Integrand size = 15, antiderivative size = 93 \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}+\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {arctanh}(\sin (x))}{8 (a+b)^3}+\frac {(3 a+7 b) \sec (x) \tan (x)}{8 (a+b)^2}+\frac {\sec ^3(x) \tan (x)}{4 (a+b)} \]
[Out]
Time = 0.16 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3269, 425, 541, 536, 212, 211} \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {arctanh}(\sin (x))}{8 (a+b)^3}+\frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}+\frac {\tan (x) \sec ^3(x)}{4 (a+b)}+\frac {(3 a+7 b) \tan (x) \sec (x)}{8 (a+b)^2} \]
[In]
[Out]
Rule 211
Rule 212
Rule 425
Rule 536
Rule 541
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\sin (x)\right ) \\ & = \frac {\sec ^3(x) \tan (x)}{4 (a+b)}+\frac {\text {Subst}\left (\int \frac {3 a+4 b+3 b x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\sin (x)\right )}{4 (a+b)} \\ & = \frac {(3 a+7 b) \sec (x) \tan (x)}{8 (a+b)^2}+\frac {\sec ^3(x) \tan (x)}{4 (a+b)}+\frac {\text {Subst}\left (\int \frac {3 a^2+7 a b+8 b^2+b (3 a+7 b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\sin (x)\right )}{8 (a+b)^2} \\ & = \frac {(3 a+7 b) \sec (x) \tan (x)}{8 (a+b)^2}+\frac {\sec ^3(x) \tan (x)}{4 (a+b)}+\frac {b^3 \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\sin (x)\right )}{(a+b)^3}+\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\sin (x)\right )}{8 (a+b)^3} \\ & = \frac {b^{5/2} \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^3}+\frac {\left (3 a^2+10 a b+15 b^2\right ) \text {arctanh}(\sin (x))}{8 (a+b)^3}+\frac {(3 a+7 b) \sec (x) \tan (x)}{8 (a+b)^2}+\frac {\sec ^3(x) \tan (x)}{4 (a+b)} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(214\) vs. \(2(93)=186\).
Time = 1.33 (sec) , antiderivative size = 214, normalized size of antiderivative = 2.30 \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=-\frac {\frac {8 b^{5/2} \arctan \left (\frac {\sqrt {a} \csc (x)}{\sqrt {b}}\right )}{\sqrt {a}}-\frac {8 b^{5/2} \arctan \left (\frac {\sqrt {b} \sin (x)}{\sqrt {a}}\right )}{\sqrt {a}}+2 \left (3 a^2+10 a b+15 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )-2 \left (3 a^2+10 a b+15 b^2\right ) \log \left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )-\frac {(a+b)^2}{\left (\cos \left (\frac {x}{2}\right )-\sin \left (\frac {x}{2}\right )\right )^4}+\frac {(a+b)^2}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^4}+\frac {(a+b) (3 a+7 b)}{\left (\cos \left (\frac {x}{2}\right )+\sin \left (\frac {x}{2}\right )\right )^2}+\frac {(a+b) (3 a+7 b)}{-1+\sin (x)}}{16 (a+b)^3} \]
[In]
[Out]
Time = 1.16 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.66
method | result | size |
default | \(-\frac {1}{2 \left (8 a +8 b \right ) \left (1+\sin \left (x \right )\right )^{2}}-\frac {3 a +7 b}{16 \left (a +b \right )^{2} \left (1+\sin \left (x \right )\right )}+\frac {\left (3 a^{2}+10 a b +15 b^{2}\right ) \ln \left (1+\sin \left (x \right )\right )}{16 \left (a +b \right )^{3}}+\frac {1}{2 \left (8 a +8 b \right ) \left (\sin \left (x \right )-1\right )^{2}}-\frac {3 a +7 b}{16 \left (a +b \right )^{2} \left (\sin \left (x \right )-1\right )}+\frac {\left (-3 a^{2}-10 a b -15 b^{2}\right ) \ln \left (\sin \left (x \right )-1\right )}{16 \left (a +b \right )^{3}}+\frac {b^{3} \arctan \left (\frac {b \sin \left (x \right )}{\sqrt {a b}}\right )}{\left (a +b \right )^{3} \sqrt {a b}}\) | \(154\) |
