Optimal. Leaf size=161 \[ \frac {a \log (1-\sin (c+d x))}{2 (a+b)^3 d}+\frac {a \log (1+\sin (c+d x))}{2 (a-b)^3 d}-\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}+\frac {a^3}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (a^2+b^2-2 a b \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 161, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2800, 1661,
1643} \begin {gather*} -\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^3}+\frac {\sec ^2(c+d x) \left (a^2-2 a b \sin (c+d x)+b^2\right )}{2 d \left (a^2-b^2\right )^2}+\frac {a^3}{d \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {a \log (1-\sin (c+d x))}{2 d (a+b)^3}+\frac {a \log (\sin (c+d x)+1)}{2 d (a-b)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 1643
Rule 1661
Rule 2800
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x)}{(a+b \sin (c+d x))^2} \, dx &=\frac {\text {Subst}\left (\int \frac {x^3}{(a+x)^2 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^2(c+d x) \left (a^2+b^2-2 a b \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d}+\frac {\text {Subst}\left (\int \frac {\frac {2 a^3 b^4}{\left (a^2-b^2\right )^2}-\frac {2 a^2 b^2 x}{a^2-b^2}-\frac {2 a b^4 x^2}{\left (a^2-b^2\right )^2}}{(a+x)^2 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 b^2 d}\\ &=\frac {\sec ^2(c+d x) \left (a^2+b^2-2 a b \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d}+\frac {\text {Subst}\left (\int \left (-\frac {a b^2}{(a+b)^3 (b-x)}-\frac {2 a^3 b^2}{(a-b)^2 (a+b)^2 (a+x)^2}-\frac {2 a^2 b^2 \left (a^2+3 b^2\right )}{(a-b)^3 (a+b)^3 (a+x)}+\frac {a b^2}{(a-b)^3 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 b^2 d}\\ &=\frac {a \log (1-\sin (c+d x))}{2 (a+b)^3 d}+\frac {a \log (1+\sin (c+d x))}{2 (a-b)^3 d}-\frac {a^2 \left (a^2+3 b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3 d}+\frac {a^3}{\left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}+\frac {\sec ^2(c+d x) \left (a^2+b^2-2 a b \sin (c+d x)\right )}{2 \left (a^2-b^2\right )^2 d}\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 145, normalized size = 0.90 \begin {gather*} \frac {\frac {2 a \log (1-\sin (c+d x))}{(a+b)^3}+\frac {2 a \log (1+\sin (c+d x))}{(a-b)^3}-\frac {4 a^2 \left (a^2+3 b^2\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^3}-\frac {1}{(a+b)^2 (-1+\sin (c+d x))}+\frac {1}{(a-b)^2 (1+\sin (c+d x))}+\frac {4 a^3}{\left (a^2-b^2\right )^2 (a+b \sin (c+d x))}}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.46, size = 143, normalized size = 0.89
method | result | size |
derivativedivides | \(\frac {\frac {a^{3}}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {1}{4 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {a \ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {a \ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(143\) |
default | \(\frac {\frac {a^{3}}{\left (a +b \right )^{2} \left (a -b \right )^{2} \left (a +b \sin \left (d x +c \right )\right )}-\frac {a^{2} \left (a^{2}+3 b^{2}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3}}+\frac {1}{4 \left (a -b \right )^{2} \left (1+\sin \left (d x +c \right )\right )}+\frac {a \ln \left (1+\sin \left (d x +c \right )\right )}{2 \left (a -b \right )^{3}}-\frac {1}{4 \left (a +b \right )^{2} \left (\sin \left (d x +c \right )-1\right )}+\frac {a \ln \left (\sin \left (d x +c \right )-1\right )}{2 \left (a +b \right )^{3}}}{d}\) | \(143\) |
risch | \(-\frac {i a x}{a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}}-\frac {i a c}{\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}-\frac {i a x}{a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}}-\frac {i a c}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}+\frac {2 i a^{4} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {2 i a^{4} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {6 i a^{2} b^{2} x}{a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}}+\frac {6 i a^{2} b^{2} c}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}+\frac {2 i \left (i a^{2} b \,{\mathrm e}^{4 i \left (d x +c \right )}-i b^{3} {\mathrm e}^{4 i \left (d x +c \right )}+a \,b^{2} {\mathrm e}^{5 i \left (d x +c \right )}-i a^{2} b \,{\mathrm e}^{2 i \left (d x +c \right )}+i b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+4 