Integrand size = 14, antiderivative size = 279 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{128 a^{9/2} (a+b)^{9/2} d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )} \]
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Time = 0.36 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {3263, 3252, 12, 3260, 211} \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {7 b (2 a+b) \sin (c+d x) \cos (c+d x)}{48 a^2 d (a+b)^2 \left (a+b \sin ^2(c+d x)\right )^3}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \sin (c+d x) \cos (c+d x)}{384 a^4 d (a+b)^4 \left (a+b \sin ^2(c+d x)\right )}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \sin (c+d x) \cos (c+d x)}{192 a^3 d (a+b)^3 \left (a+b \sin ^2(c+d x)\right )^2}+\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{128 a^{9/2} d (a+b)^{9/2}}+\frac {b \sin (c+d x) \cos (c+d x)}{8 a d (a+b) \left (a+b \sin ^2(c+d x)\right )^4} \]
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Rule 12
Rule 211
Rule 3252
Rule 3260
Rule 3263
Rubi steps \begin{align*} \text {integral}& = \frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}-\frac {\int \frac {-8 a-7 b+6 b \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^4} \, dx}{8 a (a+b)} \\ & = \frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}-\frac {\int \frac {-48 a^2-76 a b-35 b^2+28 b (2 a+b) \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^3} \, dx}{48 a^2 (a+b)^2} \\ & = \frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}-\frac {\int \frac {-192 a^3-392 a^2 b-340 a b^2-105 b^3+2 b \left (104 a^2+104 a b+35 b^2\right ) \sin ^2(c+d x)}{\left (a+b \sin ^2(c+d x)\right )^2} \, dx}{192 a^3 (a+b)^3} \\ & = \frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}-\frac {\int -\frac {3 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right )}{a+b \sin ^2(c+d x)} \, dx}{384 a^4 (a+b)^4} \\ & = \frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}+\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \int \frac {1}{a+b \sin ^2(c+d x)} \, dx}{128 a^4 (a+b)^4} \\ & = \frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )}+\frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \text {Subst}\left (\int \frac {1}{a+(a+b) x^2} \, dx,x,\tan (c+d x)\right )}{128 a^4 (a+b)^4 d} \\ & = \frac {\left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{128 a^{9/2} (a+b)^{9/2} d}+\frac {b \cos (c+d x) \sin (c+d x)}{8 a (a+b) d \left (a+b \sin ^2(c+d x)\right )^4}+\frac {7 b (2 a+b) \cos (c+d x) \sin (c+d x)}{48 a^2 (a+b)^2 d \left (a+b \sin ^2(c+d x)\right )^3}+\frac {b \left (104 a^2+104 a b+35 b^2\right ) \cos (c+d x) \sin (c+d x)}{192 a^3 (a+b)^3 d \left (a+b \sin ^2(c+d x)\right )^2}+\frac {5 b (2 a+b) \left (40 a^2+40 a b+21 b^2\right ) \cos (c+d x) \sin (c+d x)}{384 a^4 (a+b)^4 d \left (a+b \sin ^2(c+d x)\right )} \\ \end{align*}
Time = 11.90 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {24 \left (128 a^4+256 a^3 b+288 a^2 b^2+160 a b^3+35 b^4\right ) \arctan \left (\frac {\sqrt {a+b} \tan (c+d x)}{\sqrt {a}}\right )}{(a+b)^{9/2}}+\frac {2 \sqrt {a} b \left (24576 a^6+73728 a^5 b+97280 a^4 b^2+71680 a^3 b^3+32272 a^2 b^4+8720 a b^5+1050 b^6-b \left (27648 a^5+69120 a^4 b+73616 a^3 b^2+41304 a^2 b^3+12310 a b^4+1575 b^5\right ) \cos (2 (c+d x))+2 b^2 \left (2816 a^4+5632 a^3 b+4816 a^2 b^2+2000 a b^3+315 b^4\right ) \cos (4 (c+d x))-400 a^3 b^3 \cos (6 (c+d x))-600 a^2 b^4 \cos (6 (c+d x))-410 a b^5 \cos (6 (c+d x))-105 b^6 \cos (6 (c+d x))\right ) \sin (2 (c+d x))}{(a+b)^4 (2 a+b-b \cos (2 (c+d x)))^4}}{3072 a^{9/2} d} \]
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Time = 3.55 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.20
method | result | size |
