Optimal. Leaf size=214 \[ \frac {\left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {(b c-a d)^2}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {2 (b c-a d) (a c+b d)}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))} \]
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Rubi [A]
time = 0.25, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3623, 3610,
3612, 3611} \begin {gather*} -\frac {(b c-a d)^2}{2 b f \left (a^2+b^2\right ) (a+b \tan (e+f x))^2}-\frac {2 (a c+b d) (b c-a d)}{f \left (a^2+b^2\right )^2 (a+b \tan (e+f x))}-\frac {\left (2 a^3 c d-3 a^2 b \left (c^2-d^2\right )-6 a b^2 c d+b^3 \left (c^2-d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{f \left (a^2+b^2\right )^3}+\frac {x \left (a^3 \left (c^2-d^2\right )+6 a^2 b c d-3 a b^2 \left (c^2-d^2\right )-2 b^3 c d\right )}{\left (a^2+b^2\right )^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 3610
Rule 3611
Rule 3612
Rule 3623
Rubi steps
\begin {align*} \int \frac {(c+d \tan (e+f x))^2}{(a+b \tan (e+f x))^3} \, dx &=-\frac {(b c-a d)^2}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}+\frac {\int \frac {2 b c d+a \left (c^2-d^2\right )+\left (2 a c d-b \left (c^2-d^2\right )\right ) \tan (e+f x)}{(a+b \tan (e+f x))^2} \, dx}{a^2+b^2}\\ &=-\frac {(b c-a d)^2}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {2 (b c-a d) (a c+b d)}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}+\frac {\int \frac {(a c+b c-a d+b d) (a c-b c+a d+b d)-2 (b c-a d) (a c+b d) \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^2}\\ &=\frac {\left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac {(b c-a d)^2}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {2 (b c-a d) (a c+b d)}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}-\frac {\left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \int \frac {b-a \tan (e+f x)}{a+b \tan (e+f x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=\frac {\left (6 a^2 b c d-2 b^3 c d+a^3 \left (c^2-d^2\right )-3 a b^2 \left (c^2-d^2\right )\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (2 a^3 c d-6 a b^2 c d-3 a^2 b \left (c^2-d^2\right )+b^3 \left (c^2-d^2\right )\right ) \log (a \cos (e+f x)+b \sin (e+f x))}{\left (a^2+b^2\right )^3 f}-\frac {(b c-a d)^2}{2 b \left (a^2+b^2\right ) f (a+b \tan (e+f x))^2}-\frac {2 (b c-a d) (a c+b d)}{\left (a^2+b^2\right )^2 f (a+b \tan (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 3.74, size = 291, normalized size = 1.36 \begin {gather*} \frac {\frac {b d (c+d \tan (e+f x))^2}{a+b \tan (e+f x)}-\frac {b^2 (c+d \tan (e+f x))^3}{(a+b \tan (e+f x))^2}+(b c-a d) \left (\frac {(i a+b)^3 (c+i d)^2 \log (i-\tan (e+f x))}{\left (a^2+b^2\right )^2}+\frac {i (a+i b) (c-i d)^2 \log (i+\tan (e+f x))}{(a-i b)^2}-\frac {2 \left (2 a^3 c d-6 a b^2 c d+b^3 \left (c^2-d^2\right )+3 a^2 b \left (-c^2+d^2\right )\right ) \log (a+b \tan (e+f x))}{\left (a^2+b^2\right )^2}-\frac {2 (b c-a d) \left (2 a b c-a^2 d+b^2 d\right )}{b \left (a^2+b^2\right ) (a+b \tan (e+f x))}\right )}{2 \left (a^2+b^2\right ) (b c-a d) f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.26, size = 304, normalized size = 1.42 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 437 vs.