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}+b^{2} a \,{\mathrm e}^{i \left (d x +c \right )}+a^{3} {\mathrm e}^{5 i \left (d x +c \right )}+a^{3} {\mathrm e}^{i \left (d x +c \right )}\right )}{\left (1+{\mathrm e}^{2 i \left (d x +c \right )}\right )^{2} \left (b \,{\mathrm e}^{2 i \left (d x +c \right )}-b +2 i a \,{\mathrm e}^{i \left (d x +c \right )}\right ) d \left (a^{2}-b^{2}\right )^{2}}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}{\left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) d}+\frac {\ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right ) a}{d \left (a^{3}-3 a^{2} b +3 a \,b^{2}-b^{3}\right )}-\frac {a^{4} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right )}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}-\frac {3 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+\frac {2 i a \,{\mathrm e}^{i \left (d x +c \right )}}{b}-1\right ) b^{2}}{d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}\) | \(664\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 274, normalized size = 1.70 \begin {gather*} -\frac {\frac {2 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac {a \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {a \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {3 \, a^{3} + a b^{2} - 2 \, {\left (a^{3} + a b^{2}\right )} \sin \left (d x + c\right )^{2} - {\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{a^{5} - 2 \, a^{3} b^{2} + a b^{4} - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )^{3} - {\left (a^{5} - 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )^{2} + {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 388 vs.
\(2 (159) = 318\).
time = 0.45, size = 388, normalized size = 2.41 \begin {gather*} \frac {a^{5} - 2 \, a^{3} b^{2} + a b^{4} + 2 \, {\left (a^{5} - a b^{4}\right )} \cos \left (d x + c\right )^{2} - 2 \, {\left ({\left (a^{4} b + 3 \, a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{5} + 3 \, a^{3} b^{2}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (a^{4} b + 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} + a b^{4}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (a^{4} b - 3 \, a^{3} b^{2} + 3 \, a^{2} b^{3} - a b^{4}\right )} \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{5} - 3 \, a^{4} b + 3 \, a^{3} b^{2} - a^{2} b^{3}\right )} \cos \left (d x + c\right )^{2}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}\right )} d \cos \left (d x + c\right )^{2} \sin \left (d x + c\right ) + {\left (a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}\right )} d \cos \left (d x + c\right )^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{3}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.88, size = 248, normalized size = 1.54 \begin {gather*} -\frac {\frac {2 \, {\left (a^{4} b + 3 \, a^{2} b^{3}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac {a \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {a \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {2 \, a^{3} \sin \left (d x + c\right )^{2} + 2 \, a b^{2} \sin \left (d x + c\right )^{2} + a^{2} b \sin \left (d x + c\right ) - b^{3} \sin \left (d x + c\right ) - 3 \, a^{3} - a b^{2}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - a\right )}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.25, size = 351, normalized size = 2.18 \begin {gather*} \frac {\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{a^2-b^2}+\frac {2\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{a^2-b^2}+\frac {4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (a^2+b^2\right )}{{\left (a^2-b^2\right )}^2}-\frac {4\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^4-2\,a^2\,b^2+b^4}-\frac {4\,a^2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{{\left (a^2-b^2\right )}^2}}{d\,\left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-4\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )}{d\,{\left (a-b\right )}^3}-\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^4+3\,a^2\,b^2\right )}{d\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {a\,\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )}{d\,{\left (a+b\right )}^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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