Optimal. Leaf size=149 \[ \frac {4 \left (8 a^2+b^2\right ) \tan ^{-1}\left (\frac {2 a \tan (c+d x)+b}{\sqrt {4 a^2-b^2}}\right )}{d \left (4 a^2-b^2\right )^{5/2}}+\frac {12 a b \cos (2 c+2 d x)}{d \left (4 a^2-b^2\right )^2 (2 a+b \sin (2 c+2 d x))}+\frac {2 b \cos (2 c+2 d x)}{d \left (4 a^2-b^2\right ) (2 a+b \sin (2 c+2 d x))^2} \]
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Rubi [A] time = 0.18, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.389, Rules used = {2666, 2664, 2754, 12, 2660, 618, 204} \[ \frac {4 \left (8 a^2+b^2\right ) \tan ^{-1}\left (\frac {2 a \tan (c+d x)+b}{\sqrt {4 a^2-b^2}}\right )}{d \left (4 a^2-b^2\right )^{5/2}}+\frac {12 a b \cos (2 c+2 d x)}{d \left (4 a^2-b^2\right )^2 (2 a+b \sin (2 c+2 d x))}+\frac {2 b \cos (2 c+2 d x)}{d \left (4 a^2-b^2\right ) (2 a+b \sin (2 c+2 d x))^2} \]
Antiderivative was successfully verified.
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Rule 12
Rule 204
Rule 618
Rule 2660
Rule 2664
Rule 2666
Rule 2754
Rubi steps
\begin {align*} \int \frac {1}{(a+b \cos (c+d x) \sin (c+d x))^3} \, dx &=\int \frac {1}{\left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )^3} \, dx\\ &=\frac {2 b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^2}-\frac {2 \int \frac {-2 a+\frac {1}{2} b \sin (2 c+2 d x)}{\left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )^2} \, dx}{4 a^2-b^2}\\ &=\frac {2 b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^2}+\frac {12 a b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right )^2 d (2 a+b \sin (2 c+2 d x))}+\frac {8 \int \frac {8 a^2+b^2}{4 \left (a+\frac {1}{2} b \sin (2 c+2 d x)\right )} \, dx}{\left (4 a^2-b^2\right )^2}\\ &=\frac {2 b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^2}+\frac {12 a b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right )^2 d (2 a+b \sin (2 c+2 d x))}+\frac {\left (2 \left (8 a^2+b^2\right )\right ) \int \frac {1}{a+\frac {1}{2} b \sin (2 c+2 d x)} \, dx}{\left (4 a^2-b^2\right )^2}\\ &=\frac {2 b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^2}+\frac {12 a b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right )^2 d (2 a+b \sin (2 c+2 d x))}+\frac {\left (2 \left (8 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (2 c+2 d x)\right )\right )}{\left (4 a^2-b^2\right )^2 d}\\ &=\frac {2 b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^2}+\frac {12 a b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right )^2 d (2 a+b \sin (2 c+2 d x))}-\frac {\left (4 \left (8 a^2+b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-4 a^2+b^2-x^2} \, dx,x,b+2 a \tan \left (\frac {1}{2} (2 c+2 d x)\right )\right )}{\left (4 a^2-b^2\right )^2 d}\\ &=\frac {4 \left (8 a^2+b^2\right ) \tan ^{-1}\left (\frac {b+2 a \tan (c+d x)}{\sqrt {4 a^2-b^2}}\right )}{\left (4 a^2-b^2\right )^{5/2} d}+\frac {2 b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right ) d (2 a+b \sin (2 c+2 d x))^2}+\frac {12 a b \cos (2 c+2 d x)}{\left (4 a^2-b^2\right )^2 d (2 a+b \sin (2 c+2 d x))}\\ \end {align*}
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Mathematica [A] time = 0.93, size = 120, normalized size = 0.81 \[ \frac {2 \left (\frac {2 \left (8 a^2+b^2\right ) \tan ^{-1}\left (\frac {2 a \tan (c+d x)+b}{\sqrt {4 a^2-b^2}}\right )}{\left (4 a^2-b^2\right )^{5/2}}+\frac {b \cos (2 (c+d x)) \left (16 a^2+6 a b \sin (2 (c+d x))-b^2\right )}{\left (b^2-4 a^2\right )^2 (2 a+b \sin (2 (c+d x)))^2}\right )}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.06, size = 969, normalized size = 6.