derivativedivides | \(\frac {\frac {\frac {b \left (256 a^{3}+288 a^{2} b +160 a \,b^{2}+35 b^{3}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{128 a^{4} \left (a +b \right )}+\frac {\left (2304 a^{3}+3168 a^{2} b +1760 a \,b^{2}+385 b^{3}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{384 a^{3} \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2304 a^{3}+3744 a^{2} b +2336 a \,b^{2}+511 b^{3}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{384 a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b \left (256 a^{3}+480 a^{2} b +352 a \,b^{2}+93 b^{3}\right ) \tan \left (d x +c \right )}{128 a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{4}}+\frac {\left (128 a^{4}+256 a^{3} b +288 a^{2} b^{2}+160 a \,b^{3}+35 b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{128 a^{4} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \sqrt {a \left (a +b \right )}}}{d}\) | \(336\) |
default | \(\frac {\frac {\frac {b \left (256 a^{3}+288 a^{2} b +160 a \,b^{2}+35 b^{3}\right ) \left (\tan ^{7}\left (d x +c \right )\right )}{128 a^{4} \left (a +b \right )}+\frac {\left (2304 a^{3}+3168 a^{2} b +1760 a \,b^{2}+385 b^{3}\right ) b \left (\tan ^{5}\left (d x +c \right )\right )}{384 a^{3} \left (a^{2}+2 a b +b^{2}\right )}+\frac {\left (2304 a^{3}+3744 a^{2} b +2336 a \,b^{2}+511 b^{3}\right ) b \left (\tan ^{3}\left (d x +c \right )\right )}{384 a^{2} \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right )}+\frac {b \left (256 a^{3}+480 a^{2} b +352 a \,b^{2}+93 b^{3}\right ) \tan \left (d x +c \right )}{128 a \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right )}}{{\left (a \left (\tan ^{2}\left (d x +c \right )\right )+\left (\tan ^{2}\left (d x +c \right )\right ) b +a \right )}^{4}}+\frac {\left (128 a^{4}+256 a^{3} b +288 a^{2} b^{2}+160 a \,b^{3}+35 b^{4}\right ) \arctan \left (\frac {\left (a +b \right ) \tan \left (d x +c \right )}{\sqrt {a \left (a +b \right )}}\right )}{128 a^{4} \left (a^{4}+4 a^{3} b +6 a^{2} b^{2}+4 a \,b^{3}+b^{4}\right ) \sqrt {a \left (a +b \right )}}}{d}\) | \(336\) |
risch | \(\text {Expression too large to display}\) | \(1743\) |
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Leaf count of result is larger than twice the leaf count of optimal. 958 vs. \(2 (261) = 522\).
Time = 0.38 (sec) , antiderivative size = 2017, normalized size of antiderivative = 7.23 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\text {Timed out} \]
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Leaf count of result is larger than twice the leaf count of optimal. 588 vs. \(2 (261) = 522\).
Time = 0.40 (sec) , antiderivative size = 588, normalized size of antiderivative = 2.11 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {3 \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} \arctan \left (\frac {{\left (a + b\right )} \tan \left (d x + c\right )}{\sqrt {{\left (a + b\right )} a}}\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt {{\left (a + b\right )} a}} + \frac {3 \, {\left (256 \, a^{6} b + 1056 \, a^{5} b^{2} + 1792 \, a^{4} b^{3} + 1635 \, a^{3} b^{4} + 873 \, a^{2} b^{5} + 265 \, a b^{6} + 35 \, b^{7}\right )} \tan \left (d x + c\right )^{7} + {\left (2304 \, a^{6} b + 7776 \, a^{5} b^{2} + 10400 \, a^{4} b^{3} + 7073 \, a^{3} b^{4} + 2530 \, a^{2} b^{5} + 385 \, a b^{6}\right )} \tan \left (d x + c\right )^{5} + {\left (2304 \, a^{6} b + 6048 \, a^{5} b^{2} + 6080 \, a^{4} b^{3} + 2847 \, a^{3} b^{4} + 511 \, a^{2} b^{5}\right )} \tan \left (d x + c\right )^{3} + 3 \, {\left (256 \, a^{6} b + 480 \, a^{5} b^{2} + 352 \, a^{4} b^{3} + 93 \, a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{12} + 4 \, a^{11} b + 6 \, a^{10} b^{2} + 4 \, a^{9} b^{3} + a^{8} b^{4} + {\left (a^{12} + 8 \, a^{11} b + 28 \, a^{10} b^{2} + 56 \, a^{9} b^{3} + 70 \, a^{8} b^{4} + 56 \, a^{7} b^{5} + 28 \, a^{6} b^{6} + 8 \, a^{5} b^{7} + a^{4} b^{8}\right )} \tan \left (d x + c\right )^{8} + 4 \, {\left (a^{12} + 7 \, a^{11} b + 21 \, a^{10} b^{2} + 35 \, a^{9} b^{3} + 35 \, a^{8} b^{4} + 21 \, a^{7} b^{5} + 7 \, a^{6} b^{6} + a^{5} b^{7}\right )} \tan \left (d x + c\right )^{6} + 6 \, {\left (a^{12} + 6 \, a^{11} b + 15 \, a^{10} b^{2} + 20 \, a^{9} b^{3} + 15 \, a^{8} b^{4} + 6 \, a^{7} b^{5} + a^{6} b^{6}\right )} \tan \left (d x + c\right )^{4} + 4 \, {\left (a^{12} + 5 \, a^{11} b + 10 \, a^{10} b^{2} + 10 \, a^{9} b^{3} + 5 \, a^{8} b^{4} + a^{7} b^{5}\right )} \tan \left (d x + c\right )^{2}}}{384 \, d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 524 vs. \(2 (261) = 522\).
