Integrand size = 27, antiderivative size = 72 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=-\frac {a b \text {arctanh}(\sin (c+d x))}{4 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) \left (b^2+a b \sin (c+d x)\right )}{4 d} \]
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Time = 0.07 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2916, 12, 835, 653, 212} \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=-\frac {a b \text {arctanh}(\sin (c+d x))}{4 d}-\frac {\sec ^2(c+d x) \left (a b \sin (c+d x)+b^2\right )}{4 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d} \]
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Rule 12
Rule 212
Rule 653
Rule 835
Rule 2916
Rubi steps \begin{align*} \text {integral}& = \frac {b^5 \text {Subst}\left (\int \frac {x (a+x)^2}{b \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {b^4 \text {Subst}\left (\int \frac {x (a+x)^2}{\left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d} \\ & = \frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {b^2 \text {Subst}\left (\int \frac {2 b^2 (a+x)}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = \frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {b^4 \text {Subst}\left (\int \frac {a+x}{\left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{2 d} \\ & = \frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) \left (b^2+a b \sin (c+d x)\right )}{4 d}-\frac {\left (a b^2\right ) \text {Subst}\left (\int \frac {1}{b^2-x^2} \, dx,x,b \sin (c+d x)\right )}{4 d} \\ & = -\frac {a b \text {arctanh}(\sin (c+d x))}{4 d}+\frac {\sec ^4(c+d x) (a+b \sin (c+d x))^2}{4 d}-\frac {\sec ^2(c+d x) \left (b^2+a b \sin (c+d x)\right )}{4 d} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(215\) vs. \(2(72)=144\).
Time = 1.95 (sec) , antiderivative size = 215, normalized size of antiderivative = 2.99 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=\frac {a b \left (a^2-b^2\right )^2 (\log (1-\sin (c+d x))-\log (1+\sin (c+d x)))+2 a^4 b^2 \sec ^2(c+d x)+2 a^4 \left (a^2-b^2\right ) \sec ^4(c+d x)+4 a^3 b \left (a^2-b^2\right ) \sec ^3(c+d x) \tan (c+d x)+b \left (-6 a^4 b+4 a^2 b^3\right ) \tan ^2(c+d x)+2 b^4 \left (-a^2+b^2\right ) \tan ^4(c+d x)-2 a b \left (a^2-b^2\right ) \sec (c+d x) \tan (c+d x) \left (a^2+b^2+2 b^2 \tan ^2(c+d x)\right )}{8 \left (a^2-b^2\right )^2 d} \]
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Time = 0.57 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.46
method | result | size |
derivativedivides | \(\frac {\frac {a^{2}}{4 \cos \left (d x +c \right )^{4}}+2 a b \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(105\) |
default | \(\frac {\frac {a^{2}}{4 \cos \left (d x +c \right )^{4}}+2 a b \left (\frac {\sin ^{3}\left (d x +c \right )}{4 \cos \left (d x +c \right )^{4}}+\frac {\sin ^{3}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )}{8}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{8}\right )+\frac {b^{2} \left (\sin ^{4}\left (d x +c \right )\right )}{4 \cos \left (d x +c \right )^{4}}}{d}\) | \(105\) |
risch | \(\frac {i \left (4 i b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+a b \,{\mathrm e}^{7 i \left (d x +c \right )}-8 i a^{2} {\mathrm e}^{4 i \left (d x +c \right )}-7 a b \,{\mathrm e}^{5 i \left (d x +c \right )}+4 i b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+7 a b \,{\mathrm e}^{3 i \left (d x +c \right )}-a b \,{\mathrm e}^{i \left (d x +c \right )}\right )}{2 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{4}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a b}{4 d}-\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a b}{4 d}\) | \(158\) |
parallelrisch | \(\frac {4 b \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )-4 b \left (\frac {3}{4}+\frac {\cos \left (4 d x +4 c \right )}{4}+\cos \left (2 d x +2 c \right )\right ) a \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+\left (-4 a^{2}-4 b^{2}\right ) \cos \left (2 d x +2 c \right )+\left (-a^{2}+b^{2}\right ) \cos \left (4 d x +4 c \right )+14 a b \sin \left (d x +c \right )-2 a b \sin \left (3 d x +3 c \right )+5 a^{2}+3 b^{2}}{4 d \left (\cos \left (4 d x +4 c \right )+4 \cos \left (2 d x +2 c \right )+3\right )}\) | \(180\) |
norman | \(\frac {\frac {\left (4 a^{2}+4 b^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {\left (4 a^{2}+4 b^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {2 a^{2} \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {4 \left (a^{2}+2 b^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d}+\frac {9 a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {11 a b \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {11 a b \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\frac {9 a b \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a b \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4} \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}+\frac {a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{4 d}-\frac {a b \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{4 d}\) | \(293\) |
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Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.44 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=-\frac {a b \cos \left (d x + c\right )^{4} \log \left (\sin \left (d x + c\right ) + 1\right ) - a b \cos \left (d x + c\right )^{4} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 4 \, b^{2} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - 2 \, b^{2} + 2 \, {\left (a b \cos \left (d x + c\right )^{2} - 2 \, a b\right )} \sin \left (d x + c\right )}{8 \, d \cos \left (d x + c\right )^{4}} \]
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Timed out. \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=\text {Timed out} \]
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Time = 0.19 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.35 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=-\frac {a b \log \left (\sin \left (d x + c\right ) + 1\right ) - a b \log \left (\sin \left (d x + c\right ) - 1\right ) - \frac {2 \, {\left (a b \sin \left (d x + c\right )^{3} + 2 \, b^{2} \sin \left (d x + c\right )^{2} + a b \sin \left (d x + c\right ) + a^{2} - b^{2}\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1}}{8 \, d} \]
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Time = 0.40 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.24 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=-\frac {a b \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right ) - a b \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right ) - \frac {2 \, {\left (a b \sin \left (d x + c\right )^{3} + 2 \, b^{2} \sin \left (d x + c\right )^{2} + a b \sin \left (d x + c\right ) + a^{2} - b^{2}\right )}}{{\left (\sin \left (d x + c\right )^{2} - 1\right )}^{2}}}{8 \, d} \]
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Time = 19.54 (sec) , antiderivative size = 183, normalized size of antiderivative = 2.54 \[ \int \sec ^4(c+d x) (a+b \sin (c+d x))^2 \tan (c+d x) \, dx=\frac {2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2}+\frac {7\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2}+\frac {7\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{2}+\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{2}+4\,b^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {a\,b\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{2\,d} \]
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