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 252, normalized size = 1.69 \[ \frac {\frac {8 \, {\left (\pi \left \lfloor \frac {d x + c}{\pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\relax (a) + \arctan \left (\frac {2 \, a \tan \left (d x + c\right ) + b}{\sqrt {4 \, a^{2} - b^{2}}}\right )\right )} {\left (8 \, a^{2} + b^{2}\right )}}{{\left (16 \, a^{4} - 8 \, a^{2} b^{2} + b^{4}\right )} \sqrt {4 \, a^{2} - b^{2}}} + \frac {20 \, a^{3} b^{2} \tan \left (d x + c\right )^{3} - 2 \, a b^{4} \tan \left (d x + c\right )^{3} + 32 \, a^{4} b \tan \left (d x + c\right )^{2} + 14 \, a^{2} b^{3} \tan \left (d x + c\right )^{2} - b^{5} \tan \left (d x + c\right )^{2} + 44 \, a^{3} b^{2} \tan \left (d x + c\right ) - 2 \, a b^{4} \tan \left (d x + c\right ) + 32 \, a^{4} b - 2 \, a^{2} b^{3}}{{\left (16 \, a^{6} - 8 \, a^{4} b^{2} + a^{2} b^{4}\right )} {\left (a \tan \left (d x + c\right )^{2} + b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.49, size = 640, normalized size = 4.30 \[ \frac {10 b^{2} a \left (\tan ^{3}\left (d x +c \right )\right )}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \left (\tan ^{3}\left (d x +c \right )\right )}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right ) a}+\frac {16 b \,a^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}+\frac {7 b^{3} \left (\tan ^{2}\left (d x +c \right )\right )}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}-\frac {b^{5} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {22 b^{2} a \tan \left (d x +c \right )}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}-\frac {b^{4} \tan \left (d x +c \right )}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} a \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}+\frac {16 b \,a^{2}}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}-\frac {b^{3}}{d \left (\left (\tan ^{2}\left (d x +c \right )\right ) a +b \tan \left (d x +c \right )+a \right )^{2} \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right )}+\frac {32 \arctan \left (\frac {b +2 a \tan \left (d x +c \right )}{\sqrt {4 a^{2}-b^{2}}}\right ) a^{2}}{d \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \sqrt {4 a^{2}-b^{2}}}+\frac {4 \arctan \left (\frac {b +2 a \tan \left (d x +c \right )}{\sqrt {4 a^{2}-b^{2}}}\right ) b^{2}}{d \left (16 a^{4}-8 a^{2} b^{2}+b^{4}\right ) \sqrt {4 a^{2}-b^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.87, size = 396, normalized size = 2.66 \[ \frac {\frac {16\,a^2\,b-b^3}{16\,a^4-8\,a^2\,b^2+b^4}+\frac {b\,\mathrm {tan}\left (c+d\,x\right )\,\left (22\,a^2\,b-b^3\right )}{a\,\left (16\,a^4-8\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (16\,a^2\,b-b^3\right )\,\left (2\,a^2+b^2\right )}{2\,a^2\,\left (16\,a^4-8\,a^2\,b^2+b^4\right )}+\frac {b\,{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (10\,a^2\,b-b^3\right )}{a\,\left (16\,a^4-8\,a^2\,b^2+b^4\right )}}{d\,\left ({\mathrm {tan}\left (c+d\,x\right )}^2\,\left (2\,a^2+b^2\right )+a^2+a^2\,{\mathrm {tan}\left (c+d\,x\right )}^4+2\,a\,b\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}+\frac {4\,\mathrm {atan}\left (\frac {\left (\frac {4\,a\,\mathrm {tan}\left (c+d\,x\right )\,\left (8\,a^2+b^2\right )}{{\left (2\,a+b\right )}^{5/2}\,{\left (2\,a-b\right )}^{5/2}}+\frac {2\,\left (8\,a^2+b^2\right )\,\left (16\,a^4\,b-8\,a^2\,b^3+b^5\right )}{{\left (2\,a+b\right )}^{5/2}\,{\left (2\,a-b\right )}^{5/2}\,\left (16\,a^4-8\,a^2\,b^2+b^4\right )}\right )\,\left (16\,a^4-8\,a^2\,b^2+b^4\right )}{16\,a^2+2\,b^2}\right )\,\left (8\,a^2+b^2\right )}{d\,{\left (2\,a+b\right )}^{5/2}\,{\left (2\,a-b\right )}^{5/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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