Optimal. Leaf size=115 \[ \frac {a^3 \cos ^2(c+d x)}{2 d}+\frac {b \left (3 a^2-b^2\right ) \sec (c+d x)}{d}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a^2 b \cos (c+d x)}{d}+\frac {3 a b^2 \sec ^2(c+d x)}{2 d}+\frac {b^3 \sec ^3(c+d x)}{3 d} \]
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Rubi [A] time = 0.24, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {4397, 2837, 12, 894} \[ \frac {b \left (3 a^2-b^2\right ) \sec (c+d x)}{d}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d}+\frac {3 a^2 b \cos (c+d x)}{d}+\frac {a^3 \cos ^2(c+d x)}{2 d}+\frac {3 a b^2 \sec ^2(c+d x)}{2 d}+\frac {b^3 \sec ^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
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Rule 12
Rule 894
Rule 2837
Rule 4397
Rubi steps
\begin {align*} \int \sec (c+d x) (a \sin (c+d x)+b \tan (c+d x))^3 \, dx &=\int (b+a \cos (c+d x))^3 \sec (c+d x) \tan ^3(c+d x) \, dx\\ &=-\frac {\operatorname {Subst}\left (\int \frac {a^4 (b+x)^3 \left (a^2-x^2\right )}{x^4} \, dx,x,a \cos (c+d x)\right )}{a^3 d}\\ &=-\frac {a \operatorname {Subst}\left (\int \frac {(b+x)^3 \left (a^2-x^2\right )}{x^4} \, dx,x,a \cos (c+d x)\right )}{d}\\ &=-\frac {a \operatorname {Subst}\left (\int \left (-3 b+\frac {a^2 b^3}{x^4}+\frac {3 a^2 b^2}{x^3}+\frac {3 a^2 b-b^3}{x^2}+\frac {a^2-3 b^2}{x}-x\right ) \, dx,x,a \cos (c+d x)\right )}{d}\\ &=\frac {3 a^2 b \cos (c+d x)}{d}+\frac {a^3 \cos ^2(c+d x)}{2 d}-\frac {a \left (a^2-3 b^2\right ) \log (\cos (c+d x))}{d}+\frac {b \left (3 a^2-b^2\right ) \sec (c+d x)}{d}+\frac {3 a b^2 \sec ^2(c+d x)}{2 d}+\frac {b^3 \sec ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.51, size = 100, normalized size = 0.87 \[ \frac {3 a^3 \cos (2 (c+d x))+2 \left (-6 b \left (b^2-3 a^2\right ) \sec (c+d x)-6 a \left (a^2-3 b^2\right ) \log (\cos (c+d x))+9 a b^2 \sec ^2(c+d x)+2 b^3 \sec ^3(c+d x)\right )+36 a^2 b \cos (c+d x)}{12 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 122, normalized size = 1.06 \[ \frac {6 \, a^{3} \cos \left (d x + c\right )^{5} + 36 \, a^{2} b \cos \left (d x + c\right )^{4} - 3 \, a^{3} \cos \left (d x + c\right )^{3} - 12 \, {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (d x + c\right )^{3} \log \left (-\cos \left (d x + c\right )\right ) + 18 \, a b^{2} \cos \left (d x + c\right ) + 4 \, b^{3} + 12 \, {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (d x + c\right )^{2}}{12 \, d \cos \left (d x + c\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 213, normalized size = 1.85 \[ -\frac {a^{3} \left (\sin ^{2}\left (d x +c \right )\right )}{2 d}-\frac {a^{3} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {3 a^{2} b \left (\sin ^{4}\left (d x +c \right )\right )}{d \cos \left (d x +c \right )}+\frac {3 \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right ) a^{2} b}{d}+\frac {6 a^{2} b \cos \left (d x +c \right )}{d}+\frac {3 a \,b^{2} \left (\tan ^{2}\left (d x +c \right )\right )}{2 d}+\frac {3 a \,b^{2} \ln \left (\cos \left (d x +c \right )\right )}{d}+\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )^{3}}-\frac {b^{3} \left (\sin ^{4}\left (d x +c \right )\right )}{3 d \cos \left (d x +c \right )}-\frac {b^{3} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{3 d}-\frac {2 b^{3} \cos \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.35, size = 109, normalized size = 0.95 \[ -\frac {3 \, {\left (\sin \left (d x + c\right )^{2} + \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} a^{3} + 9 \, a b^{2} {\left (\frac {1}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right )^{2} - 1\right )\right )} - 18 \, a^{2} b {\left (\frac {1}{\cos \left (d x + c\right )} + \cos \left (d x + c\right )\right )} + \frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{2} - 1\right )} b^{3}}{\cos \left (d x + c\right )^{3}}}{6 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.45, size = 219, normalized size = 1.90 \[ \frac {2\,a^3\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )-6\,a\,b^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-2\,a^3-12\,a^2\,b+6\,a\,b^2+\frac {4\,b^3}{3}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (6\,a^3-12\,a^2\,b+6\,a\,b^2-4\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (6\,a^3-12\,a^2\,b+6\,a\,b^2+\frac {20\,b^3}{3}\right )+12\,a^2\,b-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (6\,a\,b^2-2\,a^3\right )-\frac {4\,b^3}{3}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )}^3\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2} \]
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 \[ \int \left (a \sin {\left (c + d x \right )} + b \tan {\left (c + d x \right )}\right )^{3} \sec {\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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