risch | \(-\frac {i \left (3 a \,{\mathrm e}^{7 i x}+7 b \,{\mathrm e}^{7 i x}+11 a \,{\mathrm e}^{5 i x}+15 b \,{\mathrm e}^{5 i x}-11 a \,{\mathrm e}^{3 i x}-15 b \,{\mathrm e}^{3 i x}-3 \,{\mathrm e}^{i x} a -7 \,{\mathrm e}^{i x} b \right )}{4 \left ({\mathrm e}^{2 i x}+1\right )^{4} \left (a +b \right )^{2}}+\frac {3 \ln \left ({\mathrm e}^{i x}+i\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {5 \ln \left ({\mathrm e}^{i x}+i\right ) a b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {15 \ln \left ({\mathrm e}^{i x}+i\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {3 \ln \left ({\mathrm e}^{i x}-i\right ) a^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {5 \ln \left ({\mathrm e}^{i x}-i\right ) a b}{4 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}-\frac {15 \ln \left ({\mathrm e}^{i x}-i\right ) b^{2}}{8 \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}+\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{2 a \left (a +b \right )^{3}}-\frac {\sqrt {-a b}\, b^{2} \ln \left ({\mathrm e}^{2 i x}-\frac {2 i \sqrt {-a b}\, {\mathrm e}^{i x}}{b}-1\right )}{2 a \left (a +b \right )^{3}}\) | \(380\) |
[In]
[Out]
none
Time = 0.40 (sec) , antiderivative size = 327, normalized size of antiderivative = 3.52 \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\left [\frac {8 \, b^{2} \sqrt {-\frac {b}{a}} \cos \left (x\right )^{4} \log \left (-\frac {b \cos \left (x\right )^{2} - 2 \, a \sqrt {-\frac {b}{a}} \sin \left (x\right ) + a - b}{b \cos \left (x\right )^{2} - a - b}\right ) + {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} \log \left (\sin \left (x\right ) + 1\right ) - {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, {\left ({\left (3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right )} \sin \left (x\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4}}, \frac {16 \, b^{2} \sqrt {\frac {b}{a}} \arctan \left (\sqrt {\frac {b}{a}} \sin \left (x\right )\right ) \cos \left (x\right )^{4} + {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} \log \left (\sin \left (x\right ) + 1\right ) - {\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \cos \left (x\right )^{4} \log \left (-\sin \left (x\right ) + 1\right ) + 2 \, {\left ({\left (3 \, a^{2} + 10 \, a b + 7 \, b^{2}\right )} \cos \left (x\right )^{2} + 2 \, a^{2} + 4 \, a b + 2 \, b^{2}\right )} \sin \left (x\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{4}}\right ] \]
[In]
[Out]
\[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\int \frac {\sec ^{5}{\left (x \right )}}{a + b \sin ^{2}{\left (x \right )}}\, dx \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 199 vs. \(2 (79) = 158\).
Time = 0.33 (sec) , antiderivative size = 199, normalized size of antiderivative = 2.14 \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {b^{3} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b}} + \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\sin \left (x\right ) + 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\sin \left (x\right ) - 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (3 \, a + 7 \, b\right )} \sin \left (x\right )^{3} - {\left (5 \, a + 9 \, b\right )} \sin \left (x\right )}{8 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (x\right )^{4} - 2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sin \left (x\right )^{2} + a^{2} + 2 \, a b + b^{2}\right )}} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 177 vs. \(2 (79) = 158\).
Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.90 \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {b^{3} \arctan \left (\frac {b \sin \left (x\right )}{\sqrt {a b}}\right )}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b}} + \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (\sin \left (x\right ) + 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {{\left (3 \, a^{2} + 10 \, a b + 15 \, b^{2}\right )} \log \left (-\sin \left (x\right ) + 1\right )}{16 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} - \frac {3 \, a \sin \left (x\right )^{3} + 7 \, b \sin \left (x\right )^{3} - 5 \, a \sin \left (x\right ) - 9 \, b \sin \left (x\right )}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} {\left (\sin \left (x\right )^{2} - 1\right )}^{2}} \]
[In]
[Out]
Time = 17.83 (sec) , antiderivative size = 832, normalized size of antiderivative = 8.95 \[ \int \frac {\sec ^5(x)}{a+b \sin ^2(x)} \, dx=\frac {5\,a^3\,\sin \left (x\right )-3\,a^3\,{\sin \left (x\right )}^3+3\,a^3\,\mathrm {atanh}\left (\sin \left (x\right )\right )+9\,a\,b^2\,\sin \left (x\right )+14\,a^2\,b\,\sin \left (x\right )-6\,a^3\,\mathrm {atanh}\left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^2+3\,a^3\,\mathrm {atanh}\left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^4-7\,a\,b^2\,{\sin \left (x\right )}^3-10\,a^2\,b\,{\sin \left (x\right )}^3+15\,a\,b^2\,\mathrm {atanh}\left (\sin \left (x\right )\right )+10\,a^2\,b\,\mathrm {atanh}\left (\sin \left (x\right )\right )-30\,a\,b^2\,\mathrm {atanh}\left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^2-20\,a^2\,b\,\mathrm {atanh}\left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^2+15\,a\,b^2\,\mathrm {atanh}\left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^4+10\,a^2\,b\,\mathrm {atanh}\left (\sin \left (x\right )\right )\,{\sin \left (x\right )}^4+\mathrm {atan}\left (\frac {a\,\sin \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}-b\,\sin \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}+a^6\,b\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,9{}\mathrm {i}+a^2\,b^5\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,289{}\mathrm {i}+a^3\,b^4\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,300{}\mathrm {i}+a^4\,b^3\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,190{}\mathrm {i}+a^5\,b^2\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,60{}\mathrm {i}}{9\,a^7\,b^3+60\,a^6\,b^4+190\,a^5\,b^5+300\,a^4\,b^6+225\,a^3\,b^7+64\,a^2\,b^8}\right )\,\sqrt {-a\,b^5}\,8{}\mathrm {i}-\mathrm {atan}\left (\frac {a\,\sin \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}-b\,\sin \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}+a^6\,b\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,9{}\mathrm {i}+a^2\,b^5\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,289{}\mathrm {i}+a^3\,b^4\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,300{}\mathrm {i}+a^4\,b^3\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,190{}\mathrm {i}+a^5\,b^2\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,60{}\mathrm {i}}{9\,a^7\,b^3+60\,a^6\,b^4+190\,a^5\,b^5+300\,a^4\,b^6+225\,a^3\,b^7+64\,a^2\,b^8}\right )\,{\sin \left (x\right )}^2\,\sqrt {-a\,b^5}\,16{}\mathrm {i}+\mathrm {atan}\left (\frac {a\,\sin \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}-b\,\sin \left (x\right )\,{\left (-a\,b^5\right )}^{3/2}\,64{}\mathrm {i}+a^6\,b\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,9{}\mathrm {i}+a^2\,b^5\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,289{}\mathrm {i}+a^3\,b^4\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,300{}\mathrm {i}+a^4\,b^3\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,190{}\mathrm {i}+a^5\,b^2\,\sin \left (x\right )\,\sqrt {-a\,b^5}\,60{}\mathrm {i}}{9\,a^7\,b^3+60\,a^6\,b^4+190\,a^5\,b^5+300\,a^4\,b^6+225\,a^3\,b^7+64\,a^2\,b^8}\right )\,{\sin \left (x\right )}^4\,\sqrt {-a\,b^5}\,8{}\mathrm {i}}{8\,a^4\,{\sin \left (x\right )}^4-16\,a^4\,{\sin \left (x\right )}^2+8\,a^4+24\,a^3\,b\,{\sin \left (x\right )}^4-48\,a^3\,b\,{\sin \left (x\right )}^2+24\,a^3\,b+24\,a^2\,b^2\,{\sin \left (x\right )}^4-48\,a^2\,b^2\,{\sin \left (x\right )}^2+24\,a^2\,b^2+8\,a\,b^3\,{\sin \left (x\right )}^4-16\,a\,b^3\,{\sin \left (x\right )}^2+8\,a\,b^3} \]
[In]
[Out]