\(2 (216) = 432\).
time = 0.53, size = 437, normalized size = 2.04 \begin {gather*} \frac {\frac {2 \, {\left ({\left (a^{3} - 3 \, a b^{2}\right )} c^{2} + 2 \, {\left (3 \, a^{2} b - b^{3}\right )} c d - {\left (a^{3} - 3 \, a b^{2}\right )} d^{2}\right )} {\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d - {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (b \tan \left (f x + e\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left ({\left (3 \, a^{2} b - b^{3}\right )} c^{2} - 2 \, {\left (a^{3} - 3 \, a b^{2}\right )} c d - {\left (3 \, a^{2} b - b^{3}\right )} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (5 \, a^{2} b^{2} + b^{4}\right )} c^{2} - 2 \, {\left (3 \, a^{3} b - a b^{3}\right )} c d + {\left (a^{4} - 3 \, a^{2} b^{2}\right )} d^{2} + 4 \, {\left (a b^{3} c^{2} - a b^{3} d^{2} - {\left (a^{2} b^{2} - b^{4}\right )} c d\right )} \tan \left (f x + e\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5} + {\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (f x + e\right )}}{2 \, f} \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. 715 vs.
\(2 (216) = 432\).
time = 1.03, size = 715, normalized size = 3.34 \begin {gather*} -\frac {{\left (7 \, a^{2} b^{3} + b^{5}\right )} c^{2} - 2 \, {\left (5 \, a^{3} b^{2} - a b^{4}\right )} c d + 3 \, {\left (a^{4} b - a^{2} b^{3}\right )} d^{2} - 2 \, {\left ({\left (a^{5} - 3 \, a^{3} b^{2}\right )} c^{2} + 2 \, {\left (3 \, a^{4} b - a^{2} b^{3}\right )} c d - {\left (a^{5} - 3 \, a^{3} b^{2}\right )} d^{2}\right )} f x - {\left ({\left (5 \, a^{2} b^{3} - b^{5}\right )} c^{2} - 6 \, {\left (a^{3} b^{2} - a b^{4}\right )} c d + {\left (a^{4} b - 5 \, a^{2} b^{3}\right )} d^{2} + 2 \, {\left ({\left (a^{3} b^{2} - 3 \, a b^{4}\right )} c^{2} + 2 \, {\left (3 \, a^{2} b^{3} - b^{5}\right )} c d - {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} d^{2}\right )} f x\right )} \tan \left (f x + e\right )^{2} - {\left ({\left (3 \, a^{4} b - a^{2} b^{3}\right )} c^{2} - 2 \, {\left (a^{5} - 3 \, a^{3} b^{2}\right )} c d - {\left (3 \, a^{4} b - a^{2} b^{3}\right )} d^{2} + {\left ({\left (3 \, a^{2} b^{3} - b^{5}\right )} c^{2} - 2 \, {\left (a^{3} b^{2} - 3 \, a b^{4}\right )} c d - {\left (3 \, a^{2} b^{3} - b^{5}\right )} d^{2}\right )} \tan \left (f x + e\right )^{2} + 2 \, {\left ({\left (3 \, a^{3} b^{2} - a b^{4}\right )} c^{2} - 2 \, {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} c d - {\left (3 \, a^{3} b^{2} - a b^{4}\right )} d^{2}\right )} \tan \left (f x + e\right )\right )} \log \left (\frac {b^{2} \tan \left (f x + e\right )^{2} + 2 \, a b \tan \left (f x + e\right ) + a^{2}}{\tan \left (f x + e\right )^{2} + 1}\right ) - 2 \, {\left (3 \, {\left (a^{3} b^{2} - a b^{4}\right )} c^{2} - 2 \, {\left (2 \, a^{4} b - 3 \, a^{2} b^{3} + b^{5}\right )} c d + {\left (a^{5} - 3 \, a^{3} b^{2} + 2 \, a b^{4}\right )} d^{2} + 2 \, {\left ({\left (a^{4} b - 3 \, a^{2} b^{3}\right )} c^{2} + 2 \, {\left (3 \, a^{3} b^{2} - a b^{4}\right )} c d - {\left (a^{4} b - 3 \, a^{2} b^{3}\right )} d^{2}\right )} f x\right )} \tan \left (f x + e\right )}{2 \, {\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} f \tan \left (f x + e\right )^{2} + 2 \, {\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} f \tan \left (f x + e\right ) + {\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: AttributeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 614 vs.
