Integrand size = 21, antiderivative size = 94 \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2} d}-\frac {b \sin (c+d x)}{2 a (a-b) d \left (a-(a-b) \sin ^2(c+d x)\right )} \]
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Time = 0.10 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3757, 393, 214} \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=\frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d (a-b)^{3/2}}-\frac {b \sin (c+d x)}{2 a d (a-b) \left (a-(a-b) \sin ^2(c+d x)\right )} \]
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Rule 214
Rule 393
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1-x^2}{\left (a-(a-b) x^2\right )^2} \, dx,x,\sin (c+d x)\right )}{d} \\ & = -\frac {b \sin (c+d x)}{2 a (a-b) d \left (a-(a-b) \sin ^2(c+d x)\right )}+\frac {(2 a-b) \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\sin (c+d x)\right )}{2 a (a-b) d} \\ & = \frac {(2 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} (a-b)^{3/2} d}-\frac {b \sin (c+d x)}{2 a (a-b) d \left (a-(a-b) \sin ^2(c+d x)\right )} \\ \end{align*}
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.21 \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=\frac {-\frac {1}{2} (2 a-b) \text {arctanh}\left (\frac {\sqrt {a-b} \sin (c+d x)}{\sqrt {a}}\right ) (a+b+(a-b) \cos (2 (c+d x)))+\sqrt {a} \sqrt {a-b} b \sin (c+d x)}{2 a^{3/2} (a-b)^{3/2} d \left (-a+(a-b) \sin ^2(c+d x)\right )} \]
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Time = 0.92 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.09
method | result | size |
derivativedivides | \(\frac {\frac {b \sin \left (d x +c \right )}{2 a \left (a -b \right ) \left (\sin \left (d x +c \right )^{2} a -b \sin \left (d x +c \right )^{2}-a \right )}+\frac {\left (2 a -b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 a \left (a -b \right ) \sqrt {a \left (a -b \right )}}}{d}\) | \(102\) |
default | \(\frac {\frac {b \sin \left (d x +c \right )}{2 a \left (a -b \right ) \left (\sin \left (d x +c \right )^{2} a -b \sin \left (d x +c \right )^{2}-a \right )}+\frac {\left (2 a -b \right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \sin \left (d x +c \right )}{\sqrt {a \left (a -b \right )}}\right )}{2 a \left (a -b \right ) \sqrt {a \left (a -b \right )}}}{d}\) | \(102\) |
risch | \(\frac {i b \left ({\mathrm e}^{3 i \left (d x +c \right )}-{\mathrm e}^{i \left (d x +c \right )}\right )}{a d \left (-a +b \right ) \left (-a \,{\mathrm e}^{4 i \left (d x +c \right )}+b \,{\mathrm e}^{4 i \left (d x +c \right )}-2 a \,{\mathrm e}^{2 i \left (d x +c \right )}-2 b \,{\mathrm e}^{2 i \left (d x +c \right )}-a +b \right )}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right ) d}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right ) b}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d a}-\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right )}{2 \sqrt {a^{2}-a b}\, \left (a -b \right ) d}+\frac {\ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{\sqrt {a^{2}-a b}}-1\right ) b}{4 \sqrt {a^{2}-a b}\, \left (a -b \right ) d a}\) | \(330\) |
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none
Time = 0.30 (sec) , antiderivative size = 337, normalized size of antiderivative = 3.59 \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=\left [\frac {{\left ({\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sqrt {a^{2} - a b} \log \left (-\frac {{\left (a - b\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {a^{2} - a b} \sin \left (d x + c\right ) - 2 \, a + b}{{\left (a - b\right )} \cos \left (d x + c\right )^{2} + b}\right ) - 2 \, {\left (a^{2} b - a b^{2}\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}, -\frac {{\left ({\left (2 \, a^{2} - 3 \, a b + b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sqrt {-a^{2} + a b} \arctan \left (\frac {\sqrt {-a^{2} + a b} \sin \left (d x + c\right )}{a}\right ) + {\left (a^{2} b - a b^{2}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} d \cos \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )}}\right ] \]
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\[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=\int \frac {\sec {\left (c + d x \right )}}{\left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
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Exception generated. \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.57 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.19 \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=-\frac {\frac {{\left (2 \, a - b\right )} \arctan \left (\frac {a \sin \left (d x + c\right ) - b \sin \left (d x + c\right )}{\sqrt {-a^{2} + a b}}\right )}{{\left (a^{2} - a b\right )} \sqrt {-a^{2} + a b}} - \frac {b \sin \left (d x + c\right )}{{\left (a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right )^{2} - a\right )} {\left (a^{2} - a b\right )}}}{2 \, d} \]
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Time = 12.96 (sec) , antiderivative size = 239, normalized size of antiderivative = 2.54 \[ \int \frac {\sec (c+d x)}{\left (a+b \tan ^2(c+d x)\right )^2} \, dx=-\frac {\left (a^2\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}-\frac {b^2\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2}+a^2\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}+\frac {b^2\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2}+\frac {a\,b\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,1{}\mathrm {i}}{2}-\frac {a\,b\,\cos \left (2\,c+2\,d\,x\right )\,\mathrm {atanh}\left (\frac {\sin \left (c+d\,x\right )\,\sqrt {a-b}}{\sqrt {a}}\right )\,3{}\mathrm {i}}{2}-\sqrt {a}\,b\,\sin \left (c+d\,x\right )\,\sqrt {a-b}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,a^{3/2}\,d\,{\left (a-b\right )}^{3/2}\,\left (\frac {a}{2}+\frac {b}{2}+\frac {a\,\cos \left (2\,c+2\,d\,x\right )}{2}-\frac {b\,\cos \left (2\,c+2\,d\,x\right )}{2}\right )} \]
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