Time = 0.38 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.88 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {3 \, {\left (128 \, a^{4} + 256 \, a^{3} b + 288 \, a^{2} b^{2} + 160 \, a b^{3} + 35 \, b^{4}\right )} {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a + 2 \, b\right ) + \arctan \left (\frac {a \tan \left (d x + c\right ) + b \tan \left (d x + c\right )}{\sqrt {a^{2} + a b}}\right )\right )}}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} \sqrt {a^{2} + a b}} + \frac {768 \, a^{6} b \tan \left (d x + c\right )^{7} + 3168 \, a^{5} b^{2} \tan \left (d x + c\right )^{7} + 5376 \, a^{4} b^{3} \tan \left (d x + c\right )^{7} + 4905 \, a^{3} b^{4} \tan \left (d x + c\right )^{7} + 2619 \, a^{2} b^{5} \tan \left (d x + c\right )^{7} + 795 \, a b^{6} \tan \left (d x + c\right )^{7} + 105 \, b^{7} \tan \left (d x + c\right )^{7} + 2304 \, a^{6} b \tan \left (d x + c\right )^{5} + 7776 \, a^{5} b^{2} \tan \left (d x + c\right )^{5} + 10400 \, a^{4} b^{3} \tan \left (d x + c\right )^{5} + 7073 \, a^{3} b^{4} \tan \left (d x + c\right )^{5} + 2530 \, a^{2} b^{5} \tan \left (d x + c\right )^{5} + 385 \, a b^{6} \tan \left (d x + c\right )^{5} + 2304 \, a^{6} b \tan \left (d x + c\right )^{3} + 6048 \, a^{5} b^{2} \tan \left (d x + c\right )^{3} + 6080 \, a^{4} b^{3} \tan \left (d x + c\right )^{3} + 2847 \, a^{3} b^{4} \tan \left (d x + c\right )^{3} + 511 \, a^{2} b^{5} \tan \left (d x + c\right )^{3} + 768 \, a^{6} b \tan \left (d x + c\right ) + 1440 \, a^{5} b^{2} \tan \left (d x + c\right ) + 1056 \, a^{4} b^{3} \tan \left (d x + c\right ) + 279 \, a^{3} b^{4} \tan \left (d x + c\right )}{{\left (a^{8} + 4 \, a^{7} b + 6 \, a^{6} b^{2} + 4 \, a^{5} b^{3} + a^{4} b^{4}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right )^{2} + a\right )}^{4}}}{384 \, d} \]
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Time = 16.12 (sec) , antiderivative size = 450, normalized size of antiderivative = 1.61 \[ \int \frac {1}{\left (a+b \sin ^2(c+d x)\right )^5} \, dx=\frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (256\,a^3\,b+480\,a^2\,b^2+352\,a\,b^3+93\,b^4\right )}{128\,a\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (2304\,a^3\,b+3744\,a^2\,b^2+2336\,a\,b^3+511\,b^4\right )}{384\,a^2\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^5\,\left (2304\,a^3\,b+3168\,a^2\,b^2+1760\,a\,b^3+385\,b^4\right )}{384\,a^3\,\left (a^2+2\,a\,b+b^2\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^7\,\left (256\,a^3\,b+288\,a^2\,b^2+160\,a\,b^3+35\,b^4\right )}{128\,a^4\,\left (a+b\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^4\,\left (6\,a^4+12\,a^3\,b+6\,a^2\,b^2\right )+{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (4\,a^4+4\,b\,a^3\right )+{\mathrm {tan}\left (c+d\,x\right )}^6\,\left (4\,a^4+12\,a^3\,b+12\,a^2\,b^2+4\,a\,b^3\right )+a^4+{\mathrm {tan}\left (c+d\,x\right )}^8\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )\right )}+\frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,a+2\,b\right )\,\left (a^4+4\,a^3\,b+6\,a^2\,b^2+4\,a\,b^3+b^4\right )}{2\,\sqrt {a}\,{\left (a+b\right )}^{9/2}}\right )\,\left (128\,a^4+256\,a^3\,b+288\,a^2\,b^2+160\,a\,b^3+35\,b^4\right )}{128\,a^{9/2}\,d\,{\left (a+b\right )}^{9/2}} \]
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