Optimal. Leaf size=75 \[ \frac {\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}{3 b^3 d}+\frac {(a+b \tan (c+d x))^5}{5 b^3 d}-\frac {a (a+b \tan (c+d x))^4}{2 b^3 d} \]
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Rubi [A] time = 0.07, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3506, 697} \[ \frac {\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}{3 b^3 d}+\frac {(a+b \tan (c+d x))^5}{5 b^3 d}-\frac {a (a+b \tan (c+d x))^4}{2 b^3 d} \]
Antiderivative was successfully verified.
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Rule 697
Rule 3506
Rubi steps
\begin {align*} \int \sec ^4(c+d x) (a+b \tan (c+d x))^2 \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^2 \left (1+\frac {x^2}{b^2}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {\left (a^2+b^2\right ) (a+x)^2}{b^2}-\frac {2 a (a+x)^3}{b^2}+\frac {(a+x)^4}{b^2}\right ) \, dx,x,b \tan (c+d x)\right )}{b d}\\ &=\frac {\left (a^2+b^2\right ) (a+b \tan (c+d x))^3}{3 b^3 d}-\frac {a (a+b \tan (c+d x))^4}{2 b^3 d}+\frac {(a+b \tan (c+d x))^5}{5 b^3 d}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 54, normalized size = 0.72 \[ \frac {(a+b \tan (c+d x))^3 \left (a^2-3 a b \tan (c+d x)+6 b^2 \tan ^2(c+d x)+10 b^2\right )}{30 b^3 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 79, normalized size = 1.05 \[ \frac {15 \, a b \cos \left (d x + c\right ) + 2 \, {\left (2 \, {\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{4} + {\left (5 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 3 \, b^{2}\right )} \sin \left (d x + c\right )}{30 \, d \cos \left (d x + c\right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.50, size = 80, normalized size = 1.07 \[ \frac {6 \, b^{2} \tan \left (d x + c\right )^{5} + 15 \, a b \tan \left (d x + c\right )^{4} + 10 \, a^{2} \tan \left (d x + c\right )^{3} + 10 \, b^{2} \tan \left (d x + c\right )^{3} + 30 \, a b \tan \left (d x + c\right )^{2} + 30 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.41, size = 82, normalized size = 1.09 \[ \frac {-a^{2} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+\frac {a b}{2 \cos \left (d x +c \right )^{4}}+b^{2} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 71, normalized size = 0.95 \[ \frac {6 \, b^{2} \tan \left (d x + c\right )^{5} + 15 \, a b \tan \left (d x + c\right )^{4} + 30 \, a b \tan \left (d x + c\right )^{2} + 10 \, {\left (a^{2} + b^{2}\right )} \tan \left (d x + c\right )^{3} + 30 \, a^{2} \tan \left (d x + c\right )}{30 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.55, size = 71, normalized size = 0.95 \[ \frac {a^2\,\mathrm {tan}\left (c+d\,x\right )+{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (\frac {a^2}{3}+\frac {b^2}{3}\right )+\frac {b^2\,{\mathrm {tan}\left (c+d\,x\right )}^5}{5}+a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+\frac {a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^4}{2}}{d} \]
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 \[ \int \left (a + b \tan {\left (c + d x \right )}\right )^{2} \sec ^{4}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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