\(2 (216) = 432\).
time = 0.73, size = 614, normalized size = 2.87 \begin {gather*} \frac {\frac {2 \, {\left (a^{3} c^{2} - 3 \, a b^{2} c^{2} + 6 \, a^{2} b c d - 2 \, b^{3} c d - a^{3} d^{2} + 3 \, a b^{2} d^{2}\right )} {\left (f x + e\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (3 \, a^{2} b c^{2} - b^{3} c^{2} - 2 \, a^{3} c d + 6 \, a b^{2} c d - 3 \, a^{2} b d^{2} + b^{3} d^{2}\right )} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (3 \, a^{2} b^{2} c^{2} - b^{4} c^{2} - 2 \, a^{3} b c d + 6 \, a b^{3} c d - 3 \, a^{2} b^{2} d^{2} + b^{4} d^{2}\right )} \log \left ({\left | b \tan \left (f x + e\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {9 \, a^{2} b^{4} c^{2} \tan \left (f x + e\right )^{2} - 3 \, b^{6} c^{2} \tan \left (f x + e\right )^{2} - 6 \, a^{3} b^{3} c d \tan \left (f x + e\right )^{2} + 18 \, a b^{5} c d \tan \left (f x + e\right )^{2} - 9 \, a^{2} b^{4} d^{2} \tan \left (f x + e\right )^{2} + 3 \, b^{6} d^{2} \tan \left (f x + e\right )^{2} + 22 \, a^{3} b^{3} c^{2} \tan \left (f x + e\right ) - 2 \, a b^{5} c^{2} \tan \left (f x + e\right ) - 16 \, a^{4} b^{2} c d \tan \left (f x + e\right ) + 36 \, a^{2} b^{4} c d \tan \left (f x + e\right ) + 4 \, b^{6} c d \tan \left (f x + e\right ) - 22 \, a^{3} b^{3} d^{2} \tan \left (f x + e\right ) + 2 \, a b^{5} d^{2} \tan \left (f x + e\right ) + 14 \, a^{4} b^{2} c^{2} + 3 \, a^{2} b^{4} c^{2} + b^{6} c^{2} - 12 \, a^{5} b c d + 14 \, a^{3} b^{3} c d + 2 \, a b^{5} c d + a^{6} d^{2} - 11 \, a^{4} b^{2} d^{2}}{{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{2}}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.30, size = 367, normalized size = 1.71 \begin {gather*} -\frac {\frac {2\,\mathrm {tan}\left (e+f\,x\right )\,\left (-a^2\,b\,c\,d+a\,b^2\,c^2-a\,b^2\,d^2+b^3\,c\,d\right )}{a^4+2\,a^2\,b^2+b^4}+\frac {a^4\,d^2-6\,a^3\,b\,c\,d+5\,a^2\,b^2\,c^2-3\,a^2\,b^2\,d^2+2\,a\,b^3\,c\,d+b^4\,c^2}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{f\,\left (a^2+2\,a\,b\,\mathrm {tan}\left (e+f\,x\right )+b^2\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )-\mathrm {i}\right )\,\left (-c^2\,1{}\mathrm {i}+2\,c\,d+d^2\,1{}\mathrm {i}\right )}{2\,f\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (e+f\,x\right )+1{}\mathrm {i}\right )\,\left (-c^2+c\,d\,2{}\mathrm {i}+d^2\right )}{2\,f\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\ln \left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )\,\left (2\,c\,d\,a^3+\left (3\,d^2-3\,c^2\right )\,a^2\,b-6\,c\,d\,a\,b^2+\left (c^2-d^2\right )\,b^3\right )}{f\,\left (a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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