\left (2,\frac {1}{2} (c+d x),\frac {2 a}{a+b}\right ) \sqrt {\sec (c+d x)}}{d \sqrt {a+b \sec (c+d x)}}+\frac {2 A E\left (\frac {1}{2} (c+d x)|\frac {2 a}{a+b}\right ) \sqrt {a+b \sec (c+d x)}}{a d \sqrt {\frac {b+a \cos (c+d x)}{a+b}} \sqrt {\sec (c+d x)}} \\ \end{align*}
\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx \]
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Result contains complex when optimal does not.
Time = 8.09 (sec) , antiderivative size = 946, normalized size of antiderivative = 4.44
method | result | size |
parts | \(\frac {2 A \sqrt {\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}\, \left (\left (1-\cos \left (d x +c \right )\right )^{3} a \sqrt {\frac {a -b}{a +b}}\, \csc \left (d x +c \right )^{3}-\left (1-\cos \left (d x +c \right )\right )^{3} b \sqrt {\frac {a -b}{a +b}}\, \csc \left (d x +c \right )^{3}+\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) a \sqrt {-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{a +b}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}-\operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) a \sqrt {-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{a +b}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}+\operatorname {EllipticE}\left (\sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right ) b \sqrt {-\frac {a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b}{a +b}}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}-a \sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-b \sqrt {\frac {a -b}{a +b}}\, \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )\right )}{d \sqrt {\frac {a -b}{a +b}}\, a \left (a \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-b \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-a -b \right ) \sqrt {-\frac {\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1}{\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-1}}}-\frac {2 C \left (-\operatorname {EllipticF}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), \sqrt {-\frac {a +b}{a -b}}\right )+2 \operatorname {EllipticPi}\left (\sqrt {\frac {a -b}{a +b}}\, \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right ), \frac {a +b}{a -b}, \frac {i}{\sqrt {\frac {a -b}{a +b}}}\right )\right ) \sqrt {a +b \sec \left (d x +c \right )}\, \sec \left (d x +c \right )^{\frac {3}{2}} \sqrt {\frac {b +a \cos \left (d x +c \right )}{\left (a +b \right ) \left (\cos \left (d x +c \right )+1\right )}}\, \cos \left (d x +c \right )^{2}}{d \sqrt {\frac {a -b}{a +b}}\, \left (b +a \cos \left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}}\) | \(946\) |
default | \(\text {Expression too large to display}\) | \(1026\) |
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\text {Timed out} \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {A + C \sec ^{2}{\left (c + d x \right )}}{\sqrt {a + b \sec {\left (c + d x \right )}} \sqrt {\sec {\left (c + d x \right )}}}\, dx \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {C \sec \left (d x + c\right )^{2} + A}{\sqrt {b \sec \left (d x + c\right ) + a} \sqrt {\sec \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {A+C \sec ^2(c+d x)}{\sqrt {\sec (c+d x)} \sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {A+\frac {C}{{\cos \left (c+d\,x\right )}^2}}{\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \,d x \]
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