Optimal. Leaf size=159 \[ \frac {b^2 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac {\sec ^2(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{2 d \left (a^2-b^2\right )}-\frac {(a A+b (2 A+B)) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac {(a A-b (2 A-B)) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]
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Rubi [A] time = 0.29, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {2837, 823, 801} \[ \frac {b^2 (A b-a B) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^2}-\frac {\sec ^2(c+d x) (-(a A-b B) \sin (c+d x)-a B+A b)}{2 d \left (a^2-b^2\right )}-\frac {(a A+b (2 A+B)) \log (1-\sin (c+d x))}{4 d (a+b)^2}+\frac {(a A-b (2 A-B)) \log (\sin (c+d x)+1)}{4 d (a-b)^2} \]
Antiderivative was successfully verified.
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Rule 801
Rule 823
Rule 2837
Rubi steps
\begin {align*} \int \frac {\sec ^3(c+d x) (A+B \sin (c+d x))}{a+b \sin (c+d x)} \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {A+\frac {B x}{b}}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=-\frac {\sec ^2(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{2 \left (a^2-b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {-a^2 A+2 A b^2-a b B-(a A-b B) x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {\sec ^2(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{2 \left (a^2-b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \left (\frac {(a-b) (-a A-b (2 A+B))}{2 b (a+b) (b-x)}+\frac {2 b (-A b+a B)}{(a-b) (a+b) (a+x)}+\frac {(a+b) (-a A+b (2 A-B))}{2 (a-b) b (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{2 \left (a^2-b^2\right ) d}\\ &=-\frac {(a A+b (2 A+B)) \log (1-\sin (c+d x))}{4 (a+b)^2 d}+\frac {(a A-b (2 A-B)) \log (1+\sin (c+d x))}{4 (a-b)^2 d}+\frac {b^2 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2 d}-\frac {\sec ^2(c+d x) (A b-a B-(a A-b B) \sin (c+d x))}{2 \left (a^2-b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.77, size = 197, normalized size = 1.24 \[ \frac {\frac {4 b^2 (A b-a B) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^2}+\frac {A+B}{(a+b) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {B-A}{(a-b) \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {2 (a A+b (2 A+B)) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}{(a+b)^2}+\frac {2 (a A+b (B-2 A)) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )}{(a-b)^2}}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 234, normalized size = 1.47 \[ \frac {2 \, B a^{3} - 2 \, A a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3} - 4 \, {\left (B a b^{2} - A b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left (A a^{3} + B a^{2} b - {\left (3 \, A - 2 \, B\right )} a b^{2} - {\left (2 \, A - B\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (\sin \left (d x + c\right ) + 1\right ) - {\left (A a^{3} + B a^{2} b - {\left (3 \, A + 2 \, B\right )} a b^{2} + {\left (2 \, A + B\right )} b^{3}\right )} \cos \left (d x + c\right )^{2} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (A a^{3} - B a^{2} b - A a b^{2} + B b^{3}\right )} \sin \left (d x + c\right )}{4 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} d \cos \left (d x + c\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 260, normalized size = 1.64 \[ -\frac {\frac {4 \, {\left (B a b^{3} - A b^{4}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{4} b - 2 \, a^{2} b^{3} + b^{5}} + \frac {{\left (A a + 2 \, A b + B b\right )} \log \left ({\left | -\sin \left (d x + c\right ) + 1 \right |}\right )}{a^{2} + 2 \, a b + b^{2}} - \frac {{\left (A a - 2 \, A b + B b\right )} \log \left ({\left | -\sin \left (d x + c\right ) - 1 \right |}\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {2 \, {\left (B a b^{2} \sin \left (d x + c\right )^{2} - A b^{3} \sin \left (d x + c\right )^{2} + A a^{3} \sin \left (d x + c\right ) - B a^{2} b \sin \left (d x + c\right ) - A a b^{2} \sin \left (d x + c\right ) + B b^{3} \sin \left (d x + c\right ) + B a^{3} - A a^{2} b - 2 \, B a b^{2} + 2 \, A b^{3}\right )}}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (\sin \left (d x + c\right )^{2} - 1\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.50, size = 297, normalized size = 1.87 \[ -\frac {A}{d \left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}-\frac {B}{d \left (4 a +4 b \right ) \left (\sin \left (d x +c \right )-1\right )}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) a A}{4 d \left (a +b \right )^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) A b}{2 d \left (a +b \right )^{2}}-\frac {\ln \left (\sin \left (d x +c \right )-1\right ) B b}{4 d \left (a +b \right )^{2}}+\frac {b^{3} \ln \left (a +b \sin \left (d x +c \right )\right ) A}{d \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {b^{2} \ln \left (a +b \sin \left (d x +c \right )\right ) a B}{d \left (a +b \right )^{2} \left (a -b \right )^{2}}-\frac {A}{d \left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {B}{d \left (4 a -4 b \right ) \left (1+\sin \left (d x +c \right )\right )}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) a A}{4 d \left (a -b \right )^{2}}-\frac {\ln \left (1+\sin \left (d x +c \right )\right ) A b}{2 d \left (a -b \right )^{2}}+\frac {\ln \left (1+\sin \left (d x +c \right )\right ) B b}{4 d \left (a -b \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.73, size = 175, normalized size = 1.10 \[ -\frac {\frac {4 \, {\left (B a b^{2} - A b^{3}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{4} - 2 \, a^{2} b^{2} + b^{4}} - \frac {{\left (A a - {\left (2 \, A - B\right )} b\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{2} - 2 \, a b + b^{2}} + \frac {{\left (A a + {\left (2 \, A + B\right )} b\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{2} + 2 \, a b + b^{2}} + \frac {2 \, {\left (B a - A b + {\left (A a - B b\right )} \sin \left (d x + c\right )\right )}}{{\left (a^{2} - b^{2}\right )} \sin \left (d x + c\right )^{2} - a^{2} + b^{2}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.52, size = 197, normalized size = 1.24 \[ \frac {\ln \left (a+b\,\sin \left (c+d\,x\right )\right )\,\left (A\,b^3-B\,a\,b^2\right )}{d\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {\frac {A\,b-B\,a}{2\,\left (a^2-b^2\right )}-\frac {\sin \left (c+d\,x\right )\,\left (A\,a-B\,b\right )}{2\,\left (a^2-b^2\right )}}{d\,{\cos \left (c+d\,x\right )}^2}+\frac {\ln \left (\sin \left (c+d\,x\right )+1\right )\,\left (A\,a-b\,\left (2\,A-B\right )\right )}{d\,\left (4\,a^2-8\,a\,b+4\,b^2\right )}-\frac {\ln \left (\sin \left (c+d\,x\right )-1\right )\,\left (A\,a+b\,\left (2\,A+B\right )\right )}{d\,\left (4\,a^2+8\,a\,b+4\,b^2\right )} \]
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 \[ \int \frac {\left (A + B \sin {\left (c + d x \right )}\right ) \sec ^{3}{\left (c + d x \right )}}{a + b